Information Spillover Effect and Autoregressive Conditional Duration Models 2013050059, 9780415721684, 9781315768847

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Information Spillover Effect and Autoregressive Conditional Duration Models

This book studies the information spillover among financial markets and explores the intraday effect and ACD models with high frequency data. This book also contributes theoretically by providing a new statistical methodology with comparative advantages for analyzing comovements between two time series. It explores this new method by testing the information spillover between the Chinese stock market and the international market, futures market and spot market. Using the high frequency data, this book investigates the intraday effect and examines which type of ACD model is particularly suited in capturing financial duration dynamics. The book will be of invaluable use to scholars and graduate students interested in comovements among different financial markets and financial market microstructure and to investors and regulation departments looking to improve their risk management. Xiangli Liu received her PhD in Management Sciences and Engineering from the School of Management, Graduate University of the Chinese Academy of Sciences in 2008. She is currently Professor of the School of Finance, Central University of Finance and Economics. She has published over 20 papers in domestic and international journals. Her research interests include econometrics, financial market microstructure and financial risk management. Yanhui Liu received her PhD in Management Sciences and Engineering from the Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences in 2005. She has worked in the Development Bank of Singapore since 2005. Now she is the Chief Executive. She has published several papers in domestic and international journals. Her research interests include econometrics, financial econometrics and financial instruments.

Yongmiao Hong received his PhD in Economics from the University of California, San Diego in 1993. He joined as Assistant Professor, Economics Department, at Cornell University in 1993 and became tenured Associate Professor in 1998 and tenured Full Professor in 2001. Now he serves as a tenured Professor of Economics and Statistics at Cornell University and a Cheung Kong Lecture Professor of Wang Yanan Institute for Studies in Economics (WISE) at Xiamen University. He has been selected as a member of the Thousand Talents Program to promote the recruitment of first-class international talents for the development of national key disciplines. His current research interests include econometrics, time series analysis and application, financial econometrics, Chinese economics and empirical research in financial markets in China. Shouyang Wang received his PhD in Operations Research from the Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences in 1986. He is currently a Bairen Distinguished Professor of Management Science at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He is also an adjunct professor of over 30 universities around the world. He is the editor-in-chief, an area editor or a co-editor of 12 journals. He has published 30 monographs and over 250 papers in leading journals. His current research interests include financial engineering, economic forecasting and financial risk management.

Routledge Advances in Risk Management Edited by Kin Keung Lai and Shouyang Wang

1 Volatility Surface and Term Structure High-profit options trading strategies Shifei Zhou, Hao Wang, Kin Keung Lai and Jerome Yen 2 China’s Financial Markets Issues and opportunities Ming Wang, Jerome Yen and Kin Keung Lai 3 Managing Risk of Supply Chain Disruptions Tong Shu, Shou Chen, Shouyang Wang, Kin Keung Lai and Xizheng Zhang 4 Information Spillover Effect and Autoregressive Conditional Duration Models Xiangli Liu, Yanhui Liu, Yongmiao Hong and Shouyang Wang

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Information Spillover Effect and Autoregressive Conditional Duration Models Xiangli Liu, Yanhui Liu, Yongmiao Hong and Shouyang Wang

First published 2015 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 711 Third Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2015 Xiangli Liu, Yanhui Liu, Yongmiao Hong and Shouyang Wang The right of Xiangli Liu, Yanhui Liu, Yongmiao Hong and Shouyang Wang to be identified as authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patent Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Liu, Xiangli Information spillover in financial markets / Xiangli Liu, Yanhui Liu, Yongmiao Hong and Shouyang Wang. pages cm. – (Routledge advances in risk management; 4) Includes bibliographical references and index. 1. Finance–Mathematical models. 2. Financial risk–Mathematical models. 3. Capital market–Mathematical models. 4. Information theory in finance. I. Title. HG106.L58 2015 332.01′154–dc23 2013050059 ISBN: 978-0-415-72168-4 (hbk) ISBN: 978-1-315-76884-7 (ebk) Typeset in Times New Roman by Out of House Publishing

Contents

List of figures List of tables Preface

ix x xiii

1

Introduction 1.1 Review of recent developments 2 1.2 Organization and major conclusions 5

1

2

Methodology to detect extreme risk spillover 2.1 Granger causality in risk 11 2.2 Method and test statistics 14 2.3 Asymptotic theory 18 2.4 Two-way Granger causality in risk 22 2.5 Finite-sample performance 26 2.6 Conclusion 35 Notes 35

9

3

VaR estimation 3.1 Upside VaR and Downside VaR 39 3.2 Parametric conditional VaR estimation 40 3.3 Semi-parametric VaR estimation based on volatility, skewness and kurtosis 41 3.4 Nonparametric VaR estimation based on kernel function 42 3.5 Backtest 44 3.6 Data 44 3.7 Empirical analysis in Chinese futures market 46 3.8 Conclusion 51

37

viii 4

Contents Extreme risk spillover between Chinese stock markets and international stock markets 4.1 The Chinese stock market 53 4.2 Data 55 4.3 Evidence on Granger causality in risk 60 4.4 Conclusion 83 Notes 83

52

5

Information spillover effects between Chinese futures market and spot market 85 5.1 Granger causality test 86 5.2 Data 91 5.3 VaR estimation 93 5.4 Empirical results for information spillover between futures market and spot market 97 5.5 Conclusion 101

6

How well can autoregressive duration models capture the price durations dynamics of foreign exchanges? 6.1 Nonparametric density forecast evaluation 102 6.2 ACD models 109 6.3 Data and estimation 114 6.4 Empirical evidence 120 6.5 Conclusion 150 Notes 151

102

7

Intraday effect 7.1 Calendar Effect 152 7.2 Data 154 7.3 Intraday trends of yield and volume 157 7.4 Analysis of correlation among yield, volume and open interest 161 7.5 Conclusion 176

152

8

Conclusions and perspective studies

178

Appendix: mathematical proof Bibliography Index

182 195 207

Figures

3.1 3.2 3.3 4.1 4.2 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 7.1 7.2 7.3 7.4 7.5

Downside VaR Upside VaR Return rate of futures copper Stock price indices Return rate of stock price indices Time series of return rate of futures copper Time series of return rate of spot copper Time series of Downside VaR at the 95 per cent confidence level Time series of Upside VaR at the 95 per cent confidence level Histograms of raw quote durations Expected quote durations conditional on time of day Histograms of observed spreads Histograms of raw quote durations Expected price durations conditional on time of day Seasonally adjusted price duration Histogram of seasonally adjusted price duration Sample autocorrelations of seasonally adjusted price duration Histogram of in-sample generalized residuals of Euro/Dollar Histogram of in-sample generalized residuals of Yen/Dollar Histogram of out-of-sample generalized residuals of Euro/Dollar Histogram of out-of-sample forecasted generalized residuals of Yen/Dollar Time series of yield Intraday pattern of absolute yield Intraday pattern of volume Variance decomposition map Impulse response map

39 40 45 56 58 92 92 96 96 115 116 116 118 118 119 119 120 132 133 148 149 156 158 160 170 173

Tables

2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2

Size at the 10 per cent and 5 per cent significant levels Power at the 10 per cent and 5 per cent significant levels under ALTER1 Power at the 10 per cent and 5 per cent significant levels under ALTER2 Power at the 10 per cent and 5 per cent significant levels under ALTER3 Summary descriptive statistics for the yield of the futures market The stationary test for the yield of the futures market The diagnostic test statistics of the EGARCH model sufficiency – Generalized Box–Pierce Q statistics The diagnostic test statistics of the TGARCH model sufficiency – Generalized Box–Pierce Q statistics VaR estimation in the futures market by EGARCH model VaR estimation in the futures market by TGARCH model VaR estimation in the futures market by semi-parametric model VaR estimation in the futures market by nonparametric model Summary descriptive statistics for daily stock price changes Quasi-maximum likelihood estimation of univariate GARCH models for daily stock price changes Risk spillover among Chinese stock markets Risk spillover among greater China Risk spillover between Chinese stock markets and Asian stock markets Risk spillover between Chinese stock markets and major international stock markets Summary descriptive statistics for the rate of return series The stationary test

30 31 32 34 46 46 47 47 48 48 49 50 59 61 65 69 74 79 93 93

Tables 5.3 The diagnostic test statistics for the model adequacy of the futures market – Generalized Box–Pierce Q statistics 5.4 The diagnostic test statistics for the model adequacy of the spot market – Generalized Box–Pierce Q statistics 5.5 VaR estimation for the futures market based on TGARCH model 5.6 VaR estimation for the spot market based on GARCH model 5.7 The correlation coefficient between the futures market and the spot market 5.8 The linear-Granger causality test 5.9 Test for Granger causality in mean 5.10 Test for Granger causality in volatility 5.11 Test for Granger causality in downside risk at 10 per cent risk level 5.12 Test for Granger causality in downside risk at 5 per cent risk level 5.13 Test for Granger causality in upside risk at 10 per cent risk level 5.14 Test for Granger causality in upside risk at 5 per cent risk level 5.15 The information spillover effect between the futures market and the spot market 6.1 Summary statistics for price durations of Euro/Dollar and Yen/Dollar 6.2 Parameter estimates of ACD models for Euro/Dollar and Yen/Dollar price durations 6.3 Nonparametric portmanteau density evaluation statistics for in-sample and out-of-sample performance of ACD models 6.4 Separate diagnostic statistics for in-sample and out-of-sample generalized residuals of ACD models 6.5 Separate inference statistics for in-sample and out-of-sample residuals of ACD models 7.1 Statistical description of yield sequences 7.2 Granger causality test between volume and absolute yield 7.3 Granger causality test between open interest and absolute yield 7.4 Granger causality test between open interest and volume 7.5 Variance decomposition

xi 94 94 95 95 96 97 98 98 98 98 99 99 100 117 121 128 134 141 155 163 164 165 166

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Preface

Controlling and monitoring financial risk have been receiving increasing attention from business practitioners, decision-makers and academic researchers. When monitoring financial risk, there are two important issues: one is how to estimate VaR (Value at Risk), the other is how the risk of spillover affects different markets (e.g. financial contagion). When large market movements occur, extreme market movements imply a change of hands of a huge amount of capital among investors, leading to bankruptcy for some of them. They can even cause collapse of financial systems and social instability. For effective financial risk management and investment/portfolio diversification, it is important to understand the mechanism of how information spillover occurs across different markets. There was a surge interest in high-frequency financial transaction data with the fast development of technology. Its availability has provided unprecedented opportunities to study various issues related to financial trading processes. The autoregressive conditional duration (ACD) model, introduced by Engle and Russell, is one of the most promising new econometric tools that focus on the time intervals between the occurrences of trading events. The ACD model is tailor-made for the analysis of microstructure market issues and has been almost exclusively used to analyze high frequency financial data. Given a wide variety of available ACD models, it is important to examine whether some ACD models perform better than others, and which type of model, if any, is particularly suited in capturing financial duration dynamics. An abnormal phenomenon in theories of microstructure of financial markets is the ‘Calendar Effect’, on yield and other financial variables. Past studies have paid little attention to the internal structure and operational characteristics of the Chinese futures market. As a new booming market, the Chinese futures market is important to the international market; its microstructure has attracted a great deal of scholarly attention. Compared with the developed markets and markets with market maker mechanisms, the microstructure of the Chinese futures market is quite different. It is important to study the patterns of intraday effect of yield and volume, and explain the reason of the formation of the pattern, from the theories of microstructure of markets.

xiv

Preface

The core of this book is to study the information spillover among financial markets and explore the intraday effect and ACD models with high-frequency data. Theoretically, this book contributes towards providing a new statistical methodology with comparative advantages for analyzing comovements between two time series. Empirically, this book explores this new method to test the information spillover between the Chinese stock market and the international market, futures market and spot market. Using the high-frequency data, this book investigates the intraday effect and examines which type of ACD model is particularly suited in capturing financial duration dynamics. The book consists of eight chapters. The contents of the book are outlined in the following. Chapter 1 briefly reviews the recent developments on the two topics and concisely introduces the contents and organization of the book. In Chapter 2, based on a new concept of Granger causality in risk which focuses on the comovements between the tails of the two distributions, a class of kernel-based statistical tests are proposed to test whether a large downside risk in one market will Granger-cause a large downside risk in another market. The proposed test checks a large number of lags but avoids suffering from loss of power due to the loss of a large number of degree of freedom, thanks to the use of the downward weighting kernel function. This downward weighting is consistent with the stylized fact that financial markets are more influenced by more recent events than by remote past events, thus enhances the power of the proposed tests. It is proved that the proposed tests have a convenient asymptotically standard normal distribution under the null hypothesis of no Granger causality in risk and certain regular conditions. A simulation study shows that the proposed tests have reasonable levels and power against a variety of empirically plausible alternatives in finite samples, including the spillover from the dynamics in mean, variance, skewness and kurtosis respectively and further demonstrate them to have relative advantages compared with traditional regression based tests and are useful in investigating adverse large market comovements between financial markets such as financial contagions. Risk measure is the key factor for risk spillover. Chapter 3 adopts parametric, semi-parametric and nonparametric approaches to estimate the unconditional VaR and conditional VaR of the copper futures in order to find an appropriate risk measure in China’s futures market. In respect that good news and bad news have different impacts on the market volatility, EGARCH and TGARCH models are adopted in parametric estimation; volatility, skewness and kurtosis are employed in semi-parametric estimation; and nonparametric estimation is based on kernel function. Generally, financial markets consider Downside VaR only. However, Upside VaR, as a new concept, is introduced for the first time in this chapter and both Downside VaR and Upside VaR are estimated with the preceding three approaches. Moreover, Kupiec’s backtest has been done to evaluate the efficiency of each approach. In Chapter 4, we provide an empirical study on spillover of extreme downside market risk among Shares A, B and H in the Chinese stock market,

Preface

xv

between different stock markets in greater China, and between the Chinese stock market and other international capital markets. Our findings suggest that the market segmentation between Share A and Share B is effective in avoiding large adverse shocks from international capital markets. In terms of large adverse market movements, the Chinese stock market has some ties with the Asian stock markets, but its link with leading international capital markets is still weak. Considering the short selling mechanism in the futures market, Chapter 5 introduces two new notions: Upside VaR and extreme upside risk spillover. Downside VaR and Upside VaR are examined by using TGARCH and GARCH models. In addition, we investigate information spillover effects between the futures market and the spot market by employing a linear Granger causality test, and Granger causality tests in mean, volatility and risk respectively. Empirical results indicate that there exist significant twoway spillover effects between the futures market and the spot market, and the spillover effects from the futures market to the spot market are much more striking. In Chapter 6, using a new omnibus density forecast evaluation procedure, we examine various commonly used ACD models in capturing the price duration dynamics of exchange rates. The ACD models under investigation include Linear, Logarithmic, Box–Cox, Exponential, Threshold and Markov Switching ACD models with the Exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. We find that none of the ACD models can adequately capture the full dynamics of foreign exchange rate price durations, either in sample or out of sample. However, some ACD models, particularly the Markov switching ACD model with Burr innovations, have not only the best in-sample fit, but also the best out-of-sample performance. It is found that sophisticated nonlinear specifications for the conditional mean duration do not help much over linear ACD models in capturing the full dynamics of price durations, but the specification of the innovation distribution is important: the generalized Gamma or Burr distribution performs much better than the Weibull and Exponential distributions. Moreover, it is important to relax the independence assumption for innovations and model higher order conditional moments of price durations. Chapter 7 considers high-frequency data series to study patterns of intraday movement of yields and volumes and discovers the ‘L’pattern of intraday absolute yield and volume. We apply the financial market microstructure theory, traders’ psychology and the trading mechanism to explain the different intraday patterns in commodities futures market and the stock market, which has a distinctive intraday movement pattern of ‘U’-type. Therefore, using the Granger causality test theories and Vector Autoregressive models (VAR), we study the factors that influence volatility of yields and the lagged orders. The results show that there is a two-way Granger causality among any two of the absolute yield, volume and open interest, and it is different from the empirical results of the stock market, in the sense that there is only a one-way Granger

xvi

Preface

causal relationship from volume to absolute yield; the difference is accounted for by the different mechanism of the futures market. Using decomposition of variance of VAR models and the impulse response analysis, we analyze the dynamic relationship among the three factors. Chapter 8 concludes the book and suggests some topics for future research. Within the perspectives of market microstructure, analysis on intraday effect and ACD models still being in their infancy, more effort is needed from academia. The innovation of this book is to contribute on spillover methodology and ACD models in following ways: (1) Providing a new statistical methodology with comparative advantages for analyzing comovements between two time series. The proposed test checks a large number of lags but avoids suffering from loss of power due to the loss of a large number of degree of freedom, thanks to the use of the downward weighting kernel function. (2) Considering the short selling mechanism in the futures market, there is still price rise risk. We introduce two new notions: Upside VaR and extreme upside risk spillover. (3) Using a new omnibus density forecast evaluation procedure, we examine various commonly used ACD models in capturing the price duration dynamics of exchange rates. (4) Investigating patterns of intraday effect of yields and volumes, and discovers the ‘L’ pattern of intraday absolute yield and volume, which is different from the stock market, which has a distinctive intraday movement pattern of ‘U’-type. This book is intended for researchers interested in comovements among different financial markets and financial market microstructure. The book is also intended for investors and regulation departments to improve risk management. Therefore, this book can be useful not only for researchers but also for graduate students in financial management, market anticipants and regulation departments. This research is partly supported by National Science Fund of China (No.71071170); Program for New Century Excellent Talents in University (NCET-11–0750); Project 211(Phase IV) of the Central University of Finance and Economics; Program for Innovation Research in the Central University of Finance and Economics. We thank the National Natural Science Foundation of China, the ministry of education and the Central University of Finance and Economics. We also would like to thank many friends and colleagues for their help and support rendered to us in preparing this monograph. Firstly, we thank Prof. Siwei Cheng of the Graduate University of Chinese Academy of Sciences, Dr. Shujuan Pang of the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences, Dr. Gang Cheng of the University of Minnesota at twin cities, Lei Zu of the Central University of

Preface

xvii

Finance and Economics and Haibin Xie of the University of International Business and Economics for their helpful discussions, suggestions and valuable comments on research in this area. Secondly, we thank the graduates Qiang Luo, Minghan, Chen, Fangming Ma, Guangxin Zhou and Jiang Wang of the Central University of Finance and Economics for the editing and checking on this book. Finally, we would tremendously like to express our gratitude to all the people dedicated to this monograph. Xiangli Liu, Yanhui Liu, Yongmiao Hong, Shouyang Wang

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1

Introduction

Controlling and monitoring financial risk have been receiving increasing attention from business practitioners, decision-makers and academic researchers. When monitoring financial risk, the probability of a large adverse market movement is always of the great practical concern. Large market movements can occur due to market uncertainty, policy changes, surprising news or shocks, speculative attacks and spillover from other markets (e.g. financial contagion). When they occur, extreme market movements imply a change of hands of a huge amount of capital among investors, leading to bankruptcy for some of them. They can even cause collapse of financial systems and social instability. Investors have been aware of painful experiences when extreme adverse market movements occur. Large market movements have become commonplace nowadays. For example, on Black Monday (October 19, 1987), US stocks collapsed by 23 per cent, wiping out US$1 trillion in capital. Around 1990, Japanese stock prices fell with the Nikkei 225 index sliding from 39,000 at the end of 1989 to 17,000 three years later, resulting in a total of US$ 2.7 trillion in capital loss. In the bond debacle of 1994, the US Federal Reserve Board, after having kept interest rates low for three years, started a series of six consecutive interest rate hikes that erased US$ 1.5 trillion in global capital. The US Orange County Investment Pool, a portfolio of US$ 7.5 billion belonging to municipal investors, including the county, cities and schools, lost US$ 1.64 billion in December, 1994. This is the largest municipal failure in history. During the 1997–1998 Asian financial crises, several Asian currencies devaluated dramatically within a very short period (e.g. Woo et al. 2000). Other examples of large market movements include recent large price adverse movements in the US stock market after the September 11 Terror Incident, the bankruptcies of the Long Term Capital Management, Enron and WorldCom. An example of extreme market movements in China is the sharp drop in Share B stock prices in 2001, due to the massive withdrawal of overseas investors after China opened Share B markets to domestic investors holding foreign currencies. For effective financial risk management and investment/portfolio diversification, it is important to understand the mechanism of how risk spillover occurs across different markets. And there has been a substantial amount of theoretical and empirical

2

Introduction

work on documenting and analyzing how stock returns and volatility are transmitted across countries. There was a surge interest in high-frequency financial transaction data with the fast development of technology. Its availability has provided unprecedented opportunities to study various issues related to financial trading processes. A distinct feature of such data is that they arrive at irregular random time intervals. As emphasized in Goodhart and O’Hara (1997); Madhaven (2000), the waiting time between intraday market events plays a key role for understanding the processing of private and public information in financial markets. In finance, the models of Diamond and Verrecchia (1987); Easley and O’Hara (1992) provide theoretical justifications for developing time series models of inter-trade-arrival times. The autoregressive conditional duration (ACD) model, introduced by Engle and Russell (1998), is one of the most promising new econometric tools that focus on the time intervals between the occurrences of trading events. It is tailor made for the analysis of microstructure market issues and has been almost exclusively used to analyze high frequency financial data. Following Engle and Russell (1998), a number of substantive extensions to the original linear ACD model have been made in the literature. Given a wide variety of available ACD models, it is important to examine whether some ACD models perform better than others, and which type of model, if any, is particularly suited in capturing financial duration dynamics. This chapter will give a brief overview of the developments in these two areas and concisely introduce the contents and organization of the book.

1.1 1.1.1

Review of recent developments Information spillover effect

In finance, volatility has been used as a standard quantitative measure for financial risk, particularly market risk. Markowitz’s (1952) portfolio theory, for example, characterizes the trade-off between return and risk by using the mean and variance of asset returns. Most of the existing literature use volatility to measure risk and concentrate on modeling volatility spillover. Theoretically, Granger, Robins and Engle (1986) first introduce a concept of Granger causality in variance, which can be used to investigate volatility spillover across financial markets (e.g. Engle et al. 1990; Cheung and Ng 1996). A Granger (1969)-type test, namely by regressing the squared residual of one variable on the squared residuals from its own lagged and other lagged variables in the framework of multivariate GARCH models, is proposed and has been extensively used in empirical analysis. Cheung and Ng (1996) proposed a test for volatility spillover using the sample cross-correlation function between two squared residuals standardized by their conditional variance estimators respectively. This test is relatively simple and convenient to implement, and

Introduction

3

can provide valuable information in building multivariate GARCH models. Hong (2001a) further proposes a class of asymptotic N (0,1) tests for volatility spillover based on a weighted sum of squared sample cross-correlations between two squared standardized residuals. The Cheung and Ng (1996) test and Granger (1969)-type regression-based test can be viewed as the special case where uniform weighting is used. Empirically, Hamao, Masulis and Ng (1990) study volatility spillovers across the New York, Tokyo and London stock markets using univariate GARCH models. They found volatility spillovers only for the period following the October 1987 crash and identified an asymmetry, that is, Tokyo stock markets were affected by the London and New York stock markets but did not have any effect on them in turn. Other related works include Lin et al. (1994); Cheung and Ng (1996); Booth et al. (1997); Cheung and Ng (1990, 1996); Engle et al. (1990); Engle and Susmel (1993); King and Wadhwani (1990); King et al. (1994). Lee (2009) examined the volatility spillover effects among six Asian countries’ stock markets using a bivariate vector autoregression-generalized autoregressive conditional heteroskedasticity [VAR(p)-GARCH(1,1)] model and found that there are statistically significant volatility spillover effects within the stock markets of these countries. Stock market integration and volatility spillover between India and its major Asian counterparts is studied by Mukherjee and Mishra (2010). Hong Kong, Korea, Singapore and Thailand are found to be four Asian markets from where there is significant flow of information in India. Singha, Kumara and Pandeyb (2010) examined price and volatility spillovers across North American, European and Asian stock markets. However, variance treats gains and losses symmetrically and cannot account for large adverse market movements. A sensible measure of risk intuitively should be associated with large losses, or large adverse market movements. Although not a perfect measure of extreme market risk, Value at Risk (VaR), originally proposed by J.P. Morgan in 1994, has become a standard synthetic measure of extreme market risk. Using VaR as a measure of extreme downside market risk, Hong (2001b) first introduce a new concept of Granger causality in risk, where a large risk is said to have occurred at a prespecified level if actual loss exceeds VaR at the given level. The concept of Granger causality in risk focuses on the comovements between the left tails of two distributions, which is more suitable than the concept of Granger causality in variance in characterizing extreme downside risk spillover between different markets, because, as noted earlier, volatility is a two-sided risk measure, and it cannot capture heavy tails due to (e.g.) jumps. Longin and Solnik (2001); Hartmann, Straetmans and de Vries (2003); Bae, Karolyi and Stulz (2003) and still many others also study on extreme market comovements. They mainly focus on estimating the probability of coexceedance based on the marginal or stationary distribution of the asset return process.

4

Introduction

1.1.2

Autoregressive conditional duration models (ACD)

Following Engle and Russell (1998), a number of substantive extensions to the original linear ACD model have been made in the literature. Bauwens and Giot (2000) propose a logarithmic ACD model, which avoids the nonnegative constraints on model parameters implied by the linear ACD model. Allen, Chan, McAleer and Peiris (2008) were concerned with the properties of the Quasi Maximum Likelihood Estimator (QMLE) of the Logarithmic Autoregressive Conditional Duration (Log-ACD) model. Dufour and Engle (2000b) suggest Box–Cox and Exponential ACD models. Fernandes and Grammig (2003) develop a family of augmented ACD models, similar in spirit to the asymmetric GARCH models introduced by Hentschel (1995). Zhang, Russell and Tsay (2001) propose a threshold ACD model, which allows for different duration dynamics across different regimes. Other important ACD models include Markov chain regime-switching ACD models (Hujer, Kokot and Vuletic 2003a), smooth transition and time-varying ACD models (Meitz and Teräsvirta 2004), stochastic volatility duration models (Ghysels, Gourieroux and Jasiak 2004) and semi-parametric ACD models (Hautsch 2002; Drost and Werker 2004). Bortoluzzoa, Morettin and Toloi (2010) generalized the autoregressive conditional duration (ACD) model applied to times between trades to the case of time-varying parameters. The use of wavelets allows that parameters vary through time and makes possible the modeling of non-stationary processes without preliminary data transformations. The time-varying ACD model estimation was done by maximum-likelihood with standard exponential distributed errors. They present a simulation exercise for a non-stationary process and an empirical application to a real series, namely the TELEMAR stock. Diagnostic and goodness of fit analysis suggest that the time-varying ACD model simultaneously modeled the dependence between durations, intraday seasonality and volatility. In ACD modeling, besides the specification of the conditional mean duration, it is also important to specify the innovation distribution in order to capture the full dynamics of financial durations. Various innovation distributions have been used in the literature, including exponential and Weibull distributions, as used in Engle and Russell (1998) and Burr and generalized Gamma distributions, as used in Grammig and Maurer (2000) and Lunde (1999) respectively. Allen, Lazarova, McAleer and Peiris (2009) compared a number of alternative autoregressive conditional duration (ACD) models by using a sample of data for three major companies traded on the Australian Stock Exchange. The comparison is performed by employing the methodology for evaluating density and interval forecasts. The common practice to check adequacy of an ACD model has been using the Ljung–Box–Pierce type tests applied to estimated standardized or squared standardized durations of an ACD model. Unfortunately, the commonly used asymptotic chi-square distribution for this test is invalid, due to the complicated nontrivial impact of parameter estimation uncertainty (see Hong 2002

Introduction

5

for more discussion). Several other specification tests for ACD models have also been proposed. They can be divided into three categories: (i) specification tests for the innovation distribution (ii) specification tests for a conditional mean duration specification (iii) specification tests for the entire conditional probability density of an ACD model. Fernandes and Grammig (2000) suggest the tests of the first type. Assuming that the conditional mean duration model is correctly specified with i.i.d. innovations, they compare a nonparametric innovation density estimator with a model-implied parametric counterpart. Hautsch (2002); Meitz and Teräsvirta (2004) advocate specification tests of the second type. Hautsch (2002) use conditional moment tests and integrated conditional moment tests, while Meitz and Teräsvirta (2004) propose Lagrange Multiplier (LM) type tests. Using Diebold, Gunther and Tay’s (1998) density evaluation procedures, Dufour and Engle (2000a); Bauwens, Giot, Grammig and Veredas (2003) consider the tests of the third type. These procedures are easy to implement, and can provide hints to sources of model mis-specification. However, they are informal and may not deliver a decisive conclusion about the relative performance of competing models. Moreover, the impact of parameter estimation uncertainty on the evaluation procedure is not considered. Dufour and Engle (2000a) propose an alternative regression-based LM type test for density forecasts. However, the power of this test depends on the choice of instrumental variables which are somehow arbitrary. Density forecasts are important in many applications. As argued by Diebold, Gunther and Tay (1998); Granger (1999); Granger and Pesaran (2000), density forecasts are important for decision-making under uncertainty when forecasters’ loss functions are asymmetric and the underlying process is non-Gaussian. Hong, Li and Zhao (2004); Egorov, Hong and Li (2004) have recently developed nonparametric evaluation methods for a conditional density model. The test has an omnibus ability to detect a wide range of suboptimal density forecasts. Furthermore, it explicitly takes into account the impact of parameter estimation uncertainty on the evaluation procedure, an issue ignored in most existing evaluation procedures for density forecasts. The new test is able to check the entire conditional density of an ACD model rather than only the innovation distribution.

1.2

Organization and major conclusions

The first part of the book focuses on the extreme risk spillover effect between financial markets. We first develop theoretical statistical tools for analyzing extreme comovements between two time series, and then do an empirical study on the extreme risk spillover between the Chinese stock market and the international market by using the proposed methodology. We also test the

6

Introduction

information spillover effect between the Chinese futures market and the spot market. Chapter 2 is devoted to developing the statistical tools to test whether a large downside risk in one market will Granger-cause a large downside risk in another market. The new concept of Granger causality in risk is first introduced and a class of kernel-based statistical tests are proposed. We further provide its asymptotic theory and finite sample performance by Monte Carlo simulation. How to regulate financial risk effciently emerges as a crucial question with the economic globalization and financial liberalization. Risk measure is the key factor for risk management. Chapter 3 adopts parametric, semiparametric and nonparametric approaches to estimate the unconditional VaR and conditional VaR of the copper futures in order to find an appropriate risk measure in China’s futures market. In respect that good news and bad news have different impacts on the market volatility, EGARCH and TGARCH models are adopted in parametric estimation, volatility, skewness and kurtosis are employed in semi-parametric estimation and nonparametric estimation is based on kernel function. Generally, financial markets consider Downside VaR only. However, Upside VaR, as a new concept, is introduced for the first time in this paper and both Downside VaR andUpside VaR are estimated with the preceding three approaches. Moreover, Kupiec’s (1995) backtest has been done to evaluate the effciency of each approach. Our findings indicate that the asymmetric factors are all significant in both of the two GARCH models. Therefore, although the trading mechanism is symmetric in China’s futures market, good news and bad news has an asymmetric effect upon the volatility of futures markets. Moreover, VaR is overestimated at the 90 per cent and 95 per cent confidence levels through both GARCH models. That means it considers more risk in the same level. This indicator suits for investor of risk aversion. Finally, though nonparametric approach performs better, nonparametric estimation has no economic explanation. So, combining parametric and nonparametric approach is a direction of studying next. An empirical study on spillover of extreme downside market risk between the Chinese stock market and other international capital markets is given in Chapter 4. We find some interesting empirical results: First, there exists strong risk spillover between Share A indices and Share B indices, and between SHSE and SZSE. Moreover, the occurrence of a large risk in Share B markets can help predict a similar future risk in Share A markets, but not vice versa. Second, there exists significant risk spillover between the mainland China stock market (particularly Shares B and H) and the stock markets in Hong Kong and Taiwan. Third, there exists some risk spillover between the Chinese stock market (particularly Shares B and H) and the stock markets in South Korea and Singapore. Fourth, there exists no risk spillover between Share A indices and major international stock markets – those in Japan, the US and Germany. It appears that the price movement of the Share A markets, the main constituent of the Chinese stock market, is driven mainly by domestic

Introduction

7

factors and its impact is, at most, regional. In particular, its interaction with international major capital markets is still weak or nonexistent in terms of large adverse market movements. Most extreme downside market risk spillover from international capital markets is absorbed in the Share B markets and particularly the Share H market. In terms of avoiding the impact of adverse large international market movements on the Chinese stock market, the segmentation between the Share A and Share B markets appears valid. This shows that the Chinese authorities’ policy of isolating the domestic A-share market proved successful and that restrictions to international capital movements are effective. Of course, this is expected to change given the recent policy changes of opening the Share B markets to domestic investors and opening the Share A markets to qualified foreign institutional investors. Chapter 5 employs a parametric approach based on TGARCH and GARCH models to estimate the VaR of the copper futures market and spot market in China. Considering the short selling mechanism in the futures market, the chapter introduces two new notions: Upside VaR and extreme upside risk spillover. Downside VaR and Upside VaR are examined by using the above approach. Also, we use Kupiec’s backtest to test the power of our approaches. In addition, we investigate information spillover effects between the futures market and the spot market by employing a linear Granger causality test, and Granger causality tests in mean, volatility and risk respectively. Moreover, we also investigate the relationship between the futures market and the spot market by using a test based on kernel function. Empirical results indicate that there exist significant two-way spillover effects between the futures market and the spot market, and the spillover effects from the futures market to the spot market are much more striking. Chapter 6 focuses on the autoregressive conditional duration (ACD) models. Applying the omnibus tests developed in Hong, Li and Zhao (2004) and Egorov, Hong and Li (2004), we provide a comprehensive empirical analysis of both in-sample and out-of-sample performances of various commonly used ACD models for price durations of Euro/Dollar and Japanese Yen/ Dollar exchange rates. The ACD models under examination include linear (LINACD), logarithmic (LOGACD), Box–Cox (BCACD), Exponential (EXPACD), Threshold (TACD) and Markov Switching (MSACD) models. For each model, we consider four commonly used innovation distributions – exponential, Weibull, generalized Gamma and Burr distributions. In contrast to earlier studies, we find that none of the ACD models can adequately capture the full dynamics of price durations of foreign exchanges, either in-sample or out-of-sample, although the MSACD model with Burr innovations performs best. Sophisticated nonlinear specifications for the conditional mean duration (e.g. logarithmic, Box–Cox and Exponential forms) do not offer substantial improvement over linear ACD models in capturing the full dynamics of price durations of foreign exchanges. However, the specification of the innovation distribution is rather important: the exponential distribution always fits data poorly while the generalized Gamma distribution performs best (except that

8

Introduction

the Burr innovation performs best for the MSACD model). Moreover, to capture the full dynamics of price durations in foreign exchange markets, it is important to relax the i.i.d. assumption for innovations and to model higher order conditional moments of price durations. Our results are similar for both Euro/Dollar and Yen/Dollar, and for both in-sample and out-of-sample. Chapter 7 considers high frequency data of six commodities futures from the three exchanges of China. It uses data series, collected by the minute, to study patterns of intraday movement of yields and volumes and discovers the ‘L’ pattern of intraday absolute yield and volume. We apply the financial market microstructure theory, traders psychology and the trading mechanism to explain the different intraday patterns in commodities futures market and the stock market, which has a distinctive intraday movement pattern of ‘U’-type. Therefore, using the Granger causality test theories and Vector Auto Regressive Models (VAR), we study the factors that influence volatility of yields and the lagged orders. The results show that there is a two-way Granger causality among any two of the absolute yield, volume and open interest, and it is different from the empirical results of the stock market, in the sense that there is only a one-way Granger causal relationship from volume to absolute yield; the difference is accounted for by the different mechanism of the futures market. Using decomposition of variance of VAR models and the impulse response analysis, we analyze the dynamic relationship among the three factors. The conclusion shows that when we treat volatility of absolute yield as an explained variable, 90 per cent of the residual disturbance can be explained by its lagged orders, and volume can explain about 10 per cent, i.e. the remaining residual disturbance. When volume is interpreted as an explained variable, about 80 per cent of the residual disturbance could be explained by its lagged orders; volatility of absolute price explains about 20 per cent. When open interest was looked at as an explained variable, lagging orders of its own could interpret about 45 per cent to 70 per cent of the residual disturbance, volume explained 25 per cent to 45 per cent and volatility of absolute price explained 5 per cent to 10 per cent. All the explanatory variables reach stability about 20 to 30 minutes later. The empirical results tell that the influence of open interest on volatility of absolute yield and volume is weak, and there is a strong correlation between volatility of absolute yield and volume. Some investment suggestions are offered from the analysis mentioned above. Chapter 8 gives conclusions and perspective studies. The appendix gives the mathematic proof of the new method we introduced in Chapter 2.

2

Methodology to detect extreme risk spillover

Recent large financial disasters have called for the need of accurate risk measures for financial institutions and regulators. Most of the existing literature use volatility to measure risk and concentrate on modeling volatility spillover. Needless to say, volatility is an important instrument in finance and macroeconomics. However, it can only adequately represent small risks. When monitoring financial risk, the probability of a large adverse market movement is always of greater practical concern (e.g. Bollerslev 2001). Volatility alone cannot satisfactorily capture risk in scenarios of occasionally occurring extreme market movement. Moreover, it includes both gains and losses in a symmetric way. Intuitively, financial risk is obviously associated with losses but not profits. Also, practical constraints often require asymmetric treatment between upside potential and downside risk. Therefore, a sensible measure of risk should be associated with large losses, or large adverse market movements. Furthermore, it is well-known in the literature that financial markets may tend to be more prone to incurring large losses during volatile periods than during tranquil periods.1 In econometrics and statistics, left tail probabilities are closely related to the likelihoods of extreme downward market movements and associated risks (e.g. Embrechts et al. 1997). VaR measures how much a certain portfolio can lose within a given time period, with a prespecified probability. It has become widely used by corporate treasures and fund managers as well as by financial institutions. It has already become an essential part of financial regulations (the Basel Committee on Banking Supervision 1996, 2001) and widely used by financial regulators for setting risk capital requirements so as to ensure that financial institutions can survive after a catastrophic event (e.g. Duffe and Pan 1997; Jorion 2000).2 VaR reduces the total risk in a portfolio of financial assets to just one monetary amount. It summarizes many complex undesired outcomes in a single number and naturally represents a compromise between the needs of different users. This conceptual simplicity and compromise has made VaR the most popular measure of risk among practitioners in spite of its weakness.3 A leading motivating example is the spillover of extreme market movements between two financial markets. When financial markets are segmented,

10

Detecting extreme risk spillover

risk cannot transmit across markets. This is exactly the reason why China, the largest emerging market in the world, could have escaped the Asian financial crisis during 1997–1998 (e.g. Lardy 1998). However, when markets are integrated and suffer from the same global shock, risk is expected to transmit across markets. Another possibility of risk spillover is the ‘market contagion’, which can occur due to the attempts of investors to infer price changes in other markets. In this case, a large price change in one market may bring about a large price change in another market, regardless of the evolution of market fundamentals (e.g. King and Wadhwani 1990). It is also possible that risk in one market is generated locally or has characteristics specific to the market, but with the integration of markets, the risk will be transmitted into another market. For example, a domestically driven decline in Japanese stock price caused a decline in the wealth of Japanese stock holders, including Japanese banks. This decline reduced Japanese banks’ capital and caused a decline in Japanese banking lending in the US credit markets, given the regulations of Bank of International Settlements or BIS (Peek and Rosengren 1997). For another example, by the end of August 1997, as several Asian currencies devalued and stock markets in the Asian-Pacific area started to decline sharply, large institutional investors began selling off Chinese stocks listed in the Hong Kong Stock Exchange, leading to a sharp decline of Chinese stateowned enterprises’ stock prices. In this chapter, we first introduce the concept of Granger causality in risk. As is well known, Granger causality (Granger 1969, 1980) is not a relationship between ‘causes’ and ‘effects’. Instead, it is defined in terms of incremental predictive ability. This concept is thus suitable for the purpose of predicting and monitoring risk spillover and provides valuable information for investment decisions, supervisor decisions, risk capital allocation and external regulation. We, then, propose a class of statistical procedures to detect Granger causality in risk between financial markets. The basic idea here is to check whether the past history of the occurrence of large risks in one market has predictive ability for the occurrence of large risks in another market. If an extreme price fall is observed in one market, the detection of Granger causality in risk can give some hints to the investors or risk managers whether a similar market movement might happen in another market, so that effective hedge strategies are implemented in advance to prevent further big losses. We emphasize that the concept of Granger causality in risk is not limited to financial markets. For example, it can be applied to finance positions (e.g. investment portfolios).4 Since the VaR calculation has been in a standard toolbox of risk managers’ desk, our procedure is easy to implement. With regard to statistical methodology, the proposed test has two appealing features. First, it checks an increasing number of lags as sample size T grows. This ensures power against a wide range of alternatives of extreme risk spillover. For example, economic variables arise from economic decisions, so it takes time to gather necessary information and then to make and implement the decision. As a consequence,

Detecting extreme risk spillover

11

there is a natural time delay in risk spillover across different financial markets. This is the case particularly when investors need time to interpret and digest news, especially when it is surprising news. As a result, the first several lags may have zero cross-correlation.5 On the other hand, it is possible that the spillover at each lag is very small, but it carries over a long distributional lag and consequently the cumulative effect all together is strong. Obviously tests with a large number of lags will be able to detect these processes. When a larger number of lags is used, the power of a test will usually suffer from the loss of a large number of degrees of freedom, such as many chi-square tests in time series analysis (e.g. Box and Pierre’s 1971 portmanteau test). See Dahl and Gozenlez-Rivera (2003) for more discussion. Fortunately this is not the case of our procedures, because our frequency domain kernel-based approach naturally discounts higher order lags. Downward weighting is consistent with the stylized fact that financial markets are often more influenced by the recent events than by the remote past events. Therefore it alleviates the loss of a large number of degrees of freedom due to the employment of many lag orders, and will enhance good power of the test. Indeed, it can be shown that over a class of non-uniform kernel functions, the Daniell kernel maximizes the power of the proposed tests. Our approach thus provides a balance in being able to check many lags while alleviating the loss of a large number of degrees of freedom caused by including many lag orders.

2.1

Granger causality in risk

2.1.1

Extreme downside market risk and Value at Risk

For a given time horizon τ and confidence level 1 B , where B ( 0,1), VaR is the loss over the time horizon τ that is not exceeded with probability 1 B . Statistically speaking, VaR, denoted by Vt y V ( I t 1 , B ) is the negative α-quantile of the conditional probability density function (pdf) of a time series (e.g. portfolio return) Yt , which satisfies the following equation: P (Yt  Vt | I t 1 )  B almost surely ( a.s.),

(2.1)

where I t 1 y {Yt 1 ,Yt 2 ,...} is the information set available at time t 1. In financial risk management, the left tail probability in (2.1) is usually called the shortfall probability. For notational simplicity, we have suppressed the dependence of Vt on level α. In practice, commonly used levels for α are 10 per cent, 5 per cent or 1 per cent. For example, BIS sets B  1 per cent and U  10 days for purposes of measuring the adequacy of bank capital, J.P. Morgan discloses its daily VaR at the 5 per cent level and Bankers Trust discloses at the 1 per cent level for its daily VaR. In economics and finance, economists have considered behavioral models of banks and insurance companies that maximize some utility criteria under a VaR-type solvency constraint (e.g.

12

Detecting extreme risk spillover

Gollier et al. (1996); Sentamero and Babbel (1997)). There have been also studies on the optimal ‘safety-first’ portfolio selection which, as an alternative to the traditional mean-variance efficient frontier, maximizes the expected return subject to a downside risk constraint (e.g. Roy (1952); Levy and Sarnet (1972); Arzac and Bawa (1977); Jansen et al. (1998)). Some recent research (e.g. Ang et al. (2002)) suggests that extreme downside risk can help explain asset returns in an extended Fama and French’s (1996) framework. To gain insight into VaR from a statistical perspective, we write the time series {Yt } as follows: Yt  Nt  T t Ft , « ¬ ­{Ft } ~ m.d .s.( 0,1) with conditional CDF Ft (–),

(2.2)

where μt y μt ( I t 1 ) and T t2 y T t2 ( I t 1 ) are the conditional mean and variance of Yt given I t 1 respectively, and Ft (–) y Ft (– | I t 1 ) is the conditional cumulative distribution function (CDF) of Ft given I t 1. By definition, the standardized innovation {Ft } is a conditionally homoskedastic martingale difference sequence (m.d.s.) with E ( Ft | I t 1 )  0 a.s. and var( Ft | I t 1 )  1 a.s., but its conditional higher order moments, such as skewness and kurtosis, may be timevarying. An example is Hansen’s (1994) autoregressive conditional density model where Ft follows a generalized Student-t-distribution with time-varying shape parameters. From (2.1) and (2.2), we can obtain the VaR Vt  Nt  T t zt (B ),

(2.3)

where zt (B ) y z ( I t 1 , B ) is the left-tailed critical value at level α of the conditional distribution Ft (–) of Ft ; that is, zt (B ) satisfies Ft [ zt (B )]  B . Obviously, Vt depends on not only the conditional mean μt and variance T t2 of Yt , but also its higher order conditional moments (e.g. skewness and kurtosis). There has been increasing empirical evidence that the conditional skewness and kurtosis of financial time series are time-varying (e.g. Gallant, Hsieh and Tauchen (1991); Hansen (1994); Harvey and Siddique (1999, 2000); Jondeau and Rockinger 2003). Some time series econometric models, such as those of Gallant and Tauchen (1996); Hansen (1994); Harvey and Siddique (2000), can capture time-varying higher order conditional moments. Note that in (2.2), Yt is not covariance-stationary if T t2 follows an integrated GARCH (IGARCH) process (Engle and Bollerslev (1986)). In this case, the unconditional variance of {Yt } does not exist, but its VaR is still well-defined. In practice, a most popular model for VaR is J.P. Morgan’s (1997) Risk Metrics:

Detecting extreme risk spillover Yt  T t Ft , « ® 2 d ¬T t  (1 M )¤ j 1 M jYt2 j , ® ­ Ft ~ i.i.d . N ( 0,1),

13

(2.4)

where the parameter λ governs the dependence of the current volatility on its past history. For daily financial series, J.P. Morgan suggests λ = 0.94. Under (2.4), the VaR is Vt  T t z (B ),

(2.5)

where z B is the one-sided N  0,1 critical value at level B . For example, z B = 1.28, 1.65 and 2.33 for α = 0.10, 0.05 and 0.01 respectively. Clearly, the higher the expected volatility T t2 , the larger the VaR. 2.1.2

Granger causality in risk

Because VaR is a threshold measure for extreme downside risk, we may say that a large downside risk has occurred if actual loss exceeds VaR. In practice, financial institutions and regulators are seriously concerned with the probability of such a risk, which is infrequent but may cause a financial disaster if it occurs. To characterize whether the occurrence of a large risk in one market Y1t can help predict the occurrence of a large risk in another market Y2t in the spirit of Granger (1969, 1980) causality, we now define a concept of Granger causality in the tail distributions. Put I t 1 y ( I1( t 1) , I 2 ( t 1) ), where I1( t 1)  {Y1t 1 ,...,Y11 } and I 2 ( t 1)  {Y2t 1 ,...,Y21 } are the information sets available at time t−1 for two time series respectively. Suppose H10 : P (Y1t  V1t | I1( t 1) )  P (Y1t  V1t | I t 1 ) a.s.,

(2.6)

we say that the time series {Y2t } does not Granger-cause the time series {Y1t } in risk at level α with respect to information set I t 1. On the other hand, if H1A : P (Y1t  V1t | I1( t 1) ) x P (Y1t  V1t | I t 1 ),

(2.7)

we say that the time series {Y2t } Granger-causes the time series {Y1t } in risk at level α with respect to I t 1. In this case, the information of the occurrence of a risk in {Y2t } can be used to predict the occurrence of a future risk in {Y1t }. In practice, level α can be determined by regulators or practitioners, depending on their objective function or risk attitude.

14

Detecting extreme risk spillover

In time series econometrics, the most commonly used Granger causality concept is Granger causality in mean, which was first introduced in Granger (1969) and Granger causality in variance. Here, Granger causality in risk can arise not only from comovements in mean and in variance, but also from the comovements in higher order conditional moments (e.g. skewness and kurtosis). In other words, Granger causality in risk may rise even in the absence of Granger causality in mean and in variance. Granger (1980) introduces a general Granger causality in terms of the entire conditional probability distribution P (Y1t b y | I1( t 1) ) x P (Y1t b y | I t 1 ) for all y  ( d, d ). Our concept of Granger causality in risk is more closely related to this general Granger causality, but again we only focus on left tail probabilities, which are more relevant to large downside market risks.

2.2

Method and test statistics

To develop tests for Granger causality in risk, we first formulate our hypotheses H10 versus H1A as hypotheses on Granger causality in mean, after a proper transformation of {Y1t ,Y2t }. Define the risk indicator Zlt y 1(Ylt  Vlt ),

l  1, 2,

(2.8)

where 1 · is the indicator function. The indicator Zlt takes value 1 when actual loss exceeds VaR and takes value 0 otherwise. Then H10 and H1A can be equivalently stated as H10 : E ( Z1t | I1( t 1) )  E ( Z1t | I t 1 ) a.s.,

(2.9)

versus the alternative hypothesis H1A : E ( Z1t | I1( t 1) )  E ( Z1t | I t 1 ) a.s.

(2.10)

Thus, Granger causality in risk between {Y1t } and {Y2t } can be viewed as Granger causality in mean between {Z1t } and {Z2t }. This observation motivates our use of the cross-spectrum of {Z1t , Z2t }, below, which is originally used in Granger (1969) to define the concept of Granger causality in mean. Cross-spectrum is a natural and powerful tool to investigate Granger causality in mean between two time series (Granger 1969). To see the implications of H10 on the cross-spectrum between {Z1t } and {Z2t }, we first recall that for a bivariate covariance-stationary process {Z1t , Z2t }, the normalized crossspectral density is

f (X ) y

1 2Q

d

¤ S( j )e

j  d

ij X

, X [ Q , Q ], i  1,

(2.11)

Detecting extreme risk spillover

15

where S( j ) y corr( Z1t , Z2 ( t j ) ) . Because S( j ) x S( j ), f (X ) is generally complex-valued. The patterns of S( j ) and f X contain valuable information on Granger causality in risk between {Y1t } and {Y2t }. Under H10, S( j )  0 for all j  0 and as a consequence, f X becomes f10 (X ) y

1 2Q

0

¤ S( j )e

ihX

, X [ Q , Q ].

(2.12)

j  d

Thus, we can compare f X and f10 (X ) to test H10. Any nontrivial difference between them is evidence against H10. It may be noted that the history of the risk indicators {Z2 s , s  t} is only a subset of I 2 ( t 1) . One can also use other information in I 2 ( t 1) to predict Granger causality in risk. However, the use of the risk indicators {Z 2s , s  t} is suitable when one is interested in the comovements of extreme market changes between two markets. Moreover, a large change in one market may be induced by a change in another market only when the latter exceeds a certain threshold. Bae, Karolyi and Stulz (2003) also consider the coincidence of extreme return shocks across countries. However, their approach is based on the marginal return distributions. In practice, spillover in the tails of the conditional distributions could be more relevant and important. A risk manager, for example, may be concerned with whether an incurred loss for a portfolio will exceed a certain pre-specified value given the current adverse changes in another market or another portfolio. Both f X and f10 (X ) are unknown, but can be estimated consistently by nonparametric methods. The kernel method is most commonly used in spectral estimation. In this paper, we will use the kernel method, which has simple and intuitive appeal in the present context. Most importantly, it naturally provides a natural and flexible downward weighting for lag orders, which is consistent with the stylized fact that financial markets are more affected by the recent events than by the remote events and is expected to enhance the power of the proposed procedure. Suppose Vlt (Rl ) y Vl ( I l ( t 1) , Rl ),

l  1, 2,

(2.13)

is a parametric VaR model for Vlt , where Rl is an unknown finite-dimensional parameter. There have been many methods to estimate VaR (e.g. Engle and Manganelli 2004; Jorion 2000). Examples are historical simulation methods, Hansen’s (1994) autoregressive conditional density model, J.P. Morgan’s (1997) Risk Metrics and Engle and Manganelli’s (2004) conditional autoregressive VaR (CAViaR) models. Suppose further we have a random sample [Y1t ,Y2t ]Tt 1 of size T and an estimator Rˆl . Put Zˆ lt y Zlt (Rˆ l ),

l  1, 2,

(2.14)

16

Detecting extreme risk spillover

where Zlt (Rl ) y 1[Ylt  Vlt (Rl )] . Then we can define the sample crosscovariance function between [Zˆ 1t ] and [Zˆ 2t ] , «T 1 T ( Zˆ 1t Bˆ 1 )( Zˆ 2 ( t j ) Bˆ 2 ), 0 b j b T 1, ¤ t 1 j ® Cˆ ( j ) y ¬ T ® T 1 ¤ t 1 j ( Zˆ 1t  j Bˆ 1 )( Zˆ 2t Bˆ 2 ), 1 T b j b 0, ­

(2.15)

where Bˆ l y T 1 ¤ t 1 Zˆ lt. The sample cross-correlation functions between [Zˆ 1t ] T

and [Zˆ 2t ] is

Sˆ ( j ) y Cˆ ( j ) / Sˆ1Sˆ2 ,

j  0, p1,… , p (T 1),

(2.16)

where Sˆ12 y Bˆ l (1 Bˆ l ) is the sample variance of [Zˆ lt ]. We can replace Bˆ l with α. This does not affect the asymptotic distribution of the proposed test statistic under H10. The kernel estimators for the cross-spectral densities f X and f10 (X ) are: 1 fˆ( X ) y 2Q

T 1

¤

k ( j / M )Sˆ ( j )e ij X ,

(2.17)

j 1 T

and 1 fˆ10 ( X ) y 2Q

0

¤

k ( j / M )Sˆ ( j )e ij X .

(2.18)

j 1 T

To compare fˆ  X and fˆ10 ( X ), we use the quadratic form:





T 1

Q L2 fˆ, fˆ10 y 2 Q ° | fˆ( X ) fˆ10 ( X ) |2 d X  ¤ k 2 ( j / M )Sˆ 2 ( j ),

Q

(2.19)

j 1

where the second equality follows by Paserval’s identity. There is no need to calculate numerical integrations over frequency X . Our test statistic for H10 versus H1A is a standardized version of the quadratic form: 1 · ¨ T 1 Q1 M y ©T ¤ k 2  j / M S 2  j C1T M ¸ / D1T M 2 , ¹ ª j 1

(2.20)

Detecting extreme risk spillover

17

where the centering and standardization constants are T 1

C1T (M ) y ¤ (1 j / T )k 2 ( j / M ), j 1

and T 1

D1T (M ) y 2¤ (1 j / T )(1 ( j  1) / T )k 4 ( j / M ). j 1

The factors (1 j / T ) and (1 ( j  1) / T ) are finite sample corrections. They could be replaced by 1. Both C1T (M ) and D1T (M ) are approximately the mean and variance of the quadratic form TL2 ( fˆ, fˆ0 ). What Q1 (M ) checks here is not the original hypothesis but only its necessary conditions. However, it captures the most important information to deliver a feasible test. To compute Q1 (M ), one can use the truncated kernel kT ( z )  1(| z |b 1),

(2.21)

where 1 · is the indicator function. This yields the following test statistic 1 · ¨ T 1 Q1TRUN (M ) y ©T ¤ S 2  j M ¸ / 2M 2 , ¹ ª j 1

(2.22)

M

Zˆ 1t y B 0  ¤ B j Zˆ 2 ( t j )  ut ,

(2.23)

j 1

M

which checks whether the coefficients [B j ] j 1 are jointly zero. This is similar to Pierce and Haugh’s (1977) residual-based test for Granger causality in mean. Here we need not include the lagged variables of Zˆ 1t because Z1t is a sequence of i.i.d. Bernoulli random variables under the null hypothesis. For the estimated Zˆ 1t, (2.23) holds asymptotically, which is almost satisfied in practical applications where usually large samples are used to estimate the parameters in VaR model. Granger (1969) proposes a popular test for causality in mean based on a regression similar to (2.23), with a fixed but arbitrarily large M. To ensure that the regression test has power against a large class of alternatives, we let M grow with the sample size T properly. This delivers a R 2 -based test statistic Q1REG  (TR 2 M ) / ( 2M )1/ 2 ,

(2.24)

18

Detecting extreme risk spillover

where R 2 is the centered squared multi-correlation coefficient from the regression in (2.23). We may view this test as a generalized version of Granger’s (1969) test for H10. This procedure is simple and intuitive. It could be shown that Q1REG (M ) is asymptotically equivalent to Q1TRUN (M ) under H10. When M is large, however, both Q1TRUN (M ) and Q1REG (M ) may not deliver good power against alternatives of practical importance. As a stylized fact, today’s financial markets are often more influenced by the recent events than by the remote events, which implies that the dependence of Z1t on the Z2( t j ) will eventually diminish as lag order j increases. Consequently, it is more efficient to discount higher order lags. The most commonly used kernels are downward weighting for higher order lags. Examples are the Bartlett, Daniell, Parzen and Quadratic-Spectral kernels.6 In contrast, the Q1TRUN (M ) and Q1REG (M ) tests are not fully efficient when M is large. See Section 2.4 for more discussion. The key step in implementing our procedure lies in the VaR estimation. This is relatively simple for practitioners in the real financial industry, because VaR can be easily calculated by most standard risk management softwares. Furthermore, VaR can be set not only at the commonly used 1 per cent or 5 per cent level, but also at any level which the investors or risk managers may be interested in. For example, investors often impose the stop-loss rule for their portfolio investments. Our procedure can be applied to investigate risk spillover at the stop-loss level.

2.3

Asymptotic theory

We now derive the limit distribution of the Q1 (M ) test under H10. Its derivation is complicated by the fact that we do not observe the true parameter values [Rl0 ] and have to estimate them. Parameter estimation uncertainty in [Rˆl ] has to be dealt with properly, as is encountered by Engle and Manganelli (2004),where the interest is in testing the adequacy of a univariate VaR model, and parameter estimation uncertainty has a nontrivial impact on the limit distribution of the test statistic, which complicates the construction of their test statistic. In particular, it involves nonparametric estimation of the conditional probability density of the underlying process. Our nonparametric cross-spectral approach fortunately enables us to get rid of the impact of Rˆl asymptotically. Intuitively, Rˆl converges to Rl0 faster than the nonparametric estimators fˆ  X and fˆ10 ( X ) to f (ω) and f10 (X ) respectively. As a consequence, the limit distribution of Q1 (M ) is solely determined by the kernel estimators fˆ  X and fˆ10 ( X ). One can proceed as if Rl0 were known and equal to Rˆl . Thus, replacing Rl0 with Rˆl has no impact on the limit distribution of Q1 (M ). This greatly simplifies the construction and implementation of our test statistic because we need not know the asymptotic expansion of [Rˆl ] and can choose any convenient T -consistent estimates. To justify the above heuristics, we impose some regularity conditions on the data generating process {Ylt }, the VaR models Vlt (Rl ) , the parameter estimators Rˆ l and the kernel function k · .

Detecting extreme risk spillover

19

Assumption 1: For l = 1,2, {Ylt } is a stochastic time series process with unknown twice continuously differentiable conditional distribution function Flt ( y ) y P (Ylt b y | I t 1 ), where y R and I t 1 is the information available at time t − 1. Assumption 2: For Rl  1l ‹ R dl , where dl is a positive integer, l = 1,2, Vlt (Rl ) y Vl ( I l ( t 1) ,Rl ) is a VaR model at level B ( 0,1) such that (i) for each Rl 1l , Vlt (Rl ) is a measurable function of I l ( t 1) ; (ii) with probability 1, Vlt ( i ) is twice T u continuously differentiable with respect to Rl 1l , with lim T 1 ¤ t 1 E sup || T md uR Rl 1l T

Flt [ Vlt (Rl )] ||4  d and lim T 1 ¤ t 1 E sup || T md

Rl 1l

u2 Flt [ Vlt (Rl )] ||2  d . uRuRa

Assumption 3: For l = 1, 2, there exists some Rl0 1l such that (i) P[Ylt  Vlt (Rl0 ) | I l ( t 1) ]  B a.s.; (ii) the risk indicator Z2t (R02 ) y 1[Y2t  V2t (R02 )] depends on an arbitrarily long but finite length of the current and past history of {Z1s (R10 ) y 1[Y1t  V1t (R10 )], s b t}. 1

Assumption 4: T 2 Rˆl R*l  OP 1 for l  1, 2, where R*l y p lim Rl and Rl*  Rl0 under the null hypothesis of interest. Assumption 5: Put St (R ) y [S a1t (R1 ), S a 2t (R2 )]a and Zt (R ) y [ Z1t (R1 ), Z2t (R2 )]a u where Slt (Rl ) y Flt [ Vlt (Rl )]. Then{St (R ), Zt (R )} is a fourth order stationary uR l process such that (i) d

d

¤

d j 0

|| ' ( j ) ||b $ , where '( j ) y Cov[St (R* ), Z at j (R* )];

d

(ii) ¤ j  d ¤ k  d ¤ l  d || L 0 ( j , k , l ) ||b $ , where L 0 ( j , k , l ) is the fourth order cumulant of the joint distribution of {St (R * ) ESt (R * ), Zt j (R * ) EZt j (R * ), St k (R * ) ESt k (R * ), Zt l (R * ) EZt l (R * )}. Assumption 6: k: R m [ 1,1] is a symmetric function that is continuous at 0 and all points except a finite number of points on R, with k  0  1 and

°

d

d

k 2 ( z )dz  d.

Assumption 1 is a standard regularity condition on the data generating processes for {Y1t ,Y2t }. We allow for some covariance-nonstationary processes {Ylt }. An example is the IGARCH process (Engle and Bollerslev 1986), which is strictly stationary but not covariance-stationary (Nelson 1991). Assumption 2 provides regularity smoothness and moment conditions on the VaR models Vlt (Rl ) . There are various VaR models in the literature (e.g. Chernozhukov and Umantsev (2001); Duffe and Pan (1997); Engle and Manganelli (2004); Jorion (2000)). Some of them essentially specify the whole conditional distribution of Ylt while others only specify the left tail of the

20

Detecting extreme risk spillover

conditional distribution. Examples of the former include Morgan’s (1996) Risk Metrics, GARCH models with i.i.d. innovations, Hansen’s (1994) autoregressive conditional density model with a generalized Student-t-distribution, and examples of the latter include Engle and Manganelli’s (2004) CAViaR models. Assumption 3 imposes some conditions on the VaR models which will be required only under H10. Assumption 3(i) is the condition on the adequacy of VaR models, which can be checked using the methods of Chernozhukov and Umantsev (2001), Christoffersen et al. (2001), Christoffersen and Jacobs (2004) and Engle and Manganelli (2004). Assumption 3(ii) allows for the possibility that under H10, although {Z2 s (R 20 ), s  t} do not affect Z1t (R10 ) , Z2t (R 20 ) may depend on the current and past history of {Z1s (R10 ), s b t}. In other words, there may exist instantaneous Granger causality between Z1t (R10 ) and Z2t (R 20 ) and/or Granger causality from {Z1s (R10 ), s  t} to Z2t (R 20 ) under H10. For simplicity, Assumption 3(ii) assumes that Z2t (R 20 ) depends on an arbitrarily long but finite history of {Z1s (R10 ), s b t}. It is possible to allow Z2t (R 20 ) to depend on the entire past history of {Z1s (R10 ), s b t}, with a suitable rate condition on the dependence of Z2t (R 20 ) on the history of {Z1s (R10 ), s b t}. However, we do not consider this possibility here for simplicity. Assumption 4 does not require any specific estimation method; in particular, Rˆl need not be asymptotically most efficient; any T -consistent estimator of Rl0 suffices under H10. An example is Engle and Manganelli’s (2004) regression quantile estimator. We do not require parameter estimation consistency under the alternative H1A . Thus, the probability limit Rl* may not coincide with Rl0 under H1A . Moreover, we need not know the asymptotic expansion of {Rl }. These features greatly simplify the construction and implementation of the test statistics. Assumption 5 is a regularity condition on the serial dependence of the process {St (R ), Zt (R )}. Under H10, we have Zlt (Rl* )  Zlt (Rl0 )  Zlt . Thus, under H10, {Z1t (R1* )} is an i.i.d. Bernoulli(α) sequence and Z1t (R1* ) is independu ent of {Z2 ( t j ) (R 2* )}dj 1, but the derivative E [ Z1t (R1* ) | I t 1 ]  S1t (R1* ) generuR1 ally depends on I t 1. We note that the fourth order cumulant condition in Assumption 5(ii) is a standard assumption in time series analysis (e.g. Hannan (1971)). Finally, Assumption 6 is a standard regularity condition on the kernel k · . Among other things, the condition that k  0  1 ensures that the asymptotic biases of the kernel-based cross-spectral density estimators fˆ  X and fˆ10 ( X ) vanish as sample size T m d. Most commonly used kernels satisfy Assumption 6 (see, e.g. Priestley 1981). We now state the asymptotic normality of Q1 (M ) under H10.

Detecting extreme risk spillover

21

Theorem 2.1 Suppose Assumptions 1–6 hold and M  cT O , where 0  c  d, 3 ´ ¥ 2 , 0  v  12 , v  min ¦§ µ if d y max( d1 , d 2 )  2, and dl is the dimension d 2 d 1¶ of Rl . Then Q1 (M ) md N ( 0,1) under H10. 3 ´ ¥ 2 , The condition that v  min ¦§ µ if d > 2 is sufficient but may not d 2 d 1¶ be necessary. This is imposed to simplify the treatment for the impact of parameter estimation uncertainty in {Rl }. It could be weakened at a cost of more a tedious proof. In the present context, the technical treatment of parameter estimation uncertainty is not trivial, because the risk indicator Zlt (Rl ) is not differentiable with respect to Rl . From the proof of Theorem 2.1 (see Theorem A.2.1 in the appendix), we find that parameter estimation uncertainty in Rˆl has no impact on the limit distribution of Q1 (M ). This occurs because Rˆ l converges to Rl0 faster than the kernel cross-spectral estimators fˆ  X and fˆ10 ( X ) to f X and f10 (X ) respectively. To understand the intuition why Q1 (M ) is asymptotically N ( 0,1) , we consider the Q1TRUN M test that is based on the truncated kernel in (2.21). First, suppose Rl0 were known. Then as T m d, we have T Sˆ ( j ) md N ( 0,1) at each lag j  0 and cov[ T S( j1 ), T S( j2 )] m 0 for any j1 x j2 under H10. M

2 Consequently, ¤ j 1T S ( j ) , being the sum of M asymptotically independent 2 D12 random variables, are asymptotically distributed as D M . By the well-known 2 normal approximation of D M when M is large, we obtain the asymptotic normality of Q1T M . The impact of parameter estimation uncertainty in Rˆl is at most a finite adjustment, which is asymptotically negligible when M is large. This intuition remains valid for nonuniform kernels. To investigate the asymptotic behavior of the Q1 (M ) test under H1A , we impose a condition on the cross-correlation S( j ) and a fourth order cumulant condition.

Assumption 7: Let S( j )  cov[ Z1t (R ), Z2 ( t j ) (R )]. Then (i) (ii)

¤

d j  d

¤

d k  d

¤

d l  d

¤

d j 1

S2 ( j )  d ;

| L 1 ( j , k , l ) | d , where L 1 ( j , k , l ) is the fourth order

cumulant of the joint distribution of {Z1t (R1* ) EZ1t (R1* ), Z2 ( t j ) (R1* ) EZ2 ( t j ) (R 2* ), Z1( t k ) (R1* ) EZ1( t k ) (R1* ),. Z2 ( t l ) (R 2* ) EZ2 ( t l ) (R 2* )}. Assumption 7(i) implies that the dependence of Z1t (R1* ) on {Z2 s (R 2* ), s  t} decays to zero at a suitable rate, but it still allows for certain strongly crossdependent processes whose cross-correlation decays to zero at a slow hyperbolic rate. We do not impose any condition on the dependence of Z2t (R 2* ) on {Z1s (R1* ), s b t} because we only check the one-way Granger causality from {Z2t (R 2* )} to {Z1t (R1* )} with respect to I t 1.

22

Detecting extreme risk spillover

Theorem 2.2 Suppose Assumptions 1–7 hold, and M  cT O for 0  c  d and 0  O  1. Then 1

1

d M2 2 Q1 (M ) m p ¨2° k 4 ( z )dz · ª© 0 ¹¸ T

d

¤S

2

( j)

j 1

under H1A . v Thus, for any sequence of constants, KT  P(T 1 2 ), we have P[Q1 (M )  KT ] m 1 whenever S( j ) x 0 for some j  0. In other words, the Q1 (M ) test has asymptotic unit power at any fixed significance level whenever S( j ) x 0 for some j  0. Because Q1 (M ) m d whenever S( j ) x 0 for some j  0, upper-tailed N ( 0,1) critical values are appropriate. For example, the critical value at the 5 per cent significance level is 1.645. We note also that the condition on M under H1A is weaker than that under H10. Hong (1996) shows that over a class of kernels d 1 k( z ) « º K(U )  ¬k (–) : k ( 0 )  1, ° k ( z )dz  d, k2 y lim  ( 0, d )» , 2

d m 0 z z ­ ¼

which includes the Parzen and Quadratic-Spectral kernels (but not the d

Bartlett kernel), the Daniell kernel kD ( z )  sin(Q z ) / Q z minimizes ° k 4 ( z )dz . 0

Using this result, it can be shown that the Daniell kernel kD (–) maximizes the asymptotic power of Q1 (M ) in terms of Bahadur’s (1966) asymptotic efficiency criterion. Of course, the relative efficiency of nonuniform kernels in K (U ) is very close each other. This implies that the choice of kernel k(–) is not important, provided the truncated (i.e. uniform) kernel in (2.21) is not used. Intuitively, the cross-dependence | S( j ) | decays to zero as j m d under Assumption 7, so it is more efficient to discount higher order lags than to put an equal weight for each lag.

2.4

Two-way Granger causality in risk

2.4.1

Bilateral Granger causality in risk

We now extend our analysis to two-way and instantaneous Granger causality in risk. We consider the hypothesis that neither {Y1t } nor {Y2t } Granger-causes each other in risk at level α with respect to I t 1. The hypotheses of interest can be stated as H 02 : P (Ylt  Vlt | I l ( t 1) )  P (Ylt  Vlt | I t 1 ) a.s. for both l = 1, 2

(2.25)

Detecting extreme risk spillover

23

versus H 2A : P (Ylt  Vlt | I l ( t 1) ) x P (Ylt  Vlt | I t 1 ) for at least one l .

(2.26)

Using the risk indicator {Zlt }, we can write these hypotheses as H 02 : E [ Zlt | I l ( t 1) ]  E [ Zlt | I t 1 ] a.s. for both l  1, 2 versus H 2A : E [ Zlt | I l ( t 1) ] x E [ Zlt | I t 1 ] for at least one l, where l =11,2. Under H 02, the past information of one series is not useful to predict the risk of the other series, and their cross-spectral density f (ω) becomes a flat spectrum: f20 (X ) y

1 S( 0 ), X [ Q , Q ]. 2Q

(2.27)

where ρ(0) is non-zero when there exists instantaneous causality between Z1t and Z2t . A consistent estimator for f20 (X ) is 1 fˆ20 ( X ) y Sˆ( 0 ), X [ Q, Q ]. 2Q

(2.28)

Our test statistic for H 02 versus. H 2A is a properly standardized version of a quadratic form between fˆ( X ) and fˆ20 ( X ): ¨ T 1 · 1 Q2 (M ) y ©T ¤ k 2 ( j / M ) S 2 ( j ) C2T (M )¸ / ; D2T (M )=2 , ª | j |1 ¹ where the centering and scaling factors are T 1

C 2T ( M ) 

¤ (1 | j | /T )k

2

( j /M)

| j | 1

and T 1

D2T (M )  2 ¨1  S4 ( 0 )· ¤ (1 | j | /T )(1 (| j | 1) / T )k 4 ( j / M ). ©ª ¸¹ | j |1

(2.29)

24

Detecting extreme risk spillover

Note that D2T (M ) involve the cross-correlation estimator Sˆ ( 0 ) which has taken into account the possible instantaneous correlation between Z1t and Z2t under H 02. The Q2 (M ) statistic is asymptotically N ( 0,1) under H 02, as is stated below. Theorem 2.3 Suppose Assumptions 1–3(i) and 4–6 hold, and M  cT O , where 3 ´ ¥ 2 , 0  v  12 , v  max ¦§ µ if d y max( d1 , d 2 )  2, and dl is the dimension d 2 d 1¶ of Rl , l  1, 2. Then Q2 (M ) md N ( 0,1) under H 02. We do not need Assumption 3(ii) here, because under H 02, neither {Z1t } nor {Z2t } Granger-causes each other with respect to I t 1. Nevertheless, we still allow for instantaneous Granger causality between Z1t and Z2t under H 02. To study the asymptotic behavior of the Q2 (M ) test under H 2A , we strengthen Assumption 7 slightly to cover two-way cross-correlations. d

Assumption 8: (i) ¤ j  d S 2 ( j )  d and (ii)

¤

d j  d

¤

d k  d

¤

d l  d

| L 1 ( j , k , l ) | d

where S( j ) and L 1 ( j , k , l ) are as in Assumption 7. Theorem 2.4 Suppose Assumptions 1, 2, 4–6 and 8 hold, and M  cT O for 0  c  d and 0  O  1. Then under H 2A , 1

2 d d [1  S 4 ( 0 )]M Q2 (M ) m p ¨2° k 4 ( z )dz · ¤ S 2 ( j ). ª© d ¹¸ | j |1 T

Thus, whenever there exists Granger causality in risk between {Y1t } and {Y2t } with respect to I t 1 such that S( j ) x 0 for some j x 0, the Q2 (M ) test will have asymptotic unit power at any given significance level. Note that the asymptotic variance depends on S( 0 ), which arises due to the presence of instantaneous risk spillover under H 02. 2.4.2

Complete non-Granger causality in risk

As noted above, the null hypothesis H 02 allows for instantaneous risk spillover between {Y1t } and {Y2t }. In practice, one may be interested in testing complete non-Granger causality in risk; i.e. there exists non-Granger causality in risk between {Y1t } and {Y2t } with respect to I t 1 and no instantaneous risk transmission between Y1t and Y2t. The hypotheses of interest are H30 : P (Ylt  Vlt | I l ( t 1) )  P (Ylt  Vlt | I l ( t 1) , I kt ) a.s. for l,k = 1,2,l x k (2.30) versus H3A : P (Ylt  Vlt | I l ( t 1) ) x P (Ylt  Vlt | I l ( t 1) , I kt ) for at least on ne l, (2.31)

Detecting extreme risk spillover

25

where l, k = 1, 2, l x k . Again, these hypotheses can be written as H30 : E [ Zlt | I l ( t 1) ]  E [ Zlt | I l ( t 1) , I kt ] a.s. for l,k = 1,2, l x k, versus the alternative H3A : E [ Zlt | I l ( t 1) ] x E [ Zlt | I l ( t 1) , I kt ] for at least one l ,where l , k  1, 2, l x k. Under H30, the two risk indicator series {Z1t } and {Z2t } are mutually independent because they are Bernoulli(α) variables. Thus, the cross-spectral density f(ω) becomes f30 (X )  0,

X [ Q , Q ].

(2.32)

The test statistic for H30 versus H3A is then ¨ T 1 · 1 Q3 (M ) y ©T ¤ k 2 ( j / M )S 2 ( j ) C3T (M )¸ / ; D3T (M )=2 , ª j 1 T ¹

(2.33)

where the centering and scale factors are T 1

C3T (M ) 

¤

(1 | j | / M )k 2 ( j / M )

j 1 T

and T 1

D3T (M ) y 2

¤

(1 | j | /T )(1 (| j | 1) / T )k 4 ( j / M ).

j 1 T

We now state the asymptotic distribution for Q3 (M ) under H30. Theorem 2.5 Suppose Assumptions 1, 2, 3(i) and 4 – 6 hold, and M  cT O , 3 ´ ¥ 2 , where 0  c  d,0  v  12 , v  max ¦§ µ if d y max( d1 , d 2 )  2, and dl d 2 d 1¶ is the dimension of Rl , Then under H30 we have Q3 (M ) md N ( 0,1).

26

Detecting extreme risk spillover

The asymptotic behavior of Q3 (M ) under the alternative of H30 is given in the below. Theorem 2.6 Suppose Assumptions 1, 2, 4 – 6 and 8 hold, and M  cT O for 0  c  d and 0  v  1. Then under H3A , M Q3 (M ) m p T

¨2 d k 4 ( z )dz · ©ª ° d ¸¹

12

d

¤S

2

( j ).

j  d

The Q3 (M ) test has asymptotic unit power whenever there exists instantaneous risk spillover and/or Granger causality in risk between {Y1t } and {Y2t } with respect to I t 1 such that S( j ) x 0 for some integer j.

2.5

Finite-sample performance

We now examine the finite sample performance of the proposed tests. For the sake of space, we focus on the Q1 (M ) test; the other tests Q2 (M ) and Q3 (M ) are expected to perform similarly. Throughout this section, we work with the following data generating process (DGP): l  1, 2, «Ylt  Cl1Y1t 1  Cl 2Y2t 1  ult , ® ult  T lt F lt , ® ¬ 2 2 2 2 ® T lt  H l 0  H l1T lt 1  H l 2 u1t 1  H l 3 u2t 1 , ®­ F lt ∼ m.d .s. ( 0,1).

(2.34)

To investigate both the level and power of our test, we consider the following cases under (2.34): NULL [No Granger causality in risk]: « ( C11 , C12 , H 10 , H 11 , H 12 , H 13 )   0.5, 0, 0.1, 0.6, 0.2, 0 , ¬ ­( C21 , C22 , H 2 0 , H 21 , H 22 , H 23 )   0, 0.5, 0.1, 0.6, 0, 0.2 , ALTER1 [Granger causality in risk from mean]: «( C11 , C12 , H 10 , H 11 , H 12 , H 13 )   0.5, 0.2, 0.1, 0.6, 0.2, 0 , ¬ ­ ( C21 , C22 , H 2 0 , H 21 , H 22 , H 23 )   0, 0.5, 0.1, 0.6, 0, 0.2 , ALTER2 [Granger causality in risk from variance]:

Detecting extreme risk spillover

27

«( C11 , C12 , H 10 , H 11 , H 12 , H 13 )   0.5, 0, 0.1, 0.5, 0.2, 0.7 , ¬ ­ ( C21 , C22 , H 2 0 , H 21 , H 22 , H 23 )   0, 0.5, 0.1, 0.5, 0.2, 0 . Under NULL, there is no Granger causality in risk between {Y1t } and {Y2t } with respect to I t 1. This allows us to examine the empirical level of the Q1 (M ) test in finite samples. On the other hand, there exists Granger causality in risk under both ALTER1 and ALTER2, but with different sources of spillover. Under ALTER1, there exists Granger causality in mean but not in any higher order conditional-moments. Under ALTER2, there exists Granger causality in variance, but not in mean and other higher order conditional- moments. Spillovers in mean and in variance are most commonly studied in the literature; ALTER1 and ALTER2 allow us to investigate how well our test can detect risk spillover from these sources. Recent empirical studies (e.g. Gallant et al. (1992); Hansen (1994); Harvey and Siddique (1999, 2000); Jondeau and Rockinger (2003); Hong (1998) find evidence of time-varying skewness and kurtosis for various financial time series. It is therefore conceivable that Granger causality in risk between financial markets may be caused by comovements in conditional skewness or in conditional kurtosis. Indeed, skewness and kurtosis are closely related to the left tail of the innovation distribution, or extreme downside risk. To investigate Granger causality in risk from higher order conditional moments, we generate data using Hansen’s (1994) autoregressive conditional density model, which is embedded in (2.34) with the innovations {F lt } following a generalized Student-t-distribution with time-varying shape parameters. More specifically, Hansen’s (1994) generalized t density is given by « ¨ 1 ®®bc ª©1  I 2 gF ( F | M , I )  ¬ ®bc ¨1  1 ®­ ª© I 2

bY  a 2 1 M

 ·¹¸ bY  a 2 1 M

 ·¹¸

( I 1) 2

if F  a b ,

( I 1) 2

if F r a b ,

where

a y 4Mc

I 2 , I 1

b 2 y 1  3M 2 a 2 ,

cy

' ((I  1) / 2 )

Q (I 2 )' (I / 2 )

.

Here, λ measures skewness and η is the degree of freedom parameter. They characterize asymmetry and fat-tailedness of F lt respectively. This density is well-defined for 1  M  1 and 2  I  d and encompasses a variety of conventional densities. For instance, if M  0, the generalized Student-t-distribution

28

Detecting extreme risk spillover

reduces to the standard Student-t-distribution. If in addition I  d, it further reduces to a normal density. We specify that {F lt } follows a generalized Studentt-distribution with time-varying parameters ( Mlt , Ilt ) , where the dynamics of Mlt and Ilt follow the specification of Jondeau and Rockinger (2003): « Mlt  1 exp( Mit ) , ® 1 exp( Mit ) ¬ 4 ®­Ilt  4  1 exp( Iit ) , where «®Mlt  E l1  E l 2 u1t 1  E l 3 u2t 1  E l 4 M1t 1  E l 5 M2t 1 , l  1, 2, ¬ Ilt  U l1  U l 2 u1t 1  U l 3 u2t 1  U l 4I1t 1  U l 5I2t 1 . ®­ We use the following parameter combinations: ALTER3 [Granger causalities in risk from skewness and kurtosis]: « ( C11 , C12 , H 10 , H 11 , H 12 , H 13 )   0.3, 0, 0.1, 0.5, 0.2, 0 , ® ( C21 , C22 , H 2 0 , H 21 , H 22 , H 23 )   0, 0.5, 0.1, 0.6, 0, 0.2 , ® ¬ ®(E11 , E12 , E13E14E15 , U11 , U12 , U13 , U14 , U15 )  ( 0.2,1, 5, 0, 0.9, 0.2,1, 5, 0, 0.9 ), ®­ (E 21 , E 22 , E 23E 24 E 25 , U 21 , U 22 , U 23 , U 24 , U 25 )  ( 0.2, 0,1, 0, 0, 0.2, 0,1, 0, 0 ). Under ALTER3, there exists no Granger causality in mean nor in variance, but there exists Granger in risk, due to the causality in skewness and in kurtosis from Y2t to Y1t with respect to I t 1. To our knowledge, the financial econometric literature has been focusing on spillover in mean and in variance; no study on spillover in skewness and kurtosis is available in the literature. For all data generating processes, we consider three sample sizes, T  500, 1 000, 2 000, which corresponds to two to eight years of daily financial data. These sample sizes may be still relatively small in view of estimating parameters involved in conditional variance and higher order conditional moments (skewness and kurtosis). For each T, we first generate T  500 observations using the GAUSS Window Version random number generator on a personal computer and then discard the first 500 to reduce the possible effect of the starting values ( hl*0 , Ml*0 , Il*0 )  (1 / (1 H 21 H 23 ), 0.2, 4.1). We choose two shortfall probabilities or risk levels: B  10 per cent and 5 per cent. To compute our test statistics, we use the Daniell kernel k ( z )  sinQ z / Q z, z  ( d, d ) which enjoys some optimal power property (see Section 2.4).7 For comparison, we also consider the truncated kernel-based test Q1TRUN M in (2.22) and the Granger-type regression test (1969) Q1REG (M ) in (2.24). To examine the

Detecting extreme risk spillover

29

impact of the choice of the lag order M, we consider M = 5, 10, 15, 20, 25 and 30, which covers a rather wide range of lag orders for the sample sizes considered here. For data generated from each of the DGPs 1–4, we use the QuasiMLE to estimate the unknown parameters in each individual null model: « Ylt  Cl1Ylt 1  ult , l  1, 2, ® ult  Tlt F lt , ® ¬ 2 2 2 ®Tlt  H l 0  H l1Tlt 1  H l 2 ult 1 , ®­ F lt ^ i.i.d. N  0,1 .

(2.35)

A BHHH algorithm is used. This delivers T -consistent estimators under NULL (cf. Bollerslev and Wooldridge 1992; Lee and Hansen 1994; Lumsdaine 1996). Table 2.1 reports the rejection rates of the Q1DAN M in (2.20) with Daniell kernel, Q1TRUN M in (2.22) and Q1REG (M ) tests in (2.24) at the 10 per cent and 5 per cent significance levels under NULL.8 Overall, the Q1DAN M test, which is based on the Daniell kernel, has reasonable sizes for all three sample sizes. It tends to over reject a little at the 5 per cent significance level, but not excessively. For each shortfall probability (α =10 per cent or 5 per cent) and each sample size T, the choice of M has little impact on the size of the Q1DAN M test. The truncated kernel-based test Q1TRUN M performs similarly to Q1DAN M at the 10 per cent significant level. The regression procedure Q1REG (M ), on the contrary, tends to a bit over reject the null hypothesis at the 10 per cent significant level. Both Q1TRUN M and Q1REG (M ) have better sizes at the 5 per cent significant level. Table 2.2 reports the power of the test under ALTER1, where there exists Granger causality in mean from Y2t to Y1t with respect to I t 1. The Q1DAN M test has good power against ALTER1 and it becomes more powerful as T increases. Given each sample size T; the power of Q1DAN M declines as the lag order M increases, but not dramatically (which is apparently due to the downward weighting of k 2 (–)). The Q1DAN M test with the 10 per cent shortfall probability (or risk level) is more powerful than the Q1DAN M test with the 5 per cent shortfall probability (or risk level). This is possibly because spillover in mean occurs in the main body of the distribution. On the other hand, Q1TRUN M and Q1REG (M ) perform similarly, and both have relatively low power. Furthermore, a larger M gives substantially smaller power. For example, at the 5 per cent risk level, the rejection rates of Q1TRUN M decrease from 70.8 per cent and 72.1 per cent to 38.2 per cent and 37.0 per cent respectively even when T = 2000. These results confirm our expectation that nonuniform weighting alleviates the impact of choosing too large a M because nonuniform weighting discounts higher order lags. Table 2.3 reports the power of the test under ALTER2, where there exists Granger causality in variance fromY2t toY1t with respect to I t 1. The Q1DAN M

500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000

9.7 9.6 9.6 9.4 7.5 8.6 12.2 10.5 11.1 11.5 9.2 10.3 11.5 11.2 10.0 11.3 11.1 9.6

10%

T

6.9 6.7 6.7 6.8 6.4 5.8 7.1 7.1 6.7 7.6 6.6 6.4 7.3 6.9 7.0 7.6 7.4 6.3

5% 10.5 10.4 10.3 10.2 10.2 10.3 10.5 12.3 10.0 12.4 11.1 9.9 10.8 11.8 10.5 11.3 11.0 10.4

10%

10

6.4 6.5 6.3 7.3 6.9 7.0 6.5 7.4 6.5 7.9 6.9 6.8 7.3 7.2 6.9 7.3 7.8 7.0

5% 11.6 10.8 10.1 10.2 11.6 9.9 11.3 11.9 10.7 12.2 12.1 10.7 10.7 11.7 8.8 10.0 11.5 10.5

10%

15

7.2 6.5 6.8 7.0 7.7 7.2 6.5 7.3 6.8 8.4 7.6 7.5 5.7 6.4 5.3 6.4 6.8 6.2

5% 12.3 10.9 10.9 10.2 11.7 11.0 9.8 11.4 12.0 11.8 11.0 10.5 10.8 11.3 8.0 9.3 10.2 9.5

10%

20

6.6 6.1 6.9 6.4 7.2 7.7 6.0 6.7 6.5 8.3 6.7 6.3 5.3 6.7 4.4 6.6 5.6 5.3

5% 11.3 10.6 10.5 10.2 11.5 11.1 10.0 11.3 12.0 12.7 11.6 10.9 10.2 11.2 8.3 10.4 10.0 8.9

10%

25

6.9 5.8 7.0 7.0 6.9 7.2 6.4 6.0 5.7 8.5 7.8 6.4 5.7 6.1 4.8 7.0 5.9 4.3

5% 12.2 10.2 11.6 10.3 11.5 11.1 10.9 11.6 11.4 11.7 11.3 10.3 11.6 9.9 9.7 10.4 9.9 9.1

10%

30

6.9 5.6 7.3 7.1 7.3 6.8 7.1 6.1 5.7 7.4 6.8 5.9 6.7 5.8 5.0 6.6 5.9 5.3

5%

Note: NULL: Yit  0.5Yit 1  uit , uit  hit Fit , hit  0.1  0.6 hit 1  0.2uit2 1, Fit ∼ m.d .s. N ( 0,1), i  1, 2 ; The sample size T=500, 1,000 and 2,000; Q110DAN , Q110TRUN , Q110REG and Q15DAN , Q15TRUN , Q15REG represent one-way tests for Granger causality in risk from Y2t to Y1t at the 10 per cent and 5 per cent risk levels respectively, where the subscripts DAN, TRUN and REG denote the Daniell kernel, the truncated kernel and the regression-based tests.

Q15REG

Q110REG

Q15TRUN

Q110TRUN

Q15DAN

Q110DAN

5

M

Table 2.1 Size at the 10 per cent and 5 per cent significant levels

500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000

58.0 80.2 96.3 43.5 56.9 77.9 48.6 69.7 92.3 38.7 50.9 70.8 48.1 72.8 93.2 42.6 49.6 72.1

10%

T 52.1 76.0 94.6 38.6 52.7 73.7 42.4 62.6 89.8 33.0 44.7 63.2 37.5 65.9 90.8 31.4 43.4 65.4

5% 52.6 74.9 94.4 41.2 54.5 75.0 41.6 58.5 86.6 34.4 43.5 60.4 40.8 62.7 87.1 32.1 43.7 61.9

10%

10

47.8 69.2 92.8 35.4 48.4 69.0 33.8 51.5 81.7 25.2 35.0 54.7 31.7 53.7 81.3 25.2 36.3 55.5

5% 50.2 70.5 92.7 39.5 51.7 70.1 37.5 52.4 81.8 28.5 38.9 55.0 34.7 55.2 82.1 27.3 37.5 55.4

10%

15

44.2 63.5 89.9 32.8 44.7 64.1 28.2 43.9 75.1 22.2 31.2 46.9 27.3 46.1 75.7 21.0 30.4 47.5

5% 47.5 66.8 90.7 37.7 48.2 67.7 33.7 49.0 77.2 25.5 35.8 52.4 30.9 49.7 77.7 22.6 34.5 50.8

10%

20

40.6 58.8 87.3 30.3 41.6 60.3 25.2 39.1 70.8 18.4 28.9 42.3 24.0 40.0 69.8 17.0 27.2 42.3

5% 44.7 63.1 88.5 35.6 46.1 64.9 32.6 46.7 75.7 23.8 35.6 48.7 27.7 44.4 73.1 19.8 32.2 48.5

10%

25

37.6 54.5 84.0 27.6 38.6 57.7 23.2 35.1 67.2 18.8 25.7 39.5 21.9 34.9 65.2 16.2 24.1 39.9

5% 43.2 60.0 86.6 33.3 45.0 61.8 30.9 41.9 73.1 23.7 32.4 46.9 27.2 41.6 69.5 18.3 30.0 46.9

10%

30

35.7 51.3 82.2 25.7 36.1 53.9 21.9 31.0 63.9 17.1 24.5 38.2 21.5 32.8 59.8 14.6 22.7 37.0

5%

causality in risk from Y2t to Y1t at the 10 per cent and 5 per cent risk levels respectively, where the subscripts DAN, TRUN and REG denote the Daniell kernel, the truncated kernel and the regression-based tests.

The sample size T=500, 1,000 and 2,000; Q110DAN , Q110TRUN , Q110REG and Q15DAN , Q15TRUN , Q15REG represent one-way tests for Granger

Note: ALTER1: Y1t  0.5Y1t 1  0.2Y2t 1  u1t , Y2t  0.5Y2t 1  u2t, uit  hit Fit, hit  0.1  0.6 hit 1  0.2uit2 1, Fit ∼ m.d .s. N ( 0,1), i  1, 2;

Q15REG

Q110REG

Q15TRUN

Q110TRUN

Q15DAN

Q

10 1DAN

5

M

Table 2.2 Power at the 10 per cent and 5 per cent significant levels under ALTER1

500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000

42.8 59.3 86.6 51.4 67.2 84.5 39.8 55.3 81.4 50.3 62.4 85.0 40.0 57.4 82.7 50.1 60.9 84.7

10%

T 36.9 53.5 83.0 46.0 62.2 85.4 33.0 47.8 76.2 44.5 56.3 80.2 33.5 50.8 77.3 44.3 55.7 80.1

5% 43.1 59.9 84.8 53.3 65.6 82.2 33.8 47.1 75.5 44.7 55.1 77.0 33.3 50.6 73.6 42.9 57.6 76.3

10%

10

36.4 52.9 80.4 46.9 59.4 83.5 25.0 38.2 68.2 36.3 48.5 71.7 25.2 42.1 67.0 34.1 48.8 70.7

5% 40.1 56.5 82.4 52.0 62.2 80.6 30.2 42.5 68.4 38.9 49.7 72.3 28.7 42.7 67.1 37.0 50.5 72.4

10%

15

33.4 49.0 77.3 45.1 56.3 80.2 22.6 32.6 60.5 30.5 42.3 66.0 20.5 33.8 57.2 29.4 42.9 65.8

5% 38.3 52.3 79.3 48.6 59.9 78.3 27.2 37.3 64.2 34.5 46.8 68.6 24.4 37.9 60.3 32.2 47.0 69.0

10%

20

30.3 44.4 73.6 42.0 52.8 77.3 19.0 27.5 53.5 26.9 37.0 60.0 16.6 28.5 50.2 24.4 38.3 60.8

5% 35.5 47.9 76.6 46.3 57.3 75.9 22.9 34.7 59.3 32.4 43.8 64.6 20.7 34.7 55.7 30.5 44.4 64.1

10%

25

27.7 41.7 71.5 38.0 50.3 74.0 16.5 25.5 48.5 26.6 34.7 55.9 14.3 27.1 44.6 24.0 33.8 56.7

5% 33.4 46.2 74.7 44.4 55.3 73.7 23.6 32.0 55.5 33.2 41.5 60.8 20.6 32.3 52.2 29.8 41.0 60.8

10%

30

26.1 40.0 68.4 35.9 48.2 72.0 16.9 23.4 43.6 25.1 32.0 51.9 13.7 24.5 40.8 22.5 31.7 53.5

5%

0.5h2t 1  0.2u22t 1 ; The sample size T=500, 1,000 and 2,000; Q110DAN , Q110TRUN , Q110REG and Q15DAN , Q15TRUN , Q15REG represent oneway tests for Granger causality in risk from Y2t to Y1t at the 10 per cent and 5 per cent risk levels respectively, where the subscripts DAN, TRUN and REG denote the Daniell kernel, the truncated kernel and the regression-based tests.

Note: ALTER2: Yit  0.5Yit 1  uit, uit  hit Fit, Fit ∼ m.d .s. N ( 0,1), i  1, 2, h1t  0.1  0.5h1t 1  0.2u12t 1 +0.7u22t 1, h2t  0.1 

Q15REG

Q110REG

Q15TRUN

Q110TRUN

Q15DAN

Q110DAN

5

M

Table 2.3 Power at the 10 per cent and 5 per cent significant levels under ALTER2

Detecting extreme risk spillover

33

test has good power against ALTER2. As under ALTER1, the power of Q1DAN M declines as M increases, but not dramatically. Again, Q1TRUN M and Q1REG (M ) perform similarly and they have relatively low power. In contrast to ALTER1, all tests have better power at the 5 per cent risk level than at the 10 per cent risk level under ALTER2. This is perhaps because under ALTER2, spillover in variance occurs mainly in the tails rather than the centers of the conditional distributions. Table 2.4 reports the power of the test under ALTER3, where there exists Granger causality in skewness and kurtosis from Y2t to Y1t with respect to I t 1. As expected, the Q1DAN M test has good power against DGP3. All power patterns are similar to those under ALTER1, where there exists Granger causality in mean. Furthermore, as under ALTER1, the Q1DAN M test with the 10 per cent shortfall probability is more powerful than the Q1DAN M test with the 5 per cent shortfall probability. With M = 5 and T = 2000, for example, the rejection rates of Q1DAN M at the 5 per cent significance level are 86.5 per cent and 39.5 per cent for the 10 per cent and 5 per cent shortfall probabilities respectively. Again this may be due to the possibility that spillover mainly comes from skewness which occurs in the main body of the distribution. All power patterns of Q1TRUN M and Q1REG (M ) are similar to those of Q1TRUN M and Q1REG (M ) under ALTER1. To better capture the empirical distribution of returns, it is proposed in existing literature to add jumps to return and volatility process (see, e.g. Hong et al. (2004) and Maheu and McCurdy (2004)). To investigate the power of our proposed test on detecting the risk spillover based on jump processes, we further consider the Poisson jump model following Hong et al. «Ylt  Cl1Y1t 1  Cl 2Y2t 1  ult  nlt Vlt , l  1, 2, ¤1 ® ® ult  T lt F lt , ® 2 ¬ T lt  H l 0  H l1T lt2 1  H l 2 u12t 1  H l 3 u22t 1 , ® F lt ∼ m.d .s. ( 0,1). ® ®­ Vlt ∼ i.i.d N ( 0, E l2 ), nlt ∼ Poisson( Ml ). Particularly, we test the alternative of Granger causality in risk from mean, however with jump added to the process: «( C11 , C12 , H 10 , H 11 , H 12 , H 13 , E1 , M1 )   0.5, 0.2, 0.1, 0.6, 0.2, 0,1, 0.2 , ¬ ­ ( C21 , C22 , H 2 0 , H 21 , H 22 , H 23 , E 2 , M2 )   0, 0.5, 0.1, 0.6, 0, 0.2,1,0.2 . We find that Q1DAN M test still has good power and it becomes more powerful as T increases. For 10 per cent risk level, Q1DAN M is 51.3 per cent, 72.7 per cent and 92.9 per cent when M = 500, 1,000 and 2,000 respectively. And

500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000

47.5 67.0 88.8 25.8 31.5 45.7 38.5 55.6 81.3 24.7 26.3 35.6 38.0 57.0 80.4 23.7 50.1 35.7

10%

T 41.1 62.1 86.5 22.9 26.9 39.5 30.9 46.9 76.1 20.7 20.2 29.4 30.9 29.1 75.4 19.8 23.0 28.7

5% 43.0 60.7 84.7 25.4 28.7 40.0 32.6 44.4 72.0 23.3 21.9 28.6 32.8 48.2 71.7 22.6 40.3 29.1

10%

10

36.6 53.8 81.6 21.9 23.8 33.8 26.3 36.6 64.3 17.8 16.7 22.3 25.8 22.9 65.3 17.2 16.9 22.3

5% 39.4 54.8 82.1 25.7 27.1 35.1 29.7 38.3 65.4 20.5 19.8 24.9 27.6 44.5 65.8 21.3 35.4 27.7

10%

15

32.3 47.8 76.5 20.6 21.2 29.1 22.0 31.4 55.6 15.7 14.9 18.2 20.8 22.6 56.5 15.7 16.1 21.3

5% 36.9 50.8 77.8 24.5 25.4 32.6 28.4 36.7 58.7 19.0 20.2 23.6 25.8 40.9 62.0 18.0 31.3 25.6

10%

20

30.2 43.7 71.9 19.6 19.4 25.8 19.9 28.4 49.6 14.7 14.0 16.0 17.2 20.9 53.2 12.5 14.6 18.9

5% 27.3 40.1 68.1 18.2 18.2 22.6 17.7 24.4 46.0 14.5 13.7 16.0 16.3 20.8 48.1 13.1 14.3 16.8

5% 34.3 44.9 71.5 22.6 23.3 29.9 25.0 33.4 52.7 19.6 18.9 20.6 23.1 36.8 54.8 17.9 25.4 22.5

10%

30

25.7 37.1 64.7 17.5 17.4 21.7 16.8 23.3 43.0 14.2 12.8 13.9 14.8 19.0 44.1 13.2 13.8 15.8

5%

( Ft | M t , It ), M it  (1 exp( M it )) /

35.7 48.0 74.9 24.2 24.5 31.1 26.9 34.1 55.5 19.3 19.9 22.6 23.6 39.3 59.0 17.5 29.1 24.4

10%

25

represent one-way tests for Granger causality in risk from Y2t to Y1t at the 10 per cent and 5 per cent risk levels respectively, where the subscripts DAN, TRUN and REG denote the Daniell kernel, the truncated kernel and the regression-based tests.

0.9I2t 1 , I2t  0.2  F 2t 1; The sample size T=500, 1,000 and 2,000; Q110DAN , Q110TRUN , Q110REG and Q15DAN , Q15TRUN , Q15REG

(1  exp( M it )) , Iit  4 / (1  exp( Iit ))  4, where M1t  0.2  F1t 1 5F 2t 1 0.9M 2t 1 , M 2t  0.2  F 2t 1, I1t  0.2  F1t 1 5F 2t 1

ALTER3: Yit  0.3Yit 1  uit , i  1, 2, uit  hit Fit , hit  0.1  0.5hit 1  0.2uit2 1 , Fit ∼ GT

Q15REG

Q110REG

Q15TRUN

Q110TRUN

Q15DAN

Q110DAN

5

M

Table 2.4 Power at the 10 per cent and 5 per cent significant levels under ALTER3

Detecting extreme risk spillover

35

for the 5 per cent risk level, the Q1DAN M test increases from 45.2 per cent to 91.1 per cent 9. In summary, the proposed tests with the Daniell kernel have reasonable size and power against a variety of empirically plausible alternatives in finite samples. The truncated kernel-based test Q1TRUN M and the Granger-type regression procedure Q1REG (M ) also have reasonable sizes for all sample sizes. However, for the alternatives under study, they often yield lower power than nonuniform weighting, especially for a larger lag order M. In contrast, the use of nonuniform weighting makes the power relatively robust to the choice of M. This suggests that our test with nonuniform weighting is a useful tool in investigating extreme risk spillover across financial markets.

2.6

Conclusion

Based on a new concept of Granger causality in risk which focuses on the comovements between the tails of the two distributions, a class of kernel-based tests are proposed to test whether a large downside risk in one market will Granger-cause a large downside risk in another market. The proposed tests check a large number of lags but avoid suffering from severe loss of power due to the loss of a large number of weighting is consistent with the stylized fact that today’s financial markets are more influenced by more recent events than by remote past events, thus enhancing the power of the proposed tests. This is expected to give good power against the alternatives with decaying crosscorrelations as the lag order increases. Indeed, nonuniform weighting often delivers better power than uniform weighting, as is illustrated in a simulation study and an application to exchange rates. A Granger-type regression-based test (1969) is equivalent to the uniform-weighting-based test. Simulation studies show that the procedures have reasonable size and good power against a number of empirically plausible alternatives in finite samples, no matter whether risk spillover arises from spillover in mean, in variance, or in skewness and kurtosis. These procedures are therefore useful for investigating the comovements between large market changes such as financial contagions.

Notes 1

2 3

For example, Login (2000); Bali (2003) point out that volatility measures based on the distribution of all returns cannot produce accurate estimates of market risks during volatile periods. Using regime-switching modeling techniques, Hong, Li and Zhao (2003) also find that the innovation distribution has a heavier tail when foreign exchange market and interest rate market have higher volatility. VaR is a measure of extreme downside risk and is similar in methodology to lower partial moments in the earlier literature (e.g. Roy 1952). For example, Artzner (1999) defines certain properties that a good risk measure should have and shows that VaR does not satisfy all of them. Many other kinds of risk measures have been proposed but none gained the popularity as VaR.

36 4

5 6

7 8 9

Detecting extreme risk spillover In a strategic portfolio allocation context, fund managers or risk-averse investors could use the concept of Granger causality in risk to select the asset classes that exhibit the lowest risk spillover. This may actually be regarded as alternative risk diversification to the classical Markowitz’s (1952) portfolio theory based on variance. The lead-lag effects between bonds and equities, or between the equities and commodities markets have long received much attention, and our procedures can shed some light on their extreme downside market movement relationship which investors and regulators care much about. In econometric analysis, whether there is a time lag in risk spillover also depends on the sampling frequency of observed data. Engle (1982), in the context of testing the existence of ARCH effects, also considers linearly declining weighting for lag orders (which is equivalent to the Bartlett kernel) to increase the power of his LM test. Here, we allow for a more general flexible weighting and allow M to grow with T. We have also considered the Bartlett kernel, which is outside the class of kernels over which the Daniell kernel has the optimal power; the results are similar to those based on the Daniell kernel. We emphasize that the significance level of our test is different from the risk level or shortfall probability level in the definition of Granger causality in risk. The Q1DAN M results for M = 10, 15, 20, 25 and 30, are available from the authors upon request.

3

VaR estimation

In recent years, with the development of economic globalization and financial liberalization, the increased volatility of financial market and the more and more complicated risk structure of financial tools induce a crucial problem, how to regulate financial risk efficiently, to be figured out. Risk measure is the key factor for risk management. One primary risk measure approach, VaR (Value at Risk) is to estimate the possible or potential loss of given finance products and portfolio at futures price volatility. It concretizes the risk of financial assets a simple number that matches income, and hence makes it convenient to measure the risk level of financial markets. Futures agreement is a financial tool with high leverage and futures trading is of high reward and high risk. The control and management of the futures trading risk are of great importance. In occident futures markets, VaR has been developed as a main approach in futures trading risk management. China’s futures market is still in its infancy and it has large volatility owning to various factors. With the increasing of uncertainty, market risk rises too. Therefore, it is necessary to estimate the VaR of China’s futures market precisely so that the risk of China’s futures market can be measured and controlled efficiently. Jorion (1997) presented the definition of VaR, which is described as the worst expected loss at the given confidence level and given time range. The precise definition is, at the given confidence level 1 B and given hold time U , VaR is the worst expected loss during time U with probability B . In statistical sense, VaR is the B quantile of the yield conditional distribution. Though the concept of VaR is simple and intuitive, how to measure it still remains a challenging question. Concerning VaR estimation, many economists made insightful discussion. The first one was introduced by J. P. Morgan, which is the variance-covariance approach by assuming that yield obeys normal distribution. Since 1999, some new findings on VaR measuring appear. Bouchaud and Potters (2001) studied how to use non-Gaussian of the financial asset volatility to calculate the VaR of nonlinear combination; Li (1999) suggested a semi-parametric approach by using the front four moments;Dowd (1999) employed an extreme value to estimate VaR; Ho, Burridge and Cadle et al. (2000) applied extreme theory in price index of six Asia regions including Japanese and Korea. Fan and Gu (2003) estimated

38

VaR estimation

volatility with semi-parametric approach, and quantile with parametric and nonparametric approaches, they also estimated VaR for the global mainly stock index. Gencay and Seluk (2004) investigate the relative performance of VaR models with the daily stock market returns of nine different emerging markets. Results indicate that EVT-based VaR estimates are more accurate at higher quantiles. Fernandez’s (2005) research was focusing on risk management by extreme tail value. Empirical literatures on the performance of various VaR measure are also emerged. Manganelli and Engle (2001) evaluate the performance of the most popular univariate VaR methodologies, results show that CAViaR models perform best with heavy-tailed DGP. Chen and Tang (2005) investigated theoretical properties of the kernel VaR estimator. Kuester, Mittnik and Paolella (2005) compared the out-of-sample performance of existing methods and some new models for predicting VaR, find that most approaches perform inadequately, although several models are acceptable under current regulatory assessment rules for model adequacy. Bao, Lee and Saltoglu (2006) made a study of five indexes of the Asian market. Results show that: though historical simulation and parametric approach didn’t perform better than other models, they perform better than EVT model. The possible reason is: compared with mature market, Asia stock markets are newly developing markets and have serial correlation and conditional heteroskedastic severity. Parallel to the international research, many Chinese economists keep working on VaR estimation. Ye (2000) used GARCH model to analyze China stock market volatility. Zhan and Tian (2000) used extreme value approach to estimate exchange rate VaR. Fan (2001) used EMWA approach to evaluate the VaR of Shanghai and Shenzhen stock market. Zhu (2001) provided empirical analysis on Hong Kong stock market with different approaches and concluded that there does not exist a dominant one. Huang (2004) utilized CAViaR model to analyze China’s stock market and probed into the model stability. To summarize previous work, VaR estimation can be derived into three kinds from statistical approaches: parametric approach including variancecovariance approach, GARCH model and EMWA; semi-parametric approach including EVT and moments approach; and nonparametric approach including historical simulation, Monte Carlo simulation, kernel function and CAViaR. Most literature concentrates on stock markets and few of them notice China’s futures market. In this chapter, three approaches, parametric estimation based on asymmetry GARCH type model and semi-parametric approach based on moments and nonparametric approach based on Kernel function are used to measure the risk in China’s futures market so that the risk could be regulated by the Chinese government. Moreover, the empirical results show that for Downside VaR and Upside VaR, nonparametric approach exhibits better than the other two approaches at all the three confidence levels. This chapter is organized as follows: section 3.1 introduces Upside VaR and Downside VaR, section 3.2 to 3.4 introduce three VaR estimations.

VaR estimation

39

f(x)




z


0

x

Figure 3.1 Downside VaR

Section 3.5 gives backtest, section 3.6 describes data selection and some basic statistical characteristics of the data. Section 3.7 provides the empirical results. Section 3.8 concludes.

3.1

Upside VaR and Downside VaR

Generally, financial markets consider the downside risk. Short selling mechanism in futures market urges the investors and supervisors concerned not only price fall but also price rise. This section introduces ‘Upside VaR’, and estimates the Downside VaR and Upside VaR with the above three approaches. Take copper, for example, left B quantile of copper yield Yt is adopted to measure the risk of price drop, whose economic meaning in futures market is that long position will face the price drop risk when buying futures contracts. And right B quantile of copper yield Yt is to measure the risk of price rise, whose economic meaning in futures market is short position will face the price rise risk when selling futures contracts. Define at the confidence level 1 B , copper price Downside VaR as Vt ( down ): P (Yt  Vt ( down ) | I t 1 )  B.

(3.1)

Similarly, define at the confidence level 1 B , copper price Upside VaR as Vt ( up ): P (Yt  Vt ( up ) | I t 1 )  B.

(3.2)

While I t 1  {Yt 1 ,Yt 2, } is the attainable information set at time t 1. Confidence level stands for the risk preference of the principle at risk. Usually, B is 0.10, 0.05 or 0.01. Figure 3.1 and 3.2 give the meaning of Upside VaR and Downside VaR.

40

VaR estimation f (x)



0

z1


x

Figure 3.2 Upside VaR

3.2

Parametric conditional VaR estimation

The core of parametric VaR estimation is to estimate the volatility of the financial market. Generally speaking, higher volatility implies higher risk. The conditional variance of the GARCH type model could be use to measure the volatility of asset or asset portfolio. Considering the existence of leverage effect, which is the information asymmetry in the financial market, the effects of good news and bad news to the volatility of the futures market are asymmetric although short selling mechanism is introduced. Hence, the parametric estimation in this paper employs two kinds of asymmetric models – EGARCH and TGARCH which are based on normal distribution to estimate the conditional VaR in the futures market. 3.2.1

Estimation based on EGARCH

Nelson (1991) threw out the EGARCH (Exponential GARCH) model to describe the asymmetric effects of good news and bad news on market volatility. The normal format is: q ¨ F F ln( ht )  X  ¤ ©B i t i  Ji t i ht i ht i i 1 © ª

· p ¸  ¤ C j ln( ht j ). ¸¹ j 1

(3.3)

If Ki , i  1, q are not all zeros, information is not symmetric and there exists leverage effect. When Ki  0, bad news has more influence on volatility than good news; when Ki  0, good news affects more than bad news on volatility. 3.2.2

Estimation based on TGARCH

Zakoian (1990); Glosten et al. (1993) developed the TGARCH (Threshold GARCH) model to explain the asymmetric effects of good news and bad news on market volatility. It is normalized as:

VaR estimation q

p

ht  X  ¤ B i Ft2 i  ¤ C j ht j H Ft2 1dt 1 . i 1

41 (3.4)

j 1

When Ft  0, dt  1, or else dt  0, threshold value dt indicates the effect of information, when dt  1, it is effect of good news, while dt  0 implies the effect of bad news. H is the parameter which reflects the asymmetric effects of good news and bad news on financial market volatility. Significant nonzero coefficient predicates the effects of good news and bad news on market volatility are asymmetric and the leverage effect is significant. H  0 shows that good news affects more than bad news on volatility and H  0 indicates contrarily. Based on the preceding two kind of asymmetric GARCH models, the Downside VaR and Upside VaR can be calculated by the following two formulas, respectively. Vt  down  Nt  zB ht

(3.5)

Vt ( up )  Nt  z1 B ht

(3.6)

Where Nt is the conditional expectation on the market, zB is the left percentile of the distribution which the normalized residual of GARCH model obeys, for instance, when B  0.10, 0.05 and 0.01, zB takes value 1.28, 1.65 and 2.33, respectively.

3.3 Semi-parametric VaR estimation based on volatility, skewness and kurtosis The yield is not normal distribution. It is biased and leptokurtic, and has a thicker tail than normal distribution. Li’s (1999) semi-parametric approach can construct the upper limit and lower limit of confidential interval by calculating the mean, variance, skewness and kurtosis of price yield sequence without any distribution hypothesis. Suppose yield r is a random variable and its mean, variance, skewness and kurtosis are as follows:

N  E ( r ),

T 2  Var( r ),

H1 

E ( r N )3 , T3

H2 

E ( r N )4

3. T4

(3.7)

If H 1 x 0, H 2 x 0, then r does not obey normal distribution. Simple proof yields that the upper limit and lower limit of the confidence interval can be attained by the following formulas:

42

VaR estimation When H 1  0 , 2 ¨ (H 2  2 )(H 2  2 H 12 ) · ¥ H  2´ H2 2 © 4

¦ 2 Z   1¸ B H1 H1 § H 1 µ¶ ¸¹ ©ª VaRl  N  T 2 (3.8) 2 ¨ (H 2  2 )(H 2  2 H 12 ) · ¥ H 2  2´ H2 2

¦  4 ©ZB  1¸ µ H1 H1 § H1 ¶ ¸¹ ©ª VaRu  N T 2 (3.9)

When H 1  0 , 2 ¨ (H 2  2 )(H 2  2 H 12 ) · ¥ H 2  2´ H2 2 ©  ¦ 4 Z   1¸ B H1 H1 § H 1 µ¶ ¸¹ ©ª VaRl  N  T 2

(3.10) 2 ¨ ( H 2  2 )( H 2  2 H 21 ) · ¥ H  2´ H2  2 ©  ¦ 2 4 Z   1¸ B H1 H1 § H 1 µ¶ ¸¹ ©ª VaRu  N T. 2 (3.11)

Where H 1 x 0, ZB is the standard significance, which is the corresponding percentile of B in standard normal distribution. For example, ZB = 1.65 under 5 per cent significance.

3.4

Nonparametric VaR estimation based on kernel function

History simulation approach assumes that the change of market factor is independent identically distributed, and is exactly the same as the change of history. This assumption disaccords with the real financial market and cannot predict and reflect the abruptness and extremeness. Moreover, this approach imparts the same weight to all observations, which conflicts with reality, too. History simulation approach needs plentiful history data. The tradeoff is that too less history data leads to the volatility and inaccuracy of VaR estimation, while long-time history sample probably violates the IID assumption even the stability of VaR estimation is getting better. Monte Carlo simulation approach, with its calculation hugeness, is accurate but not efficient. Therefore, the fresh kernel function approach is employed in this section.

VaR estimation

43

The unknown density function of futures yield makes VaR estimation very complicated. Large numbers of literature discussed this problem with respect to the density function. Nonparametric approach is used to estimate the probability density function of yield, that is to say, kernel estimation of probability density function is utilized. Kernel density estimation, which is a point-wise asymptotic unbiased estimation, has many good big sample characters. When the sample capacity is big enough, the MSE of density function will be small enough when estimating with kernel function. The kernel estimator of probability density function is: 1 n ¥ r ri ´ fˆ( r )  ¤ k ¦ µ, nh i 1 § h ¶

(3.12)

where ri is yield sequence, n is sample capacity and h denotes bandwidth. Kernel function k (t ), which is usually about origin symmetric, is the weight function. It uses the distance between point r and ri, closer points will be given higher weight. Empirical work suggests that the selection of kernel function has little influence on the estimation. Normal distribution density function, Gauss kernel function, is adopted in most cases:

k (t ) 

1 2Q

e

1

t2 2

.

(3.13)

MSE is related with the selection of bandwidth h(also called smooth parameter). Theoretically, the rule for bandwidth selection is to minimize the MSE of estimator. However, each selection has to balance the variance and the squared deviation. If the bandwidth is small, the squared deviation will be small, but the variance will be large. On the contrary, if the bandwidth is big, the squared deviation will be big while the variance will be small. Too big or too small bandwidths both enlarge MSE. Intuitively, stochastic effect adds the irregular volatility as h decreases, the curve will be over smooth and some characters will be inconspicuous as h increases. The bandwidth in this section is:

1

h  1.06 n 5 s,

(3.14)

where s is the sample standard deviation, the left percentile and right percentile will be calculated by B  °

x

d

f ( r )dr and B  °

d x

f ( r )dr , respectively.

Thus, nonparametric approach bypasses the assumption of the distribution of futures yield. At the same time, it also avoids model risk and complicated parametric estimation.

44

3.5

VaR estimation

Backtest

The estimation of risk value VaR depends on the probability distribution and given confidence level of futures yield of assets. Backtest for the VaR estimate is to check if the given confidence level matches reality when deciding VaR. If the loss exceeds VaR estimate too much, the VaR model underestimates real risk level; if the loss exceeds VaR estimate too less, the VaR model is too reserved and overestimates the real risk level. Backtest on risk value provides a way to check if the model fits reality. Kupiec (1995) developed a test, which treats the irregularity, when yield exceeds VaR estimate, as an independent event of binomial distribution. Suppose the confidence level is 1 B , sample capacity is T , failure time is N , then failure frequency is f  N T , the expectation of failure rate is a. Kupiec’s likelihood rate LR can check the null hypothesis B  f , therefore, the estimation of accuracy of the VaR model is turned to be checking if failure rate equals B significantly. T N

LR  2 ln ¨1 a ª

B N ·¹  2 ln ¨ª1 f T N  f N ·¹

(3.15)

With the null hypothesis, LR~ D 2 (1), the critical value of 99 per cent confidence level is 6.635, the critical value of 95 per cent confidence level is 3.841, the critical value of 90 per cent confidence level is 2.706. If LR is greater than the critical value, null hypothesis is rejected, which means VaR estimate is incomplete. Whereas, If LR is less than the critical value, null hypothesis is accepted, which means VaR estimate is complete and efficient.

3.6

Data

The development of China’s futures market in the past decade induces it to be regular gradually, although the kinds of futures are still not much. Among the existing all kinds of futures, copper is an active one. The trend of copper price plays an important role on Chinese economy since copper is a basilica industrial material. Government economy administration, copper exploiting corporations and many copper demanders pay great attention on the regulation of the risk in copper market, especially, the big rise or fall on copper price. Therefore, the futures price of copper is chosen to be the representative of the agent variable of futures prices. Different from stock price, each futures agreement has a due day. Usually, the transactions are more active when the due day is coming, so futures agreements on the latest month are chosen to analyze. When the latest futures agreement is in the delivery month, futures agreements in the next latest month are selected to complete a continuous sequence of futures agreements (the delivery month is not chosen to avoid the irregularity of the price volatility in the delivery month). For example, every year there are twelve copper futures

VaR estimation

45

6 4 2 0 –2 –4 –6 –8 250

500

750

1000

1250

CUF

Figure 3.3 Return rate of futures copper

agreements from January to December, therefore, those futures agreements whose delivery months are at February, 2005 is chosen at January, 2005, similarly, futures agreements whose delivery months are at March, 2005 is chosen at February, 2005, the rest may be deduced by analogy. Covering the period from July 10, 2000 to June 30, 2006, the data set used in this analysis includes 1,442 continuous futures prices after the removal of non-trading days. Futures prices are attained from the closing price per month of SHFE (Shanghai Futures Exchange) in the Reuter System. The data is logarithmized. Yield is defined as the first difference of the logarithm of price: Rt  ln( Pt ) ln( Pt 1 ).

(3.16)

Yields are illustrated as time-series plots in Figure 3.3. The change of yield and volatility are estimated primarily. There are a lot of irregular peaks in the yield sequence, and the obvious volatility clustering suggests that the daily volatility of yield sequence is significant and gusty, and has conditional heteroscedasticity characteristics. Hence, the volatility in the yield sequence is presumed not a white noise process. Some descriptive statistics for daily futures yield of copper Rt are shown in Table 3.1. It is easy to find that volatility is fluctuating around zero and the kurtosis of Rt is significantly greater than 3 and distributes with high peak and thick tail. Based on JB statistics test, it does not follow normal distribution at the 99 per cent confidence level. Then, stationary test is used towards Rt , the results are in Table 3.2. The value of ADF statistic is less than the critical value of 1 per cent, which means the null hypothesis that there exists unit root is rejected under 1 per cent significance level for the yield sequence. The time-series yield sequence is stationary.

46

VaR estimation

Table 3.1 Summary descriptive statistics for the yield of the futures market Sample capacity Copper 1441

Mean

S.D.

Maxi- Median Minimum mum

0.0854 1.2324 4.8747 0.0644

Skewness

Kurtosis JB statistics

–6.0880 –0.3573 5.6597

455.4026

Table 3.2 The stationary test for the yield of the futures market ADF –37.2107 statistic

3.7

1% critical value 5% critical value 10% critical value

–3.9644 –3.4129 –3.1285

Empirical analysis in Chinese futures market

3.7.1

EGARCH model estimation

By the variables filtering, the comparison of parameters and all kinds of statistic tests, AR(2)-EGARCH (1,1) model is chosen and checked with the p-value for the test statistic Q(M). With Maximum Likelihood Estimation, the model is shown as follows: «Yt  0.0575  0.0344Yt 2  Ft ® ( 0.0252 ) ( 0.0254 ) ®F  Y h1/ 2 , Y ~ i.i.d .N ( 0,1), t t t t ® ¬ F F ®ln ht  0.0911  0.1259 t 1  0.0240 t 1  0.9892 ln ht 1 h ht 1 ® t 1 ® ( 0.0110 ) ( 0.0154 ) ( 0.0096 ) ( 0.0030 ). ­

(3.17)

The result of power of EGARCH model is shown in Table 3.3. 3.7.2

TGARCH model estimation

Similarly, AR (2)-TGARCH(1,1) model can be derived. « ®Yt  0.0539  0.0443Yt 2  Ft ( 0.0260 ) ( 0.0261) ® ® ¬Ft  Yt ht1/ 2 , Yt ~ i.i.d .N ( 0,1), ®ht  0.0096  0.0768Ft2 1  0.9345ht 1 0.0293Ft2 1dt 1 ® ®­ ( 0.0132 ). ( 0.0027 ) ( 0.0129 ) ( 0.0083) The result of power of TGARCH model is shown in Table 3.4.

(3.18)

VaR estimation

47

Table 3.3 The diagnostic test statistics of the EGARCH model sufficiency – Generalized Box–Pierce Q statistics Q(5)

Q(10)

Q(20)

Q(30)

Q(40)

Q(50)

3.2393 [0.663]

7.0701 [0.719]

28.904 [0.090]

33.716 [0.292]

43.239 [0.335]

50.985 [0.435]

Q2(5)

Q2(10)

Q2(20)

Q2(30)

Q2(40)

Q2(50)

9.9015 [0.078]

12.351 [0.262]

17.976 [0.589]

31.308 [0.400]

35.721 [0.663]

41.658 [0.793]

Note: The numbers in the parentheses are standard errors for the estimates and the numbers in the square brackets are the p-value for test statistics. Q() is the test statistics for autocorrelation in standardized residual Fˆ t Tˆ t and Q2() is the test statistics for autocorrelation in squared standardized residual Fˆ t2 Tˆ t2 .

Table 3.4 The diagnostic test statistics of the TGARCH model sufficiency – Generalized Box–Pierce Q statistics Q(5)

Q(10)

Q(20)

Q(30)

Q(40)

Q(50)

2.9545 [0.707]

6.7176 [0.752]

29.471 [0.079]

34.639 [0.256]

44.1825 [0.299]

52.015 [0.395]

Q2(5)

Q2(10)

Q2(20)

Q2(30)

Q2(40)

Q2(50)

6.4536 [0.372]

9.1646 [0.517]

14.596 [0.799]

28.097 [0.565]

33.342 [0.763]

39.373 [0.860]

Note: The numbers in the parentheses are standard errors for the estimates and the numbers in the square brackets are the p-value for test statistics. Q() is the test statistics for autocorrelation in standardized residual Fˆ t Tˆ t and Q2() is the test statistics for autocorrelation in squared standardized residual Fˆ t2 Tˆ t2.

From Table 3.3 and 3.4, the p-values of generalized Box–Pierce type statistics for autocorrelation in standardized residuals are well above 0.05, so are the p-values of a similar test for autocorrelation in squared standardized residuals. These indicate that there does not exist autocorrelation in residuals after EGARCH or TGARCH, which means models fitting are good and the models chosen are adequate (the residuals are white noise) for each set of data. The asymmetry factors are significant in both of the two models, which imply that people could be long or short in the futures market as the price changes, but they prefer to be long than short because of their psychology. Speculators in the futures market have more risk and react stronger when the prices rise. Although the short selling mechanism is introduced to the futures market the effects of good news and bad news to the volatility of futures market are still asymmetric. Good news has more influence on volatility than bad news. There still exists leverage effect in the futures market. 3.7.3

Parametric estimation of Downside VaR and Upside VaR

Upside VaR and Downside VaR in the copper futures market are calculated by formula(3.5) and formula (3.6). The results are in Table 3.5 and 3.6.

48

VaR estimation

Table 3.5 VaR estimation in the futures market by EGARCH model EGARCH model

Confidence VaR mean VaR std level

Failure time

Failure rate

LR test statistic

Downside VaR

90% 95% 99% 90% 95% 99%

126 73 28 130 63 21

0.0876 0.0507 0.0195 0.0903 0.0438 0.0146

2.5714 0.0161 10.1880 1.5368 1.2210 2.6863

Upside VaR

1.3858 1.8039 2.5721 1.5065 1.9245 2.6928

0.5918 0.7625 1.0764 0.5943 0.7650 1.0789

Table 3.6 VaR estimation in the futures market by TGARCH model TGARCH model

Confidence level

VaR mean

VaR std

Failure time

Failure rate

LR test statistic

Downside VaR

90% 95% 99% 90% 95% 99%

1.3883 1.8062 2.5743 1.5033 1.9212 2.6893

0.5910 0.7609 1.0737 0.5937 0.7637 1.0765

127 71 30 126 64 20

0.0883 0.0493 0.0208 0.0876 0.0445 0.0139

2.2870 0.0133 13.0315 2.5714 0.9588 1.9701

Upside VaR

Table 3.5 and 3.6 show that the LR statistics of Upside VaR can pass the tests at all the three confidence levels for EGARCH and TGARCH model and the proportion of yield exceeding VaR is consistent with the confidence interval, Downside VaR pass the tests at 90 per cent and 95 per cent confidence levels, but fails at the 99 per cent confidence level the difference of failure rate and corresponding significance level, which indicates the model is insufficient to catch thick tail. VaR estimation in EGARCH and TGARCH are sufficient at the 90 per cent and 95 per cent confidence levels, but not good at 99 per cent confidence level. GARCH model behaves well on the ordinary volatility, but poorly on the extreme risk. Moreover, the two models share on common feature, VaR is overestimated at 90 per cent and 95 per cent confidence level and is underestimated at the 99 per cent confidence level. 3.7.4 Semi-parametric estimation of Downside VaR and Upside VaR Based on formulae (3.8) to (3.11), the Downside VaR and Upside VaR in the copper futures market are estimated. The results are in Table 3.7. No matter Downside VaR and Upside VaR can pass through the LR statistics test at 95 per cent and 99 per cent confidence level, that is to say, the proportion which yield surpasses VaR fits well at the corresponding significance level.While there is significant difference between the failure rate and the tail

VaR estimation

49

Table 3.7 VaR estimation in the futures market by semi-parametric model Semi-parametric Confidence model level Downside VaR Upside VaR

90% 95% 99% 90% 95% 99%

VaR

Failure time

Failure rate

LR test statistics

1.5396 2.1303 3.4980 1.7105 2.3012 3.5834

107 64 19 108 64 7

0.0743 0.0444 0.0132 0.0750 0.0444 0.0049

11.5481 0.9822 1.3424 10.9064 0.9822 4.7503

probability at 90 per cent confidence level. This indicates that semi-parametric estimation behaves well at the extreme tail, but not at the general tail. 3.7.5

Nonparametric estimation of Downside VaR and Upside VaR

Based on formula (3.12), the Downside VaR and Upside VaR in the copper futures market are estimated. The results are in Table 3.8. Table 3.8 suggests that Downside VaR and Upside VaR pass the LR statistics test at all the three confidence levels. The proportion which yield surpasses VaR fits well at the corresponding significance level. The failure rate in this approach is closer to the real significance level than the other two approaches. And the LR statistics is more significant than the other two approaches. This indicates that no matter Downside VaR and Upside VaR, no matter the extreme tail and the general tail, the VaR model based on kernel function is appropriate.The goodness of fit from nonparametric estimation is better than the parametric estimation and the semi-parametric estimation. The reasons are as follows: In the futures trading system, which is a complicated non-linear uncertainty system, there are several uncertain characters such as randomicity, fuzzy, gray and so on. Traditional parametric model is not enough to deal with the system and the model selection is subjective. Parametric estimation supposes the distribution of standardized residual is known and is IID, which does not accord with our situation. In reality, sometimes the data does not come from the population of the supposed distribution, or is polluted due to various reasons. This approach will probably leads to an inaccurate conclusion. Therefore, although being used quite simply, parametric estimation leads to quite good results when the supposed distribution is right, while if the financial sequence does not confirm the supposed distribution, serious error accompanies. Not surprisingly, different distributions result in different estimations. Semi-parametric estimation only uses the volatility, skewness and kurtosis in the sequence and cannot make full use of all sample information, serious information loss conducts imprecise estimates.

50

VaR estimation

Table 3.8 VaR estimation in the futures market by nonparametric model Nonparametric model

Confidence level

VaR

Failure time

Failure rate

LR test statistics

Downside VaR

90% 95% 99% 90% 95% 99%

1.2146 1.9536 3.7583 1.4982 2.1324 3.2313

141 73 14 143 71 13

0.0978 0.0507 0.0097 0.0992 0.0493 0.0090

0.0746 0.0131 0.0119 0.0093 0.0162 0.1440

Upside VaR

Nonparametric estimation is more suitable to analyze the uncertain phenomenon. The kernel estimation can make full use of all the sample information and it weighs more on the close spots and less on the distant spot, which is consistent with finance rule. If the bandwidth and kernel function are appropriate, the goodness of fit could be enhanced. The main reasons are as follows, (1) Nonparametric estimation does not suppose the statistical distribution of the market factor, and deal with asymmetry and thick tail effectively. It can catch the influence of each volatility, no matter it is linear or non-linear. However, the parametric estimation assumes the probability distribution of the return residual in advance, and is unable to fit some non-linear risk factors. (2) Nonparametric estimation eliminates the assumption of yield distribution, and avoids estimating the volatility, correlation and other parameters. Therefore, there does not exist the risk of parametric estimation as well as the parametric estimation error. In addition, it is also far from model risk. (3) Nonparametric estimation is a global model, while parametric estimation is a local model. The nonparametric estimation is an entire value estimation which can deal with thick tail, large volatility and the non-linear problem effectively. It can catch various risks. But parametric estimation will bring serious error when market volatility is large. (4) Nonparametric kernel estimation estimates VaR at the real confidence interval, while parametric estimation estimates VaR at the confidence interval of the supposed distribution. If the supposed distribution does not confirm the real situation, the error will then be obvious. The massive simulation test indicates that statistical result of nonparametric estimation is robust. Distribution assumption, which is a prerequisite in parametric estimation, is not necessary in nonparametric estimation. Nonparametric estimation does not examine the population parameter and it is easy to satisfy since it has fewer restraints. Therefore, besides its

VaR estimation

51

theoretical significance, the importance of nonparametric estimation is also at its application.

3.8

Conclusion

Using the daily data of copper price in SHFE, this paper introduces parametric approach of asymmetric GARCH type model, semi-parametric approach which is based on the first four moments and nonparametric approach ground on kernel function to measure the risk in China’s futures market. The empirical study reveals that: Firstly, there exists short selling mechanism in China’s futures market and the trading mechanism is symmetric, but good news and bad news has asymmetric effect upon the volatility of futures market. The leverage effect is still significant and the market reacts stronger toward good news than bad news. Secondly, VaR is overestimated at 90 per cent and 95 per cent confidence level through GARCH model. That means it considers more risk in the same level. This indicator suits for investor of risk aversion. Thirdly, Upside parametric VaR estimation passes all the three tests at different confidence levels, Downside parametric VaR estimation is acceptable at the 90 per cent and 95 per cent confidence levels, but fails at the 99 per cent confidence level, which indicates GARCH type model is insufficient to catch thick tail. Semi-parametric VaR estimation is not bad at the 95 per cent and 99 per cent confidence levels, but not good at 90 per cent confidence level test. Nonparametric estimation exhibits very well at all the tests at the three confidence levels no matter for Upside VaR or Downside VaR. Compared to the other two approaches, invalidation value is closer to the significant level. Nonparametric measures the unconditional VaR, while GARCH model measures the conditional VaR, so comparing these seems no significance. Considering nonparametric estimation has no economic explanation and GARCH model can only reflect the volatility at that time, we should combine parametric and nonparametric approach to estimate the risk in futures market accurately. The part of parametric is captured by GARCH model first, then the residual is described by non- parametric approach. It will be a direction for further study. Accurate risk estimation helps enterprise enhance risk management. At the same time, the regulation department should introduce advanced risk management model and approach to improve risk-control mechanism. Thus, we can implement the accurate estimation of the risk and quantitative study on risk management will be enhanced in China’s futures market so that the market risk could be reduced and China’s futures market could keep progressing on the right track.

4

Extreme risk spillover between Chinese stock markets and international stock markets

In order to get the benefits of the foreign capital while warding off the adverse impacts, many emerging capital markets have explicit barriers that hinder international portfolio investment. However, not much is known about whether these specific barriers play a significant role in eliminating or alleviating adverse international shocks on domestic stock capital markets associated with opening the markets. Our paper analyzes the effect of these regulations in China. The Chinese stock market is particularly well-suited for study because it maintains different kinds of listings for common stocks on both the underdeveloped mainland market and the mature market in Hong Kong and discriminates between foreign and Chinese investors. This, in turn, leads directly to the most striking feature of the Chinese stock market: its segmentation causes an internationally unique variety of Chinese share categories. The most important categories are ‘A’ and ‘B shares’, which are traded on the mainland’s two places Shanghai and Shenzhen, as well as ‘H shares’ which are traded in Hong Kong.1 A shares are dealt in Chinese Renminbi and are issued strictly to the Chinese citizens. B-shares, also known as Special Shares, are designated for foreign investors and are traded in US dollar (Shanghai) and Hong Kong dollar (Shenzhen). H shares are the Hong Kong equivalent to mainland’s B shares. This kind of legal investment restrictions set up by China is different from other emerging markets. Many emerging markets began open to foreign investment by creating a closed-end country fund listed on the New York Stock Exchange (NYSE) or the London Stock Exchange (LSE). For example, until the 1980s the closed-end Mexico Fund was the only way US investors could invest in the Mexico market. The Korea Fund partially opened the Korean equity markets to foreign investors in 1984. The design of separate equity market has some similarities to that of Thailand and Philippine. However, there are notable distinctions at the same time.2 As foreign direct and portfolio investment poured into Thailand and foreign ownership limits of many Thai firms became binding in the mid-1980s, the Stock Exchange of Thailand (SET) inaugurated its Alien Board in September 1987. For companies which have reached their foreign ownership limit, Thais continue to trade shares on the Main Board while foreigners submit orders to the Alien Board. Main and Alien Board shares are identical in all other respects,

Extreme risk spillover between markets

53

such as dividends, voting rights, and procedures for settlement and registration. Philippine companies could also issue A shares and B shares until 1994. But the segmentation between two classes of stock are only partial, where A shares may be traded only among Philippine nationals, and B shares may be traded to either Philippine nationals or foreign investors. Foreign investors are allowed to own only the foreign class of shares, but domestic investors can own both local and foreign shares. But the A share and B share markets in China were completely segmented in the early stage because the Chinese capital account is not liberalized and RMB cannot convert freely. Obviously, different shares of the Chinese common stocks have different degrees of link with international capital markets. Most international shocks are expected to be mainly absorbed in share H and B markets and their impact on share A markets will become diminished. However, with increasing integration between different categories’ markets, and between the Chinese stock market and international capital markets, mutual interaction between the Chinese stock market and overseas capital markets is expected to become stronger as time evolves. The possibility of financial contagion, such as one similar to the 1997–1998 Asian financial crisis, becomes larger. In this chapter, we will investigate the influence of international capital markets on the Chinese stock market. In particular, we will examine whether and how extreme downside market risk is transmitted between the Chinese stock market and overseas capital markets. Our detailed evidence on the effect of capital flow barriers and related market frictions has a number of policy implications concerning the design and operation of emerging capital markets and the evaluation of direct and portfolio investments in developing countries. In this chapter, applying the proposed test in the previous chapter, we investigate extreme risk spillover between different shares in the Chinese stock markets, and between the Chinese stock market and overseas stock markets, the latter including those of Hong Kong, Taiwan, Singapore, South Korea, Japan, US and Germany. Based on the daily stock indices from 1/2/1995 to 4/4/2003, we find some interesting empirical results. Below we first provide an overview of the Chinese stock market in Section 4.1 and describe the data and reports summary statistics in Section 4.2. Section 4.3 presents estimation results and empirical findings, and discusses their implications. Section 4.4 concludes.

4.1

The Chinese stock market

China’s two official national exchanges: the SHSE established on December 19, 1990, and the SZSE founded on July 3, 1991. Although started late, China’s domestic markets have developed rapidly and have experienced a rapid expansion in terms of the number of listed stocks, number of investors, trading volume and market capitalization. From 1991 to the end of September 2002, total funds raised by Chinese companies through public offering reached RMB 854 billion. From only about a dozen listed companies in 1990, there

54

Extreme risk spillover between markets

are over 1,200 companies, including 1,199 A shares and 111 B shares, listed in Shanghai and Shenzhen stock markets by the end of year 2002. The total market capitalization at the end of September 2002 was RMB 4.4 trillion, accounting about half of GDP. It ranks as the second or third largest in nonJapan Asia, and the largest emerging capital market in the world in terms of market capitalization (e.g. Chen and Hong 2003).3 At the same time, the total investor accounts reach almost 68 million. However, Chinese citizens and foreign investors do not enjoy identical investment opportunity sets. The companies listed on the two securities exchanges were originally authorized to issue A shares to domestic Chinese citizens only. From 1992, the Chinese government allowed some firms (most of which have already issued A shares) to issue B shares, which can be purchased and traded by foreign investors in securities exchange inside China, either the SHSE or SZSE. This arrangement can attract foreign funds without worrying about the loss of ownership control and adverse impact of outside shocks. The first B share was issued in the SHSE in 1992. In general, only firms that have established an excellent track record are permitted to raise funds from overseas. A and B shares of the same company are entitled to identical voting rights and cash flow rights, although A share is usually traded at a much higher price than that of B share. Some Chinese companies are also allowed to go beyond traditional domestic equity-financing channels to raise capital by listing in overseas capital markets such as Hong Kong, New York and London stock exchanges. These are the so-called Share H, Share N and L.4 As a gateway for oversea capital to China, Hong Kong was naturally identified as the primary offshore equity market for Chinese firms to raise capital. With the successful listing of six Chinese companies in 1993, the Hong Kong market become a legitimate source of equity capital for mainland China’s enterprises and helped raise the global visibility of Chinese domestic enterprises. Companies listed in Hong Kong are mainly state-owned enterprises, which had been restructured with a view to an international listing (including Hong Kong) and which usually have a strong market position in China. Some of them are the number one of the industries they belong to and have important influence to the industry. The fast developed Chinese economy is the support of the H shares company. Many H-share issuing companies have subsequently listed A shares on either the SHSE or the SZSE. H shares provide Hong Kong and international investors with opportunities to invest in Chinese stocks without having to be concerned about various investment barriers and excessive costs for investing directly in the A share market. Up to the end of 2002, there have been fifty-four H shares listed on the main board of HKSE, with some of them cross-listed on NYSE as Share N. Recently, some of these barriers have been lowered. Only starting from February 2001, China Securities Regulatory Commission (CSRC), the official institution set up in 1992 to be responsible for regulating and monitoring the Chinese stock market, allows domestic investors holding foreign currencies to invest in Share B markets and starting from December 2002, qualified foreign

Extreme risk spillover between markets

55

institutional (QFII) investors – mutual fund management institutions, insurance companies, securities companies, commercial and investment banks are allowed to invest in Share A markets. But still, QFIIs can only purchase 10 per cent of a company’s stocks and have to keep hold of the purchased shares for one year. Moreover the maximum amount of a company’s shares sold to QFIIs should not exceed 20 per cent altogether. Because of its underdeveloped financial market, China’s stock market has been featured with over speculative activity of massive individual investors, dramatic policy changes and government intervention, a huge amount of speculative capital and exhibiting excess market fluctuations. Unlike the mature stock markets in developed economies, the majority of investors in Chinese stock market are individuals, accounting for more than 99 per cent in 2002 in terms of the number of opened accounts. Institutional investors only accounts for about 0.5 per cent. The economic knowledge of Chinese investors and their access to information is limited; neither do they have much experience. This makes them being inclined to ‘follow the herd’ and sell when stock prices fall or purchase when they rise. In addition, mainland companies have poor corporate governance and lack sufficient information disclosure. As the majority of the shares belongs to the state and cannot be traded freely, a company’s poor management does not necessarily lead to falling share prices nor does it increase the company’s risk of being taken over.

4.2

Data

For convenience, in this chapter, we will call Q1 (M ) and Q2 (M ) the test for one-way and two-way Granger causality in risk respectively. Q2 (M ) here we considered is only complete non-Granger causality in risk, since one-way risk spillover will be examined in retail. We now use the Q1 (M ) and Q2 (M ) tests to investigate risk spillover between the Chinese stock market and overseas capital markets. We will consider twelve stock price indices in the Chinese stock market and those of Hong Kong, Taiwan, Singapore, South Korea, Japan, US and Germany. These indices are Shanghai A Share Index (SHA), Shanghai B Share Index (SHB), Shenzhen A Share Subindex (SZA), Shenzhen B Share Subindex (SZB), Hong Kong Hang Seng China Enterprises Index (HKH, or Share H), Hong Kong Hang Seng Index (HSI), Taiwan Weighted Index (TWI), Singapore Straits Times Index (STI), South Korea Composite Index (KOSPI), Japan Nikkei 225 Index (NK225), US S&P 500 Composite Index (S&P500) and German DAX 30 Index (DAX). To avoid the abnormal price fluctuations of the Chinese stock market a few months after 8/1/1994, during which stock price movements were mainly driven by dramatic policy changes, we select our sample from 1/2/1995 to 4/4/2003. The data are daily closing prices, with SHA, SHB, SZA and SZB obtained from SHSE and all the other from Datastream. Figure 4.1 and Figure 4.2 plot the daily series of the twelve stock price indices and their changes defined in terms of log-difference scaled by 100.

56

Extreme risk spillover between markets

2500

7000 6000

2000

5000 1500

4000

1000

3000 2000

500 0 12/19/90

1000 10/19/94

8/19/98

6/19/02

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Shanghai Composite Index

10/19/94

8/19/98

6/19/02

Shenzhen Comp. Subindex

250

3000

200

2500 2000

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1500 100

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50 0 12/19/90

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20000

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16000

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12000

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8000

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4000

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10/19/94

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Hang Seng China Enterp Index

10/19/94

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6/19/02

Shenzhen B Share Subindex

Shanghai B-Share index

6/19/02

0 12/19/90

10/19/94

8/19/98

6/19/02

Hang Seng Index

Figure 4.1 Stock price indices

Figure 4.1 shows that SHA and SZA follow a similar pattern, so do SHB and SZB. However, Share A indices and Share B indices have different patterns of price dynamics. Also, Share A indices and Share B indices exhibit different patterns from Share H and HSI. On the other hand, TWI, STI and KOSPI show similar price movements, S&P500 and DAX have similar trends. Except for Chinese A Share indices, all the other indices of Asian stock market experienced a steep fall in the middle of 1997. Figure 4.2 shows that there exists rather strong volatility clustering in all stock indices. For the Chinese stock market, SHA and SZA have a similar volatility clustering pattern, so do SHB and SZB. However, Share A indices and Share B indices have different volatility clustering patterns. For Share A indices, there were more variations in the early stage of the sample than in the later stage. This is perhaps due to the implementation of a 10 per cent band limit on daily stock price changes with a ‘T + 1’ settlement rule after 12/16/1996. In contrast, for Share B indices, there were more variations in the later stage of the sample than in the early stage. This indicates the booming of Share B markets, which might be due to

Extreme risk spillover between markets 12000

3000

10000

2500

8000

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6000

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4000

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2000 12/19/90

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Taiwan Weighted Index 30000

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25000

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400

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South Korea Composite Index 1600 1400 1200 1000 800 600 400 200 12/19/90

8/19/98

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1200

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10/19/94

57

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Japan Nikkei 225 Index 10000 8000 6000 4000 2000

10/19/94

8/19/98

Standard & Poor 500 Index

6/19/02

0 12/19/90

10/19/94

8/19/98

6/19/02

German DAX Index

Figure 4.1 (cont.)

the introduction of a legal regulation in 01/1996 to encourage foreign investment in Share B markets and to open Share B markets to domestic investors in 02/2001. Table 4.1 reports some summary statistics – sample mean, variance, skewness and kurtosis of each daily stock price change series. SHA, SZA, SHB, SZB, S&P500 and DAX all have positive average returns. Except HSI, all other Asian stock indices have negative average returns. It is clear that the Chinese stock market (except Share H) offers higher average return than other stock markets, with relatively higher standard deviation at the same time. This to some extent indicates that the Chinese stock market, as an emerging capital market, is more volatile than developed stock markets in the world. HKH has the largest standard deviation among all indices, but the average return is negative. We also note that the long-term performance of Share B markets lags far behind Share A markets, however, Share B have higher standard deviation. Compared with Shenzhen market, Shanghai market has a smaller

58

Extreme risk spillover between markets

80

30

60

20 10

40

0 20

–10

0 –20 12/19/90

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Shanghai Composite Index 30

60

20

40

10

20

0

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–10 10/19/94

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6/19/02

8/19/98

6/19/02

Shenzhen Comp. Subindex

80

–20 12/19/90

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Shanghai A Share Index

10/19/94

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15

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10

10

5

0

0 –10

–5

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–10

–10

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Hang Seng Chinaenterp Index

10/19/94

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Figure 4.2 Return rate of stock price indices

standard deviation. On the other hand, the Chinese stock indices all have positive skewness, with SHA having the largest skewness. Most of the other stock indices have negative skewness. All stock indices have a kurtosis larger than 3, indicating non-Gaussian features of stock price changes. SHA and SZA have a largest kurtosis among all indices.

10

20

5

15 10

0

5 –5

0

–10 –15 12/19/90

–5 10/19/94

8/19/98

6/19/02

–10 12/19/90

10/19/94

8/19/98

6/19/02

Singapore Straits Time Index

Taiwan Weighted Index 15

8

10 4

5 0

0

–5

–4

–10 –15 12/19/90

10/19/94

8/19/98

–8 12/19/90

6/19/02

Korea Composite Index 6 4 2 0 –2 –4 –6 –8 12/19/90

10/19/94

8/19/98

6/19/02

Japan Nikkei 225 Index 10 5 0 –5 –10

10/19/94

8/19/98

6/19/02

–15 12/19/90

Standard & Poor 500 Index

10/19/94

8/19/98

6/19/02

German DAX Index

Figure 4.2 (cont.)

Table 4.1 Summary descriptive statistics for daily stock price changes Indices

Sample size

Shanghai A Share Index Shenzhen A Share Subindex Shanghai B Share Index Shenzhen B Share Subindex Hang Seng China Enterp Index Hang Seng Index Taiwan Weighted Index

2,154

Mean

S.D.

Skewness

Kurtosis

0.040

1.953

0.979

29.909

2,154

0.043

2.030

0.305

18.380

2,154

0.032

2.324

0.412

7.892

2,154

0.025

2.503

0.212

9.200

2,154

–0.034

2.581

0.316

8.932

2,154 2,154

0.003 –0.021

1.806 1.693

0.151 –0.012

13.013 5.111

60

Extreme risk spillover between markets

Table 4.1 (cont.) Indices

Sample size

Mean

S.D.

Skewness

Kurtosis

Singapore Straits Times Index Korea Composite Index Japan Nikkei 225 Index United States S&P 500 Index Germany DAX 30 Index

2,154

–0.016

1.479

0.386

13.110

2,154

–0.028

2.245

–0.039

5.987

2,154

–0.041

1.484

0.083

5.082

2,154

0.030

1.190

–0.106

6.050

2,154

0.011

1.647

–0.237

5.796

Note: The starting date is Jan. 2nd, 1995, and the ending date is Apr. 4th, 2003. The starting date and the ending date are for index price, thus the log-price differences have one less observations. All the data of daily closing stock price indices are obtained from Datastream, except that of Shanghai A Share, B Share, Shenzhen A Share and B Share which are obtained from Shanghai Stock Exchange.

4.3

Evidence on Granger causality in risk

4.3.1

Model estimation

Let Pt be the price of a stock index at time t. We define the daily stock price change at time t as follows: Yt  100 ln( Pt / Pt 1 ).

(4.1)

To account for persistent volatility clustering and possible weak serial dependence in Yt , we specify an AR (m) model with Threshold GARCH (TGARCH) errors: m « Y b b jYt j  Ft ,   t 0 ¤ ® j 1 ®® ¬Ft  Yt ht1/ 2 , Yt ~ i.i.d.N( 0,1), n ® ®ht  X  ¤ B k ht k  C1Ft2 11( Ft -1  0 ) + C2 Ft2 11( Ft -1 b 0 ). ®­ k 1

(4.2)

In the empirical finance literature, it is often found that GARCH (1,1) and GARCH (2,1) models can capture most volatility clustering of financial time series (e.g. Engle (1986, 1993)). We use TGARCH models to capture the ‘leverage effects’. For all stock indices, we first consider an AR (3)-TGARCH (1, 1) model. If diagnostic checking shows that the model is not adequate, we

Extreme risk spillover between markets

61

Table 4.2 Quasi-maximum likelihood estimation of univariate GARCH models for daily stock price changes SHA Estimate 0.001 (0.031) b1 0.013 (0.024) –0.014 b2 (0.024) 0.019 b3 (0.024) ω 0.014 (0.005) 0.940 α1 (0.005) 0.040 β1 (0.007) 0.093 β2 (0.011) Sample size 2,154 Mean log– –1.942 likelihood Diagnostic test statistics –0.211 Q1(5) [0.583] –0.089 Q1(10) [0.536] –0.162 Q1(20) [0.564] –0.064 Q1*(5) [0.525] –0.156 Q1*(10) [0.562] –1.073 Q1*(20) [0.858]

Parameter b0

SZB

HSI

STI

Estimate –0.037 (0.041) 0.087 (0.028) –0.007 (0.026) 0.012 (0.024) 0.771 (0.080) 0.596 (0.030) 0.284 (0.037) 0.326 (0.043) 2,154 –2.166

Estimate Estimate Estimate Estimate Estimate –0.002 –0.033 –0.046 –0.045 0.039 (0.029) (0.024) (0.037) (0.029) (0.026) 0.069 0.121 0.088 –0.038 0.000 (0.023) (0.024) (0.022) (0.023) (0.023) 0.018 0.010 0.004 0.002 0.002 (0.023) (0.023) (0.022) (0.022) (0.023) 0.058 0.028 –0.014 –0.011 –0.001 (0.023) (0.023) (0.022) (0.022) (0.023) 0.041 0.038 0.011 0.051 0.030 (0.009) (0.009) (0.006) (0.014) (0.008) 0.916 0.879 0.956 0.920 0.892 (0.010) (0.013) (0.007) (0.013) (0.013) 0.013 0.059 0.024 0.017 0.054 (0.009) (0.012) (0.008) (0.008) (0.013) 0.133 0.161 0.064 0.100 0.141 (0.018) (0.020) (0.010) (0.016) (0.019) 2,154 2,154 2,154 2,154 2,154 –1.839 –1.646 –2.099 –1.763 –1.732

0.422 [0.337] 0.518 [0.302] 0.257 [0.399] –0.956 [0.831] –1.330 [0.908] –1.786 [0.963]

–1.130 [0.871] –1.332 [0.909] –1.558 [0.940] 0.293 [0.385] –0.362 [0.642] –1.278 [0.899]

–0.086 [0.534] –0.730 [0.767] –1.168 [0.879] –0.546 [0.707] –0.644 [0.740] –1.096 [0.864]

KOSPI

–0.810 [0.791] –0.878 [0.810] –0.464 [0.679] –0.509 [0.695] –0.891 [0.814] –0.861 [0.805]

N225

DAX

–1.142 [0.873] –1.262 [0.897] –1.183 [0.882] –0.169 [0.570] –0.354 [0.638] –0.431 [0.667]

–1.031 [0.849] –0.787 [0.784] –0.402 [0.656] 0.731 [0.232] 0.187 [0.426] 0.000 [0.500]

Note: The numbers in the parentheses are standard errors for the estimates and the numbers in the square brackets are the p−value for Q1 test statistics; Q1(M) and Q1*(M) are Q1 statistics for the first M autocorrelations of the standardized residual and squared standardized residuals m

n

respectively. The specified model is: Yt  b0  ¤ b jYt j  Ft , Ft  Yt ht1/ 2 , ht  X  ¤ B k ht k  C1F 2 1 t 1 j 1 k 1  Ft 1  0  C2 Ft2 11 Ft 1 b 0 .

then try models with a higher order AR and a higher order TGARCH. This is the so-called ‘from specific-to-general’ modeling approach. Table 4.2 reports the maximum likelihood estimation of univariate AR-TGARCH models for all indices. For SHA, SZB, HSI, STI, KOSPI, NK225 and DAX, diagnostic checking (reported in Table 4.2) indicates that an AR (3)-TGARCH (1, 1) model is adequate. For SZA, an AR (4)-TGARCH(2, 1) model is adequate.

Table 4.2 (cont.) SZA

HKH

TWI

SHB

Estimate –0.021 (0.033) 0.039 (0.025) –0.007 (0.023) 0.000 (0.024) 0.025 (0.024) 0.035 (0.014) 0.463 (0.127) 0.437 (0.119) 0.073 (0.010) 0.135 (0.020)

Estimate –0.038 (0.039) 0.191 (0.024) –0.026 (0.023) –0.001 (0.024)

Estimate –0.029 (0.033) 0.032 (0.023) 0.068 (0.023) 0.018 (0.022) –0.047 (0.022) 0.120 (0.034) 0.883 (0.022)

Estimate –0.086 (0.045) 0.142 (0.026) 0.011 (0.025) 0.012 (0.024)

Parameter Estimate b0 0.077 (0.020) b1 0.022 (0.023) b2 0.006 (0.023) b3 –0.048 (0.023)

0.139 (0.028) 0.733 (0.032)

ω

0.019 (0.009) 0.136 (0.025)

β

Sample size 2154 Mean log– –1.982 likelihood Diagnostic test statistics Q1(5) 0.113 [0.455] Q1(10) 0.732 [0.232] Q1(20) 0.475 [0.317] Q1*(5) 0.507 [0.306] Q1*(10) 0.942 [0.173] Q1*(20) –0.070 [0.528]

2154 –2.165

2154 –1.885

0.190 (0.028) 0.192 (0.029) 0.107 (0.068) 0.379 (0.082) 2154 –2.056

–0.309 [0.621] –0.869 [0.807] –1.230 [0.891] 0.574 [0.283] 0.054 [0.478] 0.010 [0.496]

–1.103 [0.865] –0.480 [0.684] 0.118 [0.453] –0.427 [0.665] –0.817 [0.793] –0.560 [0.712]

0.667 [0.252] 0.570 [0.284] 0.776 [0.219] –0.995 [0.840] –1.128 [0.870] –1.496 [0.933]

Parameter b0 b1 b2 b3 b4 ω α1 α2 β1 β2

0.115 (0.032) 0.349 (0.119) 0.468 (0.111) 0.128 (0.023) 0.228 (0.034)

d1 d2

S&P500

α1

0.009 (0.004) 0.918 (0.011) 0.082 (0.012)

2154 –1.481

–1.109 [0.866] –0.957 [0.831] –0.528 [0.701] 0.020 [0.492] –0.431 [0.667] –1.172 [0.879]

3

Note: For SHB, the specified model is Yt  b0  ¤ b jYt j  d11t  500  Ft , Ft  Yt ht1/ 2 , j 1

ht  X  B1ht 1  d 21t  500  C1Ft2 11Ft -1  0  C2 Ft2 11Ft -1 b 0 . 3

For S&P500, the specified model is: Yt  b0  ¤ b jYt j  Ft , Ft  Yt ht1/ 2 , ht  X  B1ht 1  CF t2 1 . j 1

Extreme risk spillover between markets

63

For HKH, an AR (3)-TGARCH (2, 1) model is adequate, and for TWI, an AR(4)-TGARCH(1,1) model is adequate. Among all indices, SHB and S&P500 are most specific. Figure 4.2 reveals the SHB volatility becomes larger from late 1996. To capture this potential structural change, we introduce two time dummy variables for the intercepts in mean and in variance respectively. Hence, the model for SHB is specified as the follows: m « Y b d t   1 (  500 )  0 1 ¤ bjYt j  Ft , ® t j 1 ®® ¬Ft  Yt ht1/ 2 , Yt ~ i.i.d.N( 0,1), n ® ®ht  X  d2 1(t  500 )  ¤ B k ht k  C1Ft2 11( Ft -1  0 )  C2 Ft2 11( Ft -1 b 0 ). ®­ k 1

(4.3) For S&P500, the estimate of C 1 in model (4.2) is almost zero, so we use the following standard AR (3)-GARCH(1, 1) model: 3 « ®®Yt  b0  ¤ b jYt j  Ft , j 1 ¬ 1/ 2 ®Ft  Yt ht , Yt ~ i.i.d.N( 0,1), ®­ht  X  B1ht 1  C1Ft2 1 .

(4.4)

Table 4.2 shows that not all parameter estimates are significant at the 5 per cent significance level, but TGARCH parameters (Bˆ 1,Bˆ 2,Cˆ 1,Cˆ 2 ) are highly significant, with Cˆ 2 Cˆ , which confirms the existence of ‘leverage effects’. 1 For SHB, the variance dummy coefficient estimate dˆ 2 is significant at the 1 per cent significance level. This is consistent with the SHB volatility pattern displayed in Figure 4.2. Table 4.2 also reports some diagnostic statistics for model adequacy. The p-values of a generalized Box and Pierce’s (1970) type portmanteau statistics for autocorrelation in standardized residuals [Fˆt / Tˆ t ] are all well above 0.10, so are the p-values of a similar test for autocorrelation in squared standardized residuals [Fˆ t2 / Tˆ t2]. These results indicate the adequacy of the specified models for each index. 4.3.2

Risk spillover among the Chinese stock markets

We first investigate risk spillover between SHA, SHB, SZA, SZB and HKH (Share H), which constitute a comprehensive picture of the Chinese stock market.

64

Extreme risk spillover between markets

Table 4.3 reports test statistics for Granger causality in risk at B = 10 per cent and B = 5 per cent levels, together with their p-values. Because commonly used non-uniform kernels deliver similar results, we only report the results based on the Daniell kernel. For the 10 per cent and 5 per cent risk levels, we find extremely significant evidence on two-way Granger causality in risk between SHA and SZA, and between SHB and SZB respectively, suggesting strong risk spillover between SHSE and SZSE. In order to identify the direction of risk spillover, we also report test statistics for one-way Granger causality in risk between SHA and SZA, and between SHB and SZB. At the 10 per cent risk level, we find that SHA Granger-causes SZA in risk with respect to I t 1 and vice versa. At the 5 per cent risk level, we find significant Granger causality in risk from SZA to SHA with respect to I t 1, but there exists only marginally significant evidence on Granger-causality in risk from SHA to SZA with respect to I t 1. On the other hand, there exists significant evidence that SHB Granger-causes SZB in risk and vice versa at the two risk levels. Next, we consider risk spillover between: (i) SHA and SHB, (ii) SHA and SZB, (iii) SZA and SHB, (iv) SZA and SZB respectively. We find extremely significant two-way Granger causality in risk between Share A and Share B. This finding is hardly surprising, because Share A and Share B are the prices of the same company in two different markets; they have some common driving forces for price dynamics despite market segmentation. Interestingly, oneway Granger causality test statistics show that Share B Granger-causes Share A in risk at the 10 per cent risk level, but not vice versa. This might be because Share B prices are more sensitive to the fundamentals of profitability of the listed companies and are not likely to be influenced by Share A prices when price changes are large. Also, among the local investors, most are liquidity traders and they are less likely to be equipped with sophisticated investment skills. Hence, the A-share market may contain a relatively low probability of information-based trading. In the B-share market, shares are traded by foreigners, most of them being institutional investors. Since foreign investors are more likely to be equipped with sophisticated portfolio management skills and have access to different sources of information, the B-share market may contain a relatively high probability of information-based trading. It challenges a widespread assumption in the finance literature that foreign investors are less informed than domestic investors (Stulz and Wasserfallen (1995); Kang and Stulz (1996); Brennan and Cao (1997)), but in line with more recent research (Froot, O’Connell and Seasholes (2001), Pan, Chan and Wright (2001)) that foreign investors portfolio inflows have noticeable ability to predict positive future returns in emerging markets. In addition, according to Easley et al. (1996), the lower the frequency of trading, the higher the probability of information-based trading. Since B shares are much less frequently traded than A shares (for example, the trading volume and turnover rate of B shares are significantly lower than those of A shares), the B-share market is thus likely to contain a higher proportion of informed

340.628 (0.000) 4.988 (0.000) 5.740 (0.000) 44.750 (0.000) 4.043 (0.000) 0.424 (0.336) 41.690 (0.000) 4.391 (0.000) –0.396 (0.654) 1.428 (0.077) 2.052 (0.020) –0.493 (0.689) 40.672 (0.000)

478.047 (0.000) 6.289 (0.000) 4.618 (0.000) 60.992 (0.000) 2.709 (0.003) 0.426 (0.335) 57.516 (0.000) 4.034 (0.000) –0.201 (0.579) 0.288 (0.387) 0.158 (0.437) –0.654 (0.744) 53.013 (0.000)

SHA඼SZA

SZA඼SHB

SHAයHKH

SHAමHKH

SHA඼HKH

SHAයSZB

SHAමSZB

SHA඼SZB

SHAයSHB

SHAමSHB

SHA඼SHB

SHAයSZA

SHAමSZA

10

5

M

241.758 (0.000) 3.499 (0.000) 4.750 (0.000) 32.161 (0.000) 3.424 (0.000) 0.323 (0.373) 29.779 (0.000) 3.686 (0.000) –0.578 (0.718) 2.177 (0.015) 3.206 (0.001) –0.433 (0.668) 29.774 (0.000)

20

α = 0.10

197.577 (0.000) 2.680 (0.004) 3.956 (0.000) 26.339 (0.000) 2.895 (0.002) 0.205 (0.419) 24.599 (0.000) 3.158 (0.000) –0.272 (0.607) 2.921 (0.002) 3.410 (0.000) 0.484 (0.314) 24.320 (0.000)

30

Table 4.3 Risk spillover among Chinese stock markets

171.244 (0.000) 2.196 (0.014) 3.477 (0.000) 22.623 (0.000) 2.223 (0.013) 0.144 (0.443) 21.380 (0.000) 2.838 (0.002) –0.272 (0.607) 3.660 (0.000) 3.754 (0.000) 1.225 (0.110) 20.782 (0.000)

40 434.681 (0.000) 0.720 (0.236) 1.381 (0.084) 33.175 (0.000) 0.307 (0.380) 0.345 (0.365) 28.016 (0.000) 1.004 (0.158) 0.077 (0.469) –0.206 (0.581) –0.853 (0.803) 0.123 (0.451) 41.000 (0.000)

5 308.696 (0.000) 1.707 (0.044) 1.445 (0.074) 25.086 (0.000) 2.246 (0.012) 0.652 (0.257) 19.699 (0.000) 0.786 (0.216) –0.278 (0.609) –0.671 (0.749) –0.920 (0.821) –0.355 (0.639) 30.459 (0.000)

10 218.966 (0.000) 1.670 (0.047) 1.160 (0.123) 18.319 (0.000) 1.901 (0.029) 0.893 (0.186) 13.970 (0.000) 0.722 (0.235) –0.338 (0.632) 0.292 (0.385) 0.270 (0.394) –0.007 (0.503) 2.089 (0.000)

20

α = 0.05

178.839 (0.000) 1.420 (0.078) 0.686 (0.246) 15.420 (0.000) 2.071 (0.019) 0.833 (0.202) 11.373 (0.000) 0.489 (0.313) –0.244 (0.596) 1.317 (0.157) 1.060 (0.145) 0.268 (0.394) 18.751 (0.000)

30

154.778 (0.000) 1.163 (0.122) 0.286 (0.387) 13.665 (0.000) 2.018 (0.022) 0.911 (0.181) 9.682 (0.000) 0.355 (0.361) –0.399 (0.655) 1.754 (0.060) 1.736 (0.041) 0.395 (0.346) 16.497 (0.000)

40

5

3.661 (0.000) 0.078 (0.469)

7.128 (0.000) 0.133 (0.447)

10 6.571 (0.000) –0.195 (0.577)

20

α = 0.10

5.512 (0.000) –0.391 (0.652)

30 4.500 (0.000) –0.526 (0.701)

40

5 2.399 (0.008) 0.163 (0.435)

3.054 (0.001) 0.820 (0.206)

10 2.261 (0.012) 1.146 (0.126)

20

α = 0.05

2.037 (0.021) 1.724 (0.0424)

30

1.615 (0.053) 1.974 (0.024)

40

SZAමHKH

SZA඼HKH

SZAයSZB

SZAමSZB

SZA඼SZB

M

5

Table 4.3 (cont.)

48.821 (0.000) 2.697 (0.003) 0.148 (0.441) 1.398 (0.081) 2.611 (0.005)

35.723 (0.000) 3.769 (0.000) –0.125 (0.550) 1.870 (0.031) 3.075 (0.001)

10 25.846 (0.000) 3.586 (0.000) –0.289 (0.614) 3.075 (0.001) 3.756 (0.000)

20

α = 0.10

21.211 (0.000) 3.060 (0.001) –0.275 (0.608) 3.680 (0.000) 3.781 (0.000)

30 18.412 (0.000) 2.737 (0.003) –0.303 (0.619) 3.809 (0.000) 3.628 (0.000)

40 41.575 (0.000) 1.107 (0.134) –0.324 (0.627) –0.657 (0.744) –0.624 (0.734)

5

29.016 (0.000) 0.571 (0.284) –0.676 (0.751) –0.664 (0.747) –0.695 (0.757)

10

20.055 (0.000) 0.450 (0.326) –1.232 (0.891) 0.253 (0.400) 0.112 (0.455)

20

α = 0.05

16.140 (0.000) 0.429 (0.334) –1.424 (0.923) 1.137 (0.128) 0.701 (0.242)

30

13.762 (0.000) 0.442 (0.329) –1.630 (0.948) 1.311 (0.095) 0.847 (0.198)

40

Note: ‘඼’ represents the two-way tests for causality in risk between the two stock price changes with respect to I t 1 ; ‘ම’ and ‘ය’ represent one-way causality in risk from the latter to the former and the former to the latter with respect to I t 1 respectively. The numbers in the parentheses are the p −values.

SZAයSHB

SZAමSHB

M

Table 4.3 (cont.)

0.214 (0.415) 141.191 (0.000) 3.783 (0.000) 0.976 (0.165) 21.590 (0.000) 5.747 (0.000) 7.137 (0.000) 10.339 (0.000) 1.261 (0.104) 1.019 (0.154)

0.091 (0.464) 102.542 (0.000) 5.179 (0.000) 1.950 (0.026) 18.100 (0.000) 5.539 (0.000) 7.068 (0.000) 8.608 (0.000) 2.282 (0.011) 1.167 (0.122)

0.967 (0.167) 73.121 (0.000) 4.258 (0.000) 1.375 (0.085) 13.499 (0.000) 3.646 (0.000) 5.920 (0.000) 6.837 (0.000) 2.513 (0.006) 0.921 (0.178)

1.722 (0.043) 59.761 (0.000) 3.571 (0.000) 0.970 (0.166) 10.950 (0.000) 2.371 (0.009) 5.223 (0.000) 6.556 (0.000) 3.035 (0.001) 1.139 (0.127)

Notes: a: the p −value is 0.053 for M = 50; b: the p −values are 0.0569 and 0.053 for M = 45, 50 respectively.

SZBමHKH

SZBයHKH

SZB඼HKH

SHBමHKH

SHBයHKH

SHB඼HKH

SHBමSZB

SHBයSZB

SHB඼SZB

SZAයHKH

2.003 (0.023) 51.987 (0.000) 3.370 (0.000) 0.804 (0.211) 9.242 (0.000) 1.486 (0.069) 4.693 (0.000) 6.427 (0.000) 3.323 (0.000) 1.344 (0.089) a

–0.290 (0.614) 88.745 (0.000) 1.474 (0.070) 1.754 (0.040) 11.654 (0.000) 3.116 (0.001) –0.644 (0.740) 8.729 (0.000) 0.214 (0.415) 0.768 (0.221)

–0.198 (0.578) 64.375 (0.000) 3.152 (0.001) 1.362 (0.087) 9.182 (0.000) 3.341 (0.000) –0.316 (0.624) 6.780 (0.000) 0.712 (0.238) 0.845 (0.199)

0.345 (0.365) 46.008 (0.000) 2.533 (0.006) 1.182 (0.119) 6.607 (0.000) 2.537 (0.006) –0.326 (0.628) 5.030 (0.000) 0.838 (0.201) 0.558 (0.288)

1.017 (0.154) 37.456 (0.000) 1.524 (0.064) 1.259 (0.104) 5.238 (0.000) 1.918 (0.028) –0.370 (0.644) 4.655 (0.000) 0.680 (0.248) 1.236 (0.108)

1.102 (0.135) 32.449 (0.000) 1.141 (0.127) 1.226 (0.110) 4.436 (0.000) 1.726 (0.0422) –0.543 (0.706)) 4.326 (0.000) 0.577 (0.282) 1.492 (0.068)b

68

Extreme risk spillover between markets

trades than the A-share market. The story is a bit different at the 5 per cent risk level: we only find instantaneous risk spillover between SZB and SZA/SHA. Finally, we consider risk spillover between Shares A/B and Share H. At the 10 per cent level, tests of two-way Granger causality in risk show strong risk spillover between Share A and Share H, and tests of one-way Granger causality in risk show that Share H Granger causes Share A in risk. We also find that SZA (but not SHA) Granger causes Share H. At the 5 per cent risk level, there is no risk spillover between SZA and HKH, but for large M, we find significant evidence that Share H Granger-causes SHA in risk with respect to I t 1. It is possible that there exists a time delay in risk spillover or risk spillover carries over a very long distributional lag from Share H to SHA. Tests for two-way Granger causality in risk show that there exists significant risk spillover between Share B and Share H at the two risk levels. Moreover, tests of one-way Granger causality in risk indicates that SHB and SZB Granger-causes HKH in risk and vice versa at the 10 per cent risk levels. To summarize our findings: (1) There exists strong risk spillover between Share A and Share B, and between SHSE and SZSE. Moreover, Share B Granger causes Share A in risk but not vice versa at the 10 per cent risk level. (2) There exists significant risk spillover between Share A and Share H, and between Share B and Share H. The latter is stronger than the former. (3) Risk spillover is more significant at the 10 per cent risk level than at the 5 per cent risk level. The existence of strong risk spillover between Share A and Share B is possibly due to the fact that both markets are influenced by domestic policy changes, government’s intervention and macroeconomic factors in China. The headquarters and the business of A Shares and H shares are in mainland China. On the other hand, Share B and Share H are both traded in foreign currencies and were restricted to foreigners before 2001. Both markets are affected by foreign investors’ sentiment toward the Chinese economy and international shocks. As a consequence, risk spillover between Share B and Share H is stronger than Share A and Share H. 4.3.3

Risk spillover among greater China

Economically, mainland China, Hong Kong and Taiwan constitute the greater China area. Panel A of Table 4.4 reports test statistics (with p-values) for Granger causality in risk between the Chinese stock market and Hong Kong Hang Seng Index (HSI). At the 10 per cent and 5 per cent risk levels, we find significant Granger causality in risk from Share A to HSI with respect to I t 1, but we find no evidence that HSI Granger causes Share A indices in risk. On the other hand, tests of two-way Granger causality in risk indicates strong risk spillover between Share B indices and HSI at both the 10 per cent and 5 per cent risk levels. Moreover, tests of one-way Granger causality in risk show that SHB Granger causes HSI in risk with respect to I t 1, and HSI Granger causes to SZB in risk with respect to I t 1. These results are perhaps

0.556 (0.411) 0.556 (0.289) –0.048 (0.519) –0.561 (0.713) –0.778 (0.782) 0.321 (0.374) 9.339 (0.000) 2.694 (0.004) 1.249 (0.106) 5.685 (0.000) 0.348 (0.364) 2.186 (0.014) 77.605 (0.000)

0.432 (0.357) 0.432 (0.333) 0.440 (0.330) 0.669 (0.748) –0.581 (0.719) 0.078 (0.469) 11.083 (0.000) 1.465 (0.071) 1.096 (0.137) 7.275 (0.000) 0.204 (0.419) 2.456 (0.007) 108.331 (0.000)

SHA඼HSI

HKH඼HSI

SZBමHSI

SZBයHSI

SZB඼HSI

SHBමHSI

SHBයHSI

SHB඼HSI

SZAමHSI

SZAයHSI

SZA඼HSI

SHAමHSI

SHAයHSI

10

5

M 1.249 (0.265) 1.249 (0.106) –0.218 (0.586) 0.089 (0.464) 0.344 (0.365) 0.061 (0.476) 6.922 (0.000) 2.256 (0.012) 0.877 (0.190) 4.073 (0.000) 0.361 (0.359) 1.430 (0.076) 55.854 (0.000)

20

α = 0.10

2.125 (0.132) 2.125 (0.017) –0.419 (0.663) 0.657 (0.256) 1.332 (0.091) –0.159 (0.563) 6.052 (0.000) 2.467 (0.007) 0.634 (0.263) 3.963 (0.000) 0.770 (0.221) 1.595 (0.055) 46.174 (0.000)

30 2.282 (0.079) 2.282 (0.011) –0.167 (0.566) 1.155 (0.124) 1.695 (0.045) 0.160 (0.436) 5.580 (0.000) 2.467 (0.007) 0.685 (0.247) 3.790 (0.000) 0.735 (0.231) 1.812 (0.035) 40.198 (0.000)

40

Table 4.4 Risk spillover among greater China Panel A: risk spillover between Chinese stock markets and that of Hong Kong

0.308 (0.379) –0.139 (0.555) –0.519 (0.698) 1.605 (0.054) 1.444 (0.074) 1.153 (0.125) 10.571 (0.000) 1.855 (0.032) –0.316 (0.624) 7.295 (0.000) 0.562 (0.287) 1.761 (0.039) 59.218 (0.000)

5 0.108 (0.457) –0.298 (0.617) –0.294 (0.616) 1.122 (0.131) 1.063 (0.144) 0.584 (0.280) 8.513 (0.000) 2.795 (0.003) –0.219 (0.587) 5.964 (0.000) 0.084 (0.467) 2.639 (0.004) 41.251 (0.000)

10 0.468 (0.320) 0.581 (0.281) –0.406 (0.658) 0.589 (0.278) 0.988 (0.161) –0.173 (0.568) 6.018 (0.000) 2.488 (0.006) –0.752 (0.774) 4.166 (0.000) –0.280 (0.610) 2.046 (0.020) 28.826 (0.000)

20

α = 0.05

1.124 (0.131) 1.487 (0.068) –0.271 (0.607) 1.142 (0.127) 1.636 (0.051) –0.022 (0.509) 5.179 (0.000) 2.505 (0.006) –0.727 (0.766) 4.080 (0.000) –0.200 (0.579) 2.604 (0.005) 24.032 (0.000)

30

1.442 (0.075) 1.832 (0.033) –0.110 (0.544) 1.389 (0.082)a 1.746 (0.040) 0.222 (0.412) 4.721 (0.000) 2.294 (0.011) –0.429 (0.667) 4.119 (0.000) –0.210 (0.583) 3.121 (0.001) 21.286 (0.000)

40

5.828 (0.000) –0.330 (0.629)

4.150 (0.000) 0.860 (0.195)

10

p −values is 0.051 for M = 50.

5 3.538 (0.000) 1.221 (0.111)

20

α = 0.10

2.910 (0.002) 1.690 (0.046)

30 2.397 (0.008) 1.811 (0.035)

40

0.055 (0.478) 0.639 (0.261) –0.409 (0.659) 0.988 (0.161) 2.019 (0.022) –0.145 (0.557)

–0.262 (0.603) –0.078 (0.531) –0.137 (0.555) 0.216 (0.414) 1.265 (0.103) –0.342 (0.634)

SHA඼TWI

SZAමTWI

SZAයTWI

SZA඼TWI

SHAමTWI

SHAයTWI

10

5

M –0.541 (0.706) 0.286 (0.388) –0.983 (0.837) 0.481 (0.315) 1.308 (0.095) –0.358 (0.640)

20

α = 0.10

–0.747 (0.773) 0.081 (0.468) –1.086 (0.861) 0.174 (0.431) 0.825 (0.205) 0.377 (0.647)

30 –0.780 (0.788) –0.005 (0.502) –1.079 (0.860) 0.191 (0.424) 0.753 (0.226) –0.309 (0.621)

40

Panel B: risk spillover between Chinese stock markets and that of Taiwan

Note: a: the

HKHමHSI

HKHයHSI

M

Table 4.4 (cont.)

–0.231 (0.591) –0.343 (0.634) –0.138 (0.555) 1.713 (0.043) 2.014 (0.022) 1.164 (0.122)

5

1.155 (0.124) –0.669 (0.748)

5

0.033 (0.487) –0.334 (0.631) 0.326 (0.372) 2.122 (0.017) 1.093 (0.137) 2.343 (0.010)

10

0.404 (0.343) –1.078 (0.860)

10

–0.502 (0.692) –0.781 (0.783) 0.004 (0.499) 1.614 (0.053) 0.127 (0.449) 2.369 (0.009)

20

α = 0.05

0.407 (0.342) –1.412 (0.921)

20

α = 0.05

–1.030 (0.848) –1.062 (0.856) –0.465 (0.679) 0.962 (0.168) –0.335 (0.631) 1.830 (0.034)

30

0.737 (0.230) –0.862 (0.806)

30

–1.078 (0.859) –1.231 (0.891) –0.351 (0.637) 0.582 (0.280) –0.622 (0.733) 1.547 (0.061)

40

1.016 (0.155) –0.470 (0.681)

40

HKHමTWI

HKHයTWI

HKH඼TWI

SZBමTWI

SZBයTWI

SZB඼TWI

SHBමTWI

SHBයTWI

SHB඼TWI

4.415 (0.000) –0.198 (0.579) 0.860 (0.195) 0.556 (0.289) –0.756 (0.775) 1.991 (0.023) 7.245 (0.000) 2.008 (0.022) –0.619 (0.732)

3.805 (0.000) –0.525 (0.700) 1.996 (0.023) 0.130 (0.448) –0.978 (0.836) 1.368 (0.086) 6.334 (0.000) 2.561 (0.005) 0.185 (0.427)

3.283 (0.001) –0.468 (0.680) 2.333 (0.010) –0.372 (0.645) –1.040 (0.851) 0.621 (0.267) 6.107 (0.000) 3.321 (0.000) 0.930 (0.176)

3.212 (0.001) –0.149 (0.559) 2.421 (0.008) –0.267 (0.605) –1.099 (0.864) 0.815 (0.207) 5.650 (0.000) 3.058 (0.001) 1.327 (0.092)

2.859 (0.002) –0.031 (0.512) 2.096 (0.018) –0.229 (0.590) –1.115 (0.868) 0.875 (0.191) 5.341 (0.000) 2.904 (0.002) 1.511 (0.065)

1.525 (0.064) 0.029 (0.489) 0.436 (0.331) 0.223 (0.412) 0.217 (0.414) –0.104 (0.541) 4.102 (0.000) 5.496 (0.000) 0.406 (0.342)

1.842 (0.033) 0.667 (0.252) 0.803 (0.211) 0.432 (0.333) 0.411 (0.340) 0.082 (0.467) 3.317 (0.000) 4.415 (0.000) 0.011 (0.496)

1.966 (0.025) 0.880 (0.189) 1.090 (0.138) 0.599 (0.274) 0.906 (0.182) –0.140 (0.556) 2.751 (0.003) 3.557 (0.000) 0.040 (0.484)

1.752 (0.040) 0.855 (0.196) 0.945 (0.172) 0.622 (0.267) 1.241 (0.107) –0.431 (0.667) 2.073 (0.019) 2.796 (0.003) –0.151 (0.561)

1.655 (0.049) 0.639 (0.261) 1.107 (0.134) 0.559 (0.288) 1.155 (0.124) –0.429 (0.667) 1.779 (0.038) 2.410 (0.008) –0.157 (0.563)

72

Extreme risk spillover between markets

due to the fact that SHB is traded in US dollars, and SZB is traded in Hong Kong dollars. On the other hand, tests of two-way Granger causality in risk show strong risk spillover between HKH and HSI at both the 10 per cent and 5 per cent levels. At the 10 per cent risk level, tests of one-way Granger causality in risk show that HKH Granger-causes HSI in risk and vice versa with respect to I t 1. At the 5 per cent level, there is only instantaneous risk spillover between HKH and HSI. We note that risk spillover between Shares B/H and HSI is stronger than risk spillover between Share A and HSI. In all, we can find that the downside movements of SHA, SZA and SHB all have a significant predictive power for the price fall of HSI, and the occurrence of a large risk in HSI help predict a similar future risk in SZB. Panel B of Table 4.4 reports test statistics for Granger causality in risk between the mainland China stock market and Taiwan stock market. There is no risk spillover between SHA and TWI, but SZA Granger-causes TWI in risk and vice versa with respect to I t 1 at the 5 per cent risk level. This is perhaps because most companies listed in SZSE are joint-ventures and foreign-trade oriented, while the majority of listed companies in SHSE are state-owned enterprises. For Share B indices and TWI, we find that TWI Granger-causes SHB in risk at the 10 per cent risk level, but at the 5 per cent level, there exists only instantaneous risk spillover between SHB and TWI. We also find that TWI Granger causes SZB in risk at the 10 per cent risk level, but at the 5 per cent level, there is no spillover between them. Finally, tests of one-way Granger causality in risk indicate strong risk spillover from Share H to TWI. To sum up, we find that the three different shares – A, B and H in the Chinese market have significant risk spillover with Hong Kong and Taiwan stock markets. This finding is consistent with the stylized fact that mainland China, Hong Kong and Taiwan have geographical proximity and close economic ties. Since economic reform, China has been embarked on a process of financial and real integration with Hong Kong and Taiwan. Even before Hong Kong’s return to China’s sovereignty in 1997, it had achieved a high degree of integration with the mainland. With respect to trade, for instance, Hong Kong intermediates the lions share of China’s external trade via reexports and offshore trade. Regarding financial activity, a substantial amount of international capital (in the forms of foreign direct investment, equity and bond financing and syndicated loans) financing China’s economic expansion is raised via Hong Kong. At the same time, Hong Kong’s role as an intermediary for trade and financial flows to China represents a major source of economic activity and greatly shapes its own economic structure. Despite political and ideological differences and occasional tensions between China and Taiwan, economic links between these two economies have proliferated since the 1990s. According to official statistics, China is the largest recipient of Taiwan’s overseas investment and Taiwan is China’s third largest source of foreign direct investment. Furthermore, it is widely believed that the official statistics under-present the overall Taiwan economic interest in China. The

Extreme risk spillover between markets

73

integration process between these three economies that comprise what is often termed ‘greater China’ is preceding more along de facto than de jure lines. Actually H shares and Red Chips of mainland companies already make up for 30 per cent of market capitalization in the Hong Kong Stock Exchange.5 Many investors of B share and H share are Hong Kong and Taiwan citizens. And one major factor moving the Hang Seng index is investor sentiment towards the Chinese economy. 4.3.4 Risk spillover between Chinese stock markets and Asian stock markets Singapore and South Korea, two newly industrialized economies in East Asia, and Japan, one of the most highly developed economies in the world, are representative of Asian countries. We now consider risk spillover between Chinese stock market and these Asian capital markets. Table 4.5(A) reports test statistics (with p-values) for risk spillover between Chinese stock market and Singapore stock market. We find a very little risk spillover between Shares A indices and STI, but there exist significant risk spillover between Share B indices and STI at both the 10 per cent and 5 per cent risk levels. For HKH and STI, tests of two-way Granger causality in risk reveal strong risk spillover between Share H and STI at the two risk levels. Tests of one-way Granger causality in risk suggests that HKH Granger-causes STI in risk and vice versa at the 10 per cent risk level, but at the 5 per cent risk level, we only find that STI Granger-causes HKH in risk with respect to I t 1. Singapore places no restrictions on equity investment for either foreigners or domestic residents and its stock exchange is one of the most international markets in Asia. Foreign companies listed in Singapore Stock Exchange account about 20 per cent, with market capitalization approximate 40 per cent. More than half of the foreign listings have relationships to China. Table 4.5(B) reports results for Granger causality in risk between the Chinese stock market and South Korean stock market. There exists significant risk spillover between Share A indices and KOSPI, and between Share B indices and KOSPI. Tests of two-way Granger causality in risk suggest strong risk spillover between HKH and KOSPI. And tests of one-way Granger causality in risk show that HKH Granger-causes KOSPI in risk and vice versa. Interestingly, a comparison between Panels A and B shows that risk spillover between the Chinese stock market and Korea stock market is stronger than risk spillover between the Chinese stock market and Singapore stock market. This might be due to the stronger economic link between China and South Korea. As two major economies in Asian, trade and direct investment between the two countries have expanded rapidly. According to South Korean statistics, South Korea’s trade with China reached US$41.2 billion in 2002, making China the third largest trade partner of South Korea. Meanwhile, South Korea’s direct investment in China totaled $1.72 billion during the same year.

0.058 (0.477) –0.115 (0.546) 0.198 (0.422) 0.013 (0.495) –0.818 (0.793) 0.018 (0.493) 8.100 (0.000) 3.890 (0.000) –0.839 (0.799) 7.385 (0.000) 0.344 (0.365) 3.907 (0.000) 19.707 (0.000)

–0.677 (0.751) –0.737 (0.769) –0.417 (0.662) 0.268 (0.394) –0.965 (0.833) 0.141 (0.444) 9.065 (0.000) 1.314 (0.094) –0.592 (0.723) 9.577 (0.000) –0.450 (0.674) 5.614 (0.000) 26.728 (0.000)

SHA඼STI

HKH඼STI

SZBමSTI

SZBයSTI

SZB඼STI

SHBමSTI

SHBයSTI

SHB඼STI

SZAමSTI

SZAයSTI

SZA඼STI

SHAමSTI

SHAයSTI

10

5

M 0.723 (0.235) 0.513 (0.304) 0.546 (0.293) 0.836 (0.202) 0.057 (0.477) 0.615 (0.269) 6.476 (0.000) 4.458 (0.000) –1.295 (0.902) 6.003 (0.000) 1.018 (0.154) 3.007 (0.001) 14.102 (0.000)

20

α = 0.10

1.387 (0.083) 1.043 (0.148) 0.966 (0.167) 1.295 (0.098) 0.394 (0.347) 1.036 (0.150) 5.462 (0.000) 4.218 (0.000) –1.422 (0.922) 5.456 (0.000) 1.747 (0.040) 2.296 (0.011) 11.272 (0.000)

30 1.710 (0.044) 1.382 (0.083) 1.080 (0.140) 1.443 (0.074) 0.538 (0.295) 1.156 (0.124) 4.678 (0.000) 3.681 (0.000) –1.354 (0.912) 5.018 (0.000) 1.973 (0.024) 1.926 (0.027) 9.370 (0.000)

40 –1.187 (0.882) –0.402 (0.656) –0.955 (0.830) –0.967 (0.833) –0.245 (0.597) –0.905 (0.817) 5.673 (0.000) –0.391 (0.652) –0.523 (0.700) 2.354 (0.009) –0.504 (0.693) 2.309 (0.010) 24.672 (0.000)

5

Table 4.5 Risk spillover between Chinese stock markets and Asian stock markets Panel A: risk spillover between Chinese stock markets and that of Singapore

–0.959 (0.831) –0.519 (0.698) –0.535 (0.704) –0.757 (0.775) –0.644 (0.740) –0.205 (0.581) 4.484 (0.000) 0.340 (0.367) –0.205 (0.581) 1.911 (0.028) –0.544 (0.707) 2.080 (0.019) 17.678 (0.000)

10 –0.608 (0.728) –0.336 (0.631) –0.276 (0.609) –0.668 (0.748) –0.470 (0.681) –0.301 (0.618) 3.417 (0.000) 0.918 (0.179) –0.469 (0.680) 1.560 (0.055) –0.764 (0.778) 2.169 (0.015) 12.804 (0.000)

20

α = 0.05

–0.561 (0.713) –0.611 (0.729) 0.025 (0.490) –0.741 (0.771) –0.285 (0.612) –0.620 (0.732) 2.587 (0.005) 0.784 (0.216) –0.721 (0.764) 1.044 (0.148) –1.198 (0.885) 1.949 (0.026) 10.668 (0.000)

30

–0.538 (0.705) –0.714 (0.762) 0.136 (0.446) –0.908 (0.818) –0.367 (0.643) –0.796 (0.787) 2.105 (0.018) 0.619 (0.268) –0.763 (0.777) 0.632 (0.264) –1.536 (0.938) 1.789 (0.037) 9.316 (0.000)

40

1.313 (0.095) 4.589 (0.000)

1.719 (0.043) 3.294 (0.000)

1.869 (0.031) 1.758 (0.039)

1.322 (0.093) 1.236 (0.108)

0.946 (0.172) 0.678 (0.249)

1.967 (0.025) 2.420 (0.008) 0.896 (0.185) 1.387 (0.083) –0.124 (0.549) 2.666 (0.004) 4.450 (0.000) 2.114 (0.017) –0.601 (0.726)

1.286 (0.099) 2.365 (0.009) 0.271 (0.393) 1.104 (0.135) 0.215 (0.415) 2.296 (0.011) 5.729 (0.000) 2.033 (0.021) –0.623 (0.733)

SHA඼KOSPI

SHBමKOSPI

SHBයKOSPI

SHB඼KOSPI

SZAමKOSPI

SZAයKOSPI

SZA඼KOSPI

SHAමKOSPI

SHAයKOSPI

10

5

M 1.531 (0.063) 1.138 (0.128) 1.316 (0.094) 1.055 (0.146) –0.619 (0.732) 2.451 (0.007) 3.644 (0.000) 1.374 (0.085) 0.380 (0.352)

20

α = 0.10

1.544 (0.061 0.471 (0.319) 1.932 (0.027) 1.059 (0.145) –0.883 (0.811) 2.646 (0.004) 3.568 (0.000) 0.681 (0.248) 1.589 (0.056)

30 1.414 (0.079) 0.051 (0.480) 2.129 (0.017) 0.960 (0.169) –1.023 (0.847) 2.602 (0.005) 3.707 (0.000) 0.637 (0.262) 2.203 (0.014)

40

Panel B: risk spillover between Chinese stock markets and that of Korea

HKHමSTI

HKHයSTI

0.967 (0.167) 2.235 (0.013) –0.809 (0.791) 0.411 (0.411) 0.398 (0.345) –0.465 (0.679) 7.069 (0.000) 1.544 (0.061) –0.791 (0.785)

5

1.370 (0.085) 3.305 (0.000)

2.361 (0.009) 4.120 (0.000) –0.633 (0.737) 0.466 (0.466) 0.589 (0.278) –0.744 (0.771) 4.835 (0.000) 1.140 (0.127) –0.911 (0.819)

10

0.795 (0.213) 2.550 (0.005)

1.844 (0.033) 3.258 (0.001) –0.638 (0.738) 0.574 (0.574) 0.179 (0.429) –0.647 (0.741) 3.137 (0.001) 0.461 (0.322) –0.725 (0.766)

20

α = 0.05

1.081 (0.140) 1.617 (0.053)

1.621 (0.052) 2.335 (0.010) –0.056 (0.522) 0.428 (0.428) –0.093 (0.537) 0.198 (0.422) 2.358 (0.009) 0.130 (0.448) –0.642 (0.740)

30

1.515 (0.065) 0.956 (0.170)

1.836 (0.033) 1.971 (0.024) 0.612 (0.270) 0.309 (0.309) –0.231 (0.591) 0.814 (0.208) 2.120 (0.017) 0.253 (0.400) –0.584 (0.720)

40

1.582 (0.057) 0.644 (0.260)

1.337 (0.091) –1.235 (0.891) 2.558 (0.005) 4.625 (0.000) 0.651 (0.258) 2.062 (0.020)

2.352 (0.009) –0.956 (0.831) 3.797 (0.000) 5.331 (0.000) –0.803 (0.789) 2.988 (0.001)

SZB඼KOSPI

0.574 (0.283) –1.310 (0.905) 1.660 (0.048) 4.022 (0.000) 1.983 (0.024) 0.958 (0.169)

20

α = 0.10

0.238 (0.406) –1.295 (0.902) 1.241 (0.107) 3.984 (0.000) 1.942 (0.026) 1.442 (0.075)

30 0.329 (0.371) –1.189 (0.883) 1.321 (0.093) 3.873 (0.000) 1.756 (0.040) 1.762 (0.039)

40

–0.007 (0.503) –0.192 (0.576) 0.439 (0.330)

–0.333 (0.630) –0.449 (0.673) 0.287 (0.387)

SHA඼N225

SHAමN225

SHAයN225

10

5

M –0.479 (0.684) –0.731 (0.768) 0.201 (0.420)

20

α = 0.10

–0.881 (0.811) –0.938 (0.826) –0.199 (0.579)

30 –0.956 (0.830) –1.097 (0.864) –0.160 (0.564)

40

Panel C: risk spillover between Chinese stock markets and that of Japan

HKHමKOSPI

HKHයKOSPI

HKH඼KOSPI

SZBමKOSPI

SZBයKOSPI

10

5

M

Table 4.5 (cont.)

–0.372 (0.645) –0.229 (0.591) 0.098 (0.461)

5

2.636 (0.004) –0.815 (0.793) 0.952 (0.171) 17.063 (0.000) 1.058 (0.145) 0.874 (0.191)

5

0.263 (0.396) 0.663 (0.254) 0.060 (0.476)

10

1.603 (0.054) –1.228 (0.890) 0.914 (0.180) 12.998 (0.000) 1.793 (0.036) 0.868 (0.193)

10

0.523 (0.301) 0.884 (0.188) 0.102 (0.460)

20

α = 0.05

0.904 (0.183) –1.827 (0.966) 1.275 (0.101) 10.018 (0.000) 1.636 (0.051) 1.373 (0.085)

20

α = 0.05

0.813 (0.208) 1.267 (0.102) 0.088 (0.465)

30

0.623 (0.267) –2.071 (0.981) 1.458 (0.072) 8.704 (0.000) 1.471 (0.071) 1.705 (0.044)

30

0.992 (0.161) 1.446 (0.074) 0.134 (0.447)

40

0.407 (0.342) –2.161 (0.985) 1.441 (0.075) 7.814 (0.000) 1.299 (0.097) 1.823 (0.034)

40

HKHමN225

HKHයN225

HKH඼N225

SZBමN225

SZBයN225

SZB඼N225

SHBමN225

SHBයN225

SHB඼N225

SZAමN225

SZAයN225

SZA඼N225

0.276 (0.391) 0.890 (0.187) –0.485 (0.686) 0.412 (0.340) 0.153 (0.439) –0.756 (0.775) 1.676 (0.047) –0.161 (0.564) 0.171 (0.432) 8.072 (0.000) 1.151 (0.125) –0.067 (0.527)

0.048 (0.481) 0.598 (0.275) –0.563 (0.713) 0.312 (0.378) 0.202 (0.420) –0.558 (0.712) 1.312 (0.095) –0.135 (0.554) 0.336 (0.369) 5.757 (0.000) 1.345 (0.089) –0.577 (0.718)

0.484 (0.314) 0.872 (0.192) –0.187 (0.574) –0.103 (0.541) –0.431 (0.667) –0.287 (0.613) 1.116 (0.132) –0.255 (0.600) 0.667 (0.253) 4.339 (0.000) 1.273 (0.102) –0.364 (0.642)

0.438 (0.331) 0.852 (0.197) –0.238 (0.594) –0.006 (0.502) –0.200 (0.579) –0.265 (0.605) 0.878 (0.190) –0.179 (0.571) 0.461 (0.322) 3.323 (0.000) 0.895 (0.186) –0.485 (0.686)

0.510 (0.305) 0.796 (0.213) –0.079 (0.531) 0.206 (0.419) 0.004 (0.499) –0.101 (0.540) 0.447 (0.328) –0.417 (0.662) 0.207 (0.418) 2.589 (0.005) 0.549 (0.291) –0.618 0.732)

–0.411 (0.659) –0.848 (0.802) 0.002 (0.499) 4.497 (0.000) 1.139 (0.127) –0.783 (0.783) 8.940 (0.000) 0.723 (0.235) –0.871 (0.808) 7.643 (0.000) 0.194 (0.423) –0.480 (0.684)

–0.518 (0.698) –0.932 (0.824) 0.040 (0.484) 3.262 (0.000) 1.399 (0.081) –1.046 (0.852) 6.858 (0.000) 1.348 (0.089) –0.626 (0.734) 5.473 (0.000) 0.375 (0.354) –0.457 (0.676)

–0.919 (0.821) –1.237 (0.892) –0.180 (0.571) 2.540 (0.006) 1.010 (0.156) –0.432 (0.667) 4.528 (0.000) 0.572 (0.284) –0.562 (0.713) 3.846 (0.000) 0.476 (0.317) –0.573 (0.717)

–0.713 (0.762) –0.735 (0.769) –0.353 (0.638) 1.767 (0.039) 0.608 (0.271) –0.590 (0.722) 3.664 (0.000) 0.588 (0.278) –0.632 (0.736) 3.268 (0.001) 0.389 (0.348) –0.286 (0.613)

–0.448 (0.673) –0.106 (0.542) –0.586 (0.721) 1.263 (0.103) 0.377 (0.354) –0.749 (0.773) 3.098 (0.001) 0.421 (0.337) –0.573 (0.717) 2.888 (0.002) 0.390 (0.348) –0.221 (0.587)

78

Extreme risk spillover between markets

This constitutes 34 per cent of South Korea’s total outward foreign direct investment. Table 4.5(C) reports test statistics for Granger causality between the Chinese stock and Japan stock market. There is no evidence on risk spillover between Shares A indices and NK225. However, at the 5 per cent risk level, tests of two-way Granger causality in risk show significant risk spillover between Share B indices and NK225. Because all one-way Granger causality in risk between Share B indices and NK225 are insignificant, we may conclude that there is only instantaneous risk spillover between Share B indices and NK225 at the 5 per cent risk level. At the 10 per cent risk level, there exists instantaneous risk spillover between SZB and NK225, but there is no evidence on risk spillover between SHB and NK225. Finally, there exists strong instantaneous risk spillover between HKH and NK225 at both the 10 per cent and 5 per cent risk levels. 4.3.5 Risk spillover between Chinese stock markets and international stock markets As is well-known, the US is the largest and most powerful economy in the world, and Germany is an advanced market economy and a world leader in export. Their stock markets are most representative of major international markets. We now investigate risk spillover between the Chinese stock market with these two mature capital markets. Table 4.6 reports test statistics for Granger causality in risk between the Chinese stock market and US and German stock markets. Both tests of two-way and one-way Granger causality in risk suggest that there is no risk spillover between Shares A indices and S&P500/DAX. However, there exists significant risk spillover between Share B indices and S&P500/DAX. There is even stronger risk spillover between Share H and S&P500/DAX. We should note that there are time lags between the opening hours of the Chinese stock market and US and German stock markets; care must be taken when one interprets the above empirical results. To summarize, we have the following observations: (i) There exists strong risk spillover between Share A and Share B, and between SHSE and SZSE. Moreover, Share B Granger causes Share A in risk with respect to I t 1 at the 10 per cent risk level, but not vice versa. (ii) There exists strong risk spillover between Shares A/B and Share H. Risk spillover between Share B and Share H is stronger than the risk spillover between Share A and Share H. (iii) There exists significant risk spillover between the mainland China stock market (particularly Shares B and H) and Hong Kong and Taiwan’s stock markets. (iv) There exists some risk spillover between the Chinese stock market (particularly Shares B and H) and Singapore stock market and between the

–0.879 (0.810) –1.299 (0.903) –0.440 (0.670) –0.292 (0.615) –0.459 (0.677) –0.509 (0.694) 2.253 (0.012) 1.264 (0.103) 2.613 (0.004) 0.800 (0.212) 1.167 (0.122) 0.448 (0.327) 11.150 (0.000)

–0.214 (0.585) –1.066 (0.857) 0.096 (0.462) 0.373 (0.354) 0.108 (0.457) –0.286 (0.612) 1.879 (0.030) 1.461 (0.072) 0.371 (0.009) 1.132 (0.129) 1.206 (0.114) 1.312 (0.095) 13.130 (0.000)

SHA඼S&P500

HKH඼S&P500

SZBමS&P500

SZBයS&P500

SZB඼S&P500

SHBමS&P500

SHBයS&P500

SHB඼S&P500

SZAමS&P500

SZAයS&P500

SZA඼S&P500

SHAමS&P500

SHAයS&P500

10

5

M –0.969 (0.867) –1.365 (0.914) –0.326 (0.628) –0.285 (0.612) –0.880 (0.811) 0.107 (0.457) 1.408 (0.080) 0.978 (0.164) 1.378 (0.084) 0.261 (0.397) 0.909 (0.182) –0.263 (0.604) 8.308 (0.000)

20

α = 0.10

–1.172 (0.834) –1.585 (0.944) –0.329 (0.629) –0.180 (0.571) –0.907 (0.818) 0.362 (0.359) 0.889 (0.187) 0.477 (0.317) 1.044 (0.148) 0.140 (0.444) 1.043 (0.149) –0.627 (0.735) 6.647 (0.000)

30 –0.892 (0.880) –1.314 (0.906) –0.157 (0.562) 0.033 (0.487) –0.515 (0.697) 0.317 (0.375) 0.728 (0.233) 0.279 (0.390) 0.969 (0.166) –0.214 (0.585) 0.784 (0.217) –0.910 (0.819) 5.910 (0.000)

40 –0.835 (0.879) –0.159 (0.563) –0.744 (0.772) –0.395 (0.654) 0.371 (0.355) –0.849 (0.802) 0.494 (0.311) –0.065 (0.526) 1.457 (0.073) 1.629 (0.052) 0.186 (0.426) 1.683 (0.046) 11.347 (0.000)

5

Table 4.6 Risk spillover between Chinese stock markets and major international stock markets Panel A: risk spillover between Chinese stock markets and that of the United States

–0.483 (0.858) –0.422 (0.664) 0.009 (0.497) –0.551 (0.709) –0.128 (0.551) –0.595 (0.724) 0.596 (0.276) 0.680 (0.248) 0.592 (0.277) 1.239 (0.108) 0.179 (0.429) 1.155 (0.124) 8.265 (0.000)

10 –0.039 (0.798) 0.469 (0.319) –0.304 (0.620) –0.434 (0.668) 0.206 (0.418) –0.762 (0.777) 0.239 (0.406) 0.843 (0.200) –0.249 (0.598) 1.021 (0.154) 0.591 (0.277) 0.526 (0.300) 5.406 (0.000)

20

α = 0.05

0.054 (0.515) 1.048 (0.147) –0.791 (0.785) –0.359 (0.640) 0.678 (0.249) –1.131 (0.871) –0.042 (0.517) 0.752 (0.226) –0.617 (0.731) 0.968 (0.167) 1.105 (0.135) –0.011 (0.505) 4.601 (0.000)

30

0.032 (0.487) 1.191 (0.117) –0.990 (0.839) –0.271 (0.607) 0.828 (0.204) –1.161 (0.877) –0.372 (0.645) 0.537 (0.296) –0.904 (0.817) 0.829 (0.204) 1.143 (0.127) –0.215 (0.585) 4.053 (0.000)

40

4.964 (0.000) 10.671 (0.000)

5.784 (0.000) 14.124 (0.000)

HKHයS&P500

3.420 (0.000) 7.800 (0.000)

20

α = 0.10

2.549 (0.005) 6.292 (0.000)

30 2.060 (0.020) 5.767 (0.000)

40

–1.337 (0.909) –1.150 (0.875) –0.488 (0.687) –1.541 (0.938) –1.160 (0.877) –1.244 (0.893) 0.287 (0.387)

–1.033 (0.849) –1.152 (0.875) 0.051 (0.480) 0.904 (0.817) –0.734 (0.768) –0.858 (0.805) –0.202 (0.580)

SHA඼DAX

SHB඼DAX

SZAමDAX

SZAයDAX

SZA඼DAX

SHAමDAX

SHAයDAX

10

5

M –1.235 (0.892) –0.988 (0.839) –0.548 (0.708) –1.306 (0.904) –1.018 (0.846) –0.931 (0.824) 0.995 (0.160)

20

α = 0.10

–0.968 (0.833) –0.951 (0.829) –0.228 (0.590) –1.021 (0.846) –1.066 (0.857) –0.438 (0.669) 0.901 (0.184)

30 –0.799 (0.788) –0.908 (0.818) –0.050 (0.520) –0.833 (0.798) –1.145 (0.874) –0.074 (0.530) 0.878 (0.190)

40

Panel B: risk spillover between Chinese stock markets and that of Germany

HKHමS&P500

10

5

M

Table 4.6 (cont.)

–0.694 (0.756) –1.098 (0.864) 0.548 (0.292) –0.518 (0.698) –0.020 (0.508) –0.208 (0.582) 3.374 (0.000)

5

–0.303 (0.619) 14.178 (0.000)

5

–0.805 (0.789) –1.417 (0.922) 0.583 (0.280) –0.943 (0.827) –0.468 (0.680) –0.562 (0.713) 2.760 (0.003)

10

–0.750 (0.773) 10.022 (0.000)

10

–0.554 (0.710) –1.014 (0.845) 0.470 (0.319) –1.633 (0.949) –1.122 (0.869) –0.997 (0.841) 2.011 (0.022)

20

α = 0.05

–1.144 (0.874) 6.781 (0.000)

20

α = 0.05

–0.435 (0.668) –0.918 (0.821) 0.505 (0.307) –2.025 (0.979) –1.571 (0.942) –1.140 (0.873) 1.288 (0.099)

30

–1.399 (0.919) 6.212 (0.000)

30

–0.378 (0.647) –0.834 (0.798) 0.478 (0.316) –2.339 (0.990) –1.825 (0.966) –1.352 (0.912) 1.209 (0.113)

40

–1.571 (0.942) 5.809 (0.000)

40

HKHමDAX

HKHයDAX

HKH඼DAX

SZBමDAX

SZBයDAX

SZB඼DAX

SHBමDAX

SHBයDAX

–0.731 (0.768) 0.785 (0.216) 5.833 (0.000) –0.841 (0.800) 9.980 (0.000) 12.282 (0.000) 1.586 (0.056) 3.617 (0.000)

–0.452 (0.674) 1.146 (0.126) 4.227 (0.000) –0.873 (0.809) 6.894 (0.000) 9.273 (0.000) 1.654 (0.049) 2.583 (0.005)

0.150 (0.440) 1.486 (0.069) 2.691 (0.004) –0.734 (0.769) 4.378 (0.000) 6.696 (0.000) 1.067 (0.143) 2.002 (0.023)

0.091 (0.464) 1.355 (0.088) 2.252 (0.012) –0.181 (0.572) 3.195 (0.001) 5.236 (0.000) 0.636 (0.262) 1.497 (0.067)

0.238 (0.406) 1.147 (0.126) 2.179 (0.015) 0.296 (0.383) 2.626 (0.004) 4.345 (0.000) 0.429 (0.334) 1.127 (0.130)

0.705 (0.241) 2.742 (0.003) 8.258 (0.000) 1.906 (0.028) 12.638 (0.000) 7.024 (0.000) 4.639 (0.000) 3.183 (0.001)

0.673 (0.251) 2.111 (0.017) 7.213 (0.000) 3.213 (0.001) 8.229 (0.000) 6.321 (0.000) 4.866 (0.000) 2.230 (0.013)

0.436 (0.331) 1.537 (0.062) 4.894 (0.000) 2.537 (0.006) 4.915 (0.000) 4.515 (0.000) 3.162 (0.001) 1.713 (0.043)

0.036 (0.486) 1.041 (0.149) 3.810 (0.000) 2.163 (0.015) 3.571 (0.000) 3.508 (0.000) 2.110 (0.017) 1.556 (0.060)

–0.040 (0.516) 1.100 (0.136) 3.230 (0.001) 1.799 (0.036) 3.03 (0.001) 2.785 (0.003) 1.420 (0.078) 1.369 (0.086)

82

Extreme risk spillover between markets

Chinese stock market and South Korean stock market. The latter is stronger than the former. (v) There is no evidence on risk spillover between Share A indices and major international stock markets – Japan, US and Germany. However, there exists risk spillover between Shares B/H and these international stock markets. (vi) Among the three shares – A, B and H in the Chinese stock market, risk spillover between Share H and overseas capital markets is the strongest, and risk spillover between Share A and overseas capital markets is the weakest or nonexistent. The findings that Share A markets have little risk spillover with leading international capital markets is possibly due to the fact that the market environments of the H and A shares are quite different. The price movement of Share A markets is largely driven by domestic policy changes and government’s market intervention. It is largely uncorrelated with the listed companies’ fundamental profitability and is expected to be insulated from global financial market turbulence given market segmentation and invertibility of Chinese currency. On the other hand, Shares B and H are both traded by foreign investors with different investment opportunities. Both markets are affected by foreign investors’ sentiment toward Chinese economy. They are influenced by global market climates and the fundamental profitability of the listed Chinese companies, in addition to domestic policy changes in China. However, there is a difference: Share B markets are traded in China’s domestic exchanges by offshore investors before 2001 and suffered from lack of liquidity from the beginning. Share H markets are traded in Hong Kong where there is a higher liquidity, more institutional investors, more disclosure and better communications with shareholders. For example, the SEHK has introduced additional listing requirements for issuers that are incorporated in mainland China. Lacking ownership restrictions and currency control, H shares are attractive to institutional and individual investors from Hong Kong and overseas. A survey by the Hong Kong Exchanges and Clearing Limited (2002) indicates that although local investors still dominate the SEHK, overseas investors, mainly institutional investors, contributed 40 per cent of the total market trading value between October 2000 and September 2001. The major origins of overseas participation are the UK and the US. Local and overseas institutional trading contributes 57 per cent of the total market trading value. Consequently, Share H market has been associated with tremendous market volatility and has more mutual interaction with international capital markets than Share B markets. Hong Kong remains the window of China to the world. As regards accounting standards, the regulatory framework and information disclosure, most investors prefer Hong Kong as a stepping stone to participate in mainland’s stunning growth.

Extreme risk spillover between markets

4.4

83

Conclusion

Monitoring extreme downside market risk is important for investment/portfolio diversification and risk management. In this chapter, we have provided probably the first empirical study on spillover of extreme downside market risk between Shares A, B and H in the Chinese stock market, between different stock markets in greater China, and between the Chinese stock market and international capital markets. It is found that there exists strong risk spillover between Share A indices and Share B indices, and the occurrence of a large downside risk in Share B markets can help predict the occurrence of a similar future risk in Share A markets. There also exist strong risk spillover between Share A and Share H, and particularly between Share B and Share H. Share B, and particularly Share H, have significant risk spillover with Asian and international stock markets. In contrast, although Share A has some risk spillover with Korean and Singapore stock markets, it has no risk spillover with leading international mature capital markets – Japan, US and Germany. These findings suggest that the market segmentation between Share A and Share B is effective in avoiding large adverse shocks from international capital markets in terms of large adverse market movements. The Chinese stock market has close ties with Asian stock markets, but its link with leading international capital markets is still weak or nonexistent. The closeness might indicate still significant barriers to international trade and investments. But on the other hand Chinese equities all the more seem to be an adequate device of portfolio diversification. However, the segmentation between Share A and Share B markets – originally restricted for local and foreign investors respectively, has blurred in recent years. Domestic investors have been allowed to invest in Share B markets since about two years before. At the end of last year, China decided to open Share A markets to qualified foreign institutional investors.6 These reforms, together with more and more Chinese companies listed in oversea stock markets, will bring the Chinese stock markets closer to international capital markets. It is conceivable that relatively weak risk spillover between Share A markets and international stock markets will become stronger, but this remains to be verified when a sufficiently amount of data becomes available as time goes. Our approach and results can be useful for further empirical study along this avenue.

Notes 1 Both Shares A and B are tradable shares. In addition, there are also non-tradable state-owned shares, which include ‘state shares’ and ‘legal person shares’. 2 See Harvey (1993) for a summary of restrictions on foreign equity investments in emerging markets. 3 Approximately 65 per cent of the market capitalization are owned by the state and not tradable.

84

Extreme risk spillover between markets

4 We will not consider the N-share and L-share market because of the small number of Chinese firms listed in NYSE and LSE. 5 Red Chips are Chinese conglomerates. They are incorporated in Hong Kong, but their main operational business is done on the mainland. For these enterprises, the trading location being outside mainland China only serves the purpose of raising foreign currency. We don’t discuss it here because of their similarities to H Shares. 6 Four major international institutional investors, UBS AG of Switzerland, Japan’s Nomura Securities, Morgan Stanley and Citigroup, have been recently approved for entering the Share A markets.

5

Information spillover effects between Chinese futures market and spot market

China’s futures market has experienced volatile fluctuations since its emergence during the late 1980s, bringing a great deal of uncertainties and risks to market participants. As a result, the futures market efficiency and the relationship between the futures market and the spot market have been major concerns for the supervision authorities and investors. A large volume of study has been done on the relationship between the futures market and the spot market. Garbade and Solber (1983) present a model to examine the price discovery role of futures prices and the effect of arbitrage on price changes in spot and the futures commodity markets. Haigh et al. (2000) use cointegration analysis to study the relationship between the prices of the futures market and the spot market. According to previous studies, for most futures products, there exists a cointegrating relationship between the prices of the futures market and the spot market. Hasbrouck (1995) defines price discovery in terms of the variance of the innovations to the common factor, based on which the futures and the spot markets’ relative contributions to this variance can be examined. Tse (1999) investigates the minute-by-minute price discovery process and volatility spillovers between the Dow Jones Industrial Average (DJIA) index and the index futures. Tse and So (2004) use data from Hong Kong’s Hang Seng Index, Hang Seng Index futures and the Tracker Fund to examine price discovery among the Hang Seng Index market via the Hasbrouck and Gonzalo and Granger information sharing techniques and the Multivariate Generalized Autoregressive Conditional Heteroskedasticity (M-GARCH) model. Empirical evidence shows that the movements of the three markets are interrelated and they have different degrees of information processing abilities. In the Chinese future markets literature, Hua and Zhong (2002) study the price discovery in the Chinese futures market using the Garbade and Solber (1983) model. Tong et al. (2005) investigate price discovery in the mainland Chinese futures market. Hua and Zhong (2005) use cointegration analysis to study the relationship between the futures price and spot price for copper and aluminum traded at SHFE (Shanghai Futures Exchange). Gao (2004) studies the relationship between soybean futures price and spot price as well

86

Information spillover effects between markets

as the relationship between SHFE copper price and LME (London Mental Exchange) copper price. It is obvious that study on the relationship between the futures market and the spot market concentrates mainly on price discovery, and little research has been done on the information spillover effect especially risk spillover effect between the two markets. To fill in this gap, this paper will examine the information spillover effects between the futures market and spot market in China. We employ the methods developed by Hong (2001a) and use their tests based on the kernel function to investigate the relationship between the futures market and the spot market for the first time. The kernel weight function ensures good power of the test method for using many lags, and Granger causality can be tested across a wide range of alternative hypotheses. Given the short selling mechanism in the futures market, we examine the risks for long and short positions separately. Considering the rising price tendency in the copper market after 2004, we firstly introduce the notions of Upside VaR and extreme upside risk spillover. Specifically, we study the VaR of the Chinese copper futures market and spot market by applying a parametric approach based on TGARCH and GARCH models. In addition, we investigate information spillover effects between the futures market and the spot market by employing a linear Granger causality test, and Granger causality tests in mean, volatility and risk respectively. This chapter is organized as follows: Section 5.1 introduces the methodology of various Granger causality tests; Section 5.2 gives a description of the variables used in the study and the descriptive statistics of their data series. Section 5.3 gives the VaR estimation. Then estimation and empirical results are presented in Section 5.4. Finally, Section 5.5 summarizes and concludes this chapter.

5.1

Granger causality test

The concept of Granger causality introduced by Granger (1969, 1980) is a very useful notion for characterizing relationships between time series in economics and econometrics. Suppose X  [xt ] ,Y  [ yt ] are two random time series, and X t  [xt s , s r 0] ,Yt  [ yt s , s r 0] are their entire time series up to time t. Then X is said to Granger-cause Y if the prediction of certain aspect of the probability distribution of Yt is better using X t 1 ∪Yt 1 than that only using Yt 1. 5.1.1

Linear Granger causality testing

Geweke, Meese and Dent (1983) introduce test models for linear Granger causality. Suppose m

Yt  a10  ¤ a1iYt i  F1t i 1

(5.1)

Information spillover effects between markets

87

and m

k

Yt  a20  ¤ a2iYt i  ¤ C j X t i  F 2t , i 1

(5.2)

j 1

where {a1i } and {a2i } are the coefficients on lagged values of Yt , {C j } are the coefficients on lagged values of X t , F1t and F 2t are normal white noises, m and k are lag lengths. Given this specification, no Granger causality from X t to Yt is equivalent to the null hypothesis that H 0 : C j  0, j  1, 2

k.

(5.3)

The associated F-Test statistic is

F

( ESS1 ESS2 ) / k , ESS2 / ( N k m 1)

(5.4)

where ESS1 and ESS2 are the sums of squared residuals in regressions (5.1) and (5.2), N is the size of sample Yt . Under the null hypothesis, the F statistic follows a F distribution with ( K , N k m 1) degrees of freedom. At significance level α, if F  FB ( K , N k m 1), where F  FB ( K , N k m 1) is the critical value at level a, the null hypothesis is rejected, and X t Granger causes Yt . 5.1.2

Granger causality in mean, volatility and risk

Granger (1969) proposes the concept of Granger causality in mean in terms of incremental predictability for the conditional mean. The test described in Section 5.1.1 is in fact a test for Granger causality in mean. Granger (1980) proposes a concept of general Granger causality in terms of incremental predictability for the conditional distribution. Granger, Robins and Engle (1986) introduce the concept of Granger causality in volatility in terms of incremental predictability. Hong, Cheng, Liu and Wang (2004) introduce the concept of Granger causality in risk and examine the extreme downside risk spillover among financial markets. In this study we extend Hong, Cheng, Liu and Wang (2004) and introduce a new concept: Granger causality in upside risk. Let I t 1 y {I1( t 1) , I 2 ( t 1), },where I1( t 1)  {Y1( t 1) , Y11, } and I 2 ( t 1)  {Y2 ( t 1) , Y21, } are the information sets available at time t-1 in market 1 and market 2 respectively. We are interested in whether the knowledge of {Y2t } increases one’s ability to forecast {Y1t } in terms of various concepts of Granger causality.

88

Information spillover effects between markets

Granger causality in mean: To test for one-way mean spillover from market 2 to market 1, we consider the following null and alternative hypotheses: H 0 : E (Y1t | I1t 1 )  E (Y1t | I t 1 )

(5.5)

H A : E (Y1t | I1t 1 ) x E (Y1t | I t 1 ).

(5.6)

and

Granger causality in volatility: To test for one-way volatility spillover from market 2 to market 1, we consider the following null and alternative hypotheses: H 0 : E [Var(Y1t | I1t 1 ) | I1t 1 ]  Var(Y1t | I t 1 )

(5.7)

H A : E [Var(Y1t | I1t 1 ) | I1t 1 ] x Var(Y1t | I t 1 ).

(5.8)

and

Granger causality in downside risk: To test for one-way downside risk spillover effect from market 2 to market 1, the null and alternative hypotheses are: H 0 : P (Y1t  V1t | I1t 1 )  P (Y1t  V1t | I t 1 ) (a.s.)

(5.9)

H A : P (Y1t  V1t | I1t 1 ) x P (Y1t  V1t | I t 1 ) (a.s.).

(5.10)

and

Granger causality in upside risk: To test for one-way upside risk spillover effects from market 2 to market 1, the null and alternative hypotheses are H 0 : P (Y1t  V1t | I1t 1 )  P (Y1t  V1t | I t 1 ) (a.s.)

(5.11)

H A : P (Y1t  V1t | I1t 1 ) x P (Y1t  V1t | I t 1 ) (a.s.).

(5.12)

and

Define the ‘downside risk indicator function’ based on Downside VaR as follows:

Information spillover effects between markets Zlt y 1(Ylt  Vlt ), l  1, 2

89

(5.13)

where 1(–) is the indicator function. When the actual loss exceeds VaR, Zlt takes value 1; otherwise Zlt takes value 0. To test for the one-way downside risk spillovers from market 2 to market 1, the null hypothesis H 0 and alternative hypothesis H A in Granger causality in downside risk can be stated equivalently as: H 0 : E ( Z1t | I1t 1 )  E ( Z1t | I t 1 )

(5.14)

H A : E ( Z1t | I1t 1 ) x E ( Z1t | I t 1 ).

(5.15)

and

Similarly, we can define ‘the upside risk indicator function’ based on the Upside VaR as: Zlt y 1(Ylt  Vlt ), l  1, 2.

(5.16)

where 1(–) is the indicator function. When actual return exceeds the Upside VaR, Zlt in (5.16) takes value 1; otherwise Zlt in (5.16) takes value 0. To test for the one-way upside risk spillover effect from market 2 to market 1, the null hypothesis H 0 and the alternative hypothesis H A in Granger causality in upside risk also can be stated equivalently as: H 0 : E ( Z1t | I1t 1 )  E ( Z1t | I t 1 )

(5.17)

H A : E ( Z1t | I1t 1 ) x E ( Z1t | I t 1 ).

(5.18)

and

SupposeVlt is an estimator forVlt (B ) orVlt (1 B ). Let Zlt be defined in the same way as Zlt in (5.13) or (5.16) with Vlt replacing Vlt (B ) or Vlt (1 B ). Then we have the sample cross-correlation function between Zˆ 1t and Zˆ 2t as follows: « 1 T ®T ¤ ( Zˆ 1t Bˆ 1)( Zˆ 2 ( t j ) Bˆ 2 ) 0 b j b T 1 ® t 1 j Cˆ ( j ) y ¬ T ®T 1 ( Zˆ 1( t  j ) Bˆ 1)( Zˆ 2t Bˆ 2 ) 1 T b j  0 ® t¤ 1 j ­

(5.19)

90

Information spillover effects between markets T

where Bˆ l y T 1 ¤ Zˆ lt , (l=1, 2) and the sample cross-correlation is: t 1

Sˆ ( j ) y Cˆ ( j ) Sˆ1 Sˆ2 , j  0, p1,

p (T 1),

(5.20)

where Sˆl2 y Bˆ l (1 Bˆ l ) is sample variance of Zˆ lt . Hong, Cheng, Liu and Wang (2004) propose a class of kernel-based tests for Granger causality in risk. To test one-way Granger causality in risk from market 2 to market 1, the test statistic is «® T 1 º® 1 2 Q1 (M )  ¬T ¤ k 2 ( j  M )S2( j ) C1T (M )»  [2D1T (M )] ­® j 1 ¼®

(5.21)

where k(.) is a kernel function that assign weights to various lags, and the centering factor and scaling factor are T 1

C1T (M )  ¤ (1 j T )k 2 ( j M ),

(5.22)

j 1

and T 2

D1T (M ) 

¤ (1 j

T ){1 ( j  1) T }k 4 ( j M ).

(5.23)

j 1

Hong, Cheng, Liu and Wang (2004) also construct tests for downside risk spillover effects (including instantaneous downside risk spillover effects) between two markets. The test statistic is: «® T 1 º® 1 2 Q2 (M )  ¬T ¤ k 2 ( j  M )S2( j ) C2T (M )»  [2D2T (M )] ­® j 1 T ¼®

(5.24)

Where the centering factor and scaling factor are T 1

C 2T ( M ) 

¤

j 1 T

(1 j T )k 2 ( j M )

(5.25)

Information spillover effects between markets

91

and T 2

D2T (M ) 

¤

(1 j T ){1 ( j  1) T }k 4 ( j M ).

(5.26)

j  2 T

Under regular conditions, if H 0 holds, then Q M and Q M follow an 1 2 asymptotic N(0,1) distribution. Therefore, if the values of Q M and 1 Q M is larger than the right-tailed N (0, 1) critical value at a prespecified 2 significance level, the null hypothesis is rejected.

5.2

Data

In the Chinese futures market, nearly 20 commodities are currently traded. Among them copper futures is one of the most actively traded products. Meanwhile, China has exceeded the United States as the largest copper consuming country in 2002, and China’s copper consumption accounts for 21 per cent of the world total consumption in 2004. Therefore, as an important industrial raw material, the price fluctuations of copper have significant influence on Chinese and world’s economies. As a result, risk management in the copper market has become a serious concern for regulators, firms and traders. Following Foster (1995); Moosa and Korczak (1999); Sequtira, Chiat and Aleer (2004); Sarno and Valente (2005), the futures series are conducted from the daily closing prices on futures contracts one month prior to the expiration month (we also draw the same conclusion by using the data from the daily closing prices on futures contracts three months prior to the expiration month). We consider the period from 10 July 2000 to 30 June 2006, containing 1442 daily observations. The prices of copper futures are SHFE (Shanghai Futures Exchange) onemonth copper futures closing prices, obtained from Reuters system. Copper spot prices are average daily prices of #1 copper cathodes obtained from www. smm.com.cn. Both futures series and spot series are expressed in the natural logarithm form, which are denoted as LnF and LnS respectively. We defined the rate of return as follows: RFt  DLNF  ln( Ft ) ln( Ft 1 )

(5.27)

RSt  DLNS  ln(St ) ln(St 1 ),

(5.28)

and

where, Ft denotes the price of futures copper and St denotes the price of spot copper. RFt and RSt are the one order difference of the natural logarithm price, denoting the return rate of futures copper and spot copper.

92

Information spillover effects between markets

6 4 2 0 –2 –4 –6 –8 250

500

750

1000

1250

CUF

Figure 5.1 Time series of return rate of futures copper 12 8 4 0 –4 –8 250

500

750

1000

1250

CUS

Figure 5.2 Time series of return rate of spot copper

Figure 5.1 and Figure 5.2 give the time series of return of the futures market and the spot market. According to Figures 5.1 and Figure 5.2, there are some irregular peaks in both series. The obvious volatility clustering suggests that the daily volatility in the two markets is significant and abrupt, and displays conditional heteroskedasticity. In addition, when abnormal volatility and volatility clustering appear, the two series show a similar volatility pattern, indicating that there may exist correlation and information spillover between them. Some descriptive statistics for the futures series RFt and the spot series RSt are given in Table 5.1. According to Table 5.1, the average return of the futures market is larger than that of the spot market, and the volatility range in the

Information spillover effects between markets

93

Table 5.1 Summary descriptive statistics for the rate of return series Sample Mean size Copper 1441 futures Copper 1441 spot

S.D.

Maximum

Median Minimum

Skewness

Kurto- JB sis statistics

0.0854 1.2324 4.8747 0.0644

–6.0880 –0.3573 5.6597 455.4026

0.0869 1.1878 9.3744 0.0331

–7.4208

0.2073 11.3668 4213.45

Table 5.2 The stationary tests ADF test statistics (futures) ADF test statistics (spot)

–37.2107

1% critical value

–3.9644

–35.1714

5% critical value

–3.4129

10% critical value

–3.1285

futures market is larger than that of the spot market as well. Both series have the kurtosis significantly higher than 3, showing a high peak and fat-tails. The Jarque–Bera test confirms the non-normal distribution of both return series at the 1 per cent significance level. Table 5.2 reports the results of the Augmented Dicky-Fuller test for stationarity of both the futures and spot return series. The results of ADF tests show that the statisitics are above all the critical values Therefore the two series are stationary.

5.3

VaR estimation

We first use the partial autocorrelation function and autocorrelation function to decide the order of an AR process in the mean equation. According to the characteristics of residual series, we decide the order of ARCH and GARCH in the variance equation. Then, by filtering variables, comparing parameters and various test statistics, particularly examining the p-values of the Box– Pierce portmanteau test statistics Q(M), the use of the TGARCH or GARCH model is warranted (see Tables 5.3 and 5.4). Specifically, we build the following AR(2)-TGARCH(1,1) model to study the copper futures market: «Yt  0.0539  0.0443Yt 2  Ft ® ( 0.0260 ) ( 0.0261) ® ® ¬Ft  Yt ht1/ 2 , Yt ~ i.i.d .N ( 0,1), ®h  0.0096  0.0768 F 2  0.9345 h 0.0293 F 2 d t 1 t 1 t 1 t 1 ® t ®­ ( 0.0132 ) ( 0.0027 ) ( 0.0129 ) ( 0.0083)

(5.29)

94

Information spillover effects between markets

Table 5.3 The diagnostic test statistics for the model adequacy of the futures market – Generalized Box–Pierce Q statisitics Q(5)

Q(10)

Q(20)

Q(30)

Q(40)

Q(50)

2.9545 [0.707]

6.7176 [0.752]

29.471 [0.079]

34.639 [0.256]

44.1825 [0.299]

52.015 [0.395]

Q2(5)

Q2(10)

Q2(20)

Q2(30)

Q2(40)

Q2(50)

6.4536 [0.372]

9.1646 [0.517]

14.596 [0.799]

28.097 [0.565]

33.342 [0.763]

39.373 [0.860]

Note: Numbers in [ ] are p-values for General Box–Pierce test statistics.

Table 5.4 The diagnostic test statistics for the model adequacy of the spot market – Generalized Box–Pierce Q statistics Q(5)

Q(10)

Q(20)

Q(30)

Q(40)

Q(50)

6.0448 [0.196]

10.665 [0.299]

27.233 [0.099]

37.349 [0.138]

44.285 [0.258]

58.612 [0.163]

Q2(5)

Q2(10)

Q2(20)

Q2(30)

Q2(40)

Q2(50)

6.0759 [0.194]

12.570 [0.183]

17.897 [0.529]

22.685 [0.791]

24.883 [0.961]

28.186 [0.993]

Note: Numbers in [ ] are p-values for General Box–Pierce test statistics.

Loglikelihood = –2115.306. Similarly, we build the following AR(1)-GARCH(1,1) model for the copper spot market: «Yt  0.0549  0.0731Yt 1  Ft ® ( 0.0242 ) ( 0.0270 ) ® 1/ 2 F  Y ¬ t t ht , Yt ~ i .i .d .N ( 0,1), ®ht  0.0098  0.0783Ft2 1  0.9186 ht 1 ® ( 0.0020 ) ( 0.0076 ) ( 0.0076 ) ­

(5.30)

Loglikelihood = –1945.855. From Table 5.3 and 5.4, the p-values of the Box–Pierce statistics based on ^

autocorrelations in standardized residuals Fˆ t / ht and squared standardized 2

residuals Ft / ht are larger than 0.05, indicating that there exists no autocorrelation in standardized innovations after a TGARCH or GARCH model is fitted. Thus, for each data series, the model selected is adequate. It can be seen that asymmetry factors are significant in the futures market, which implies that in the futures market, people prefer to take long positions due to psychological activities. When the price goes up, the number of

Information spillover effects between markets

95

Table 5.5 VaR estimation for the futures market based on TGARCH model TGARCH model

Confidence level

Mean

S.D.

Failure time

Failure rate

LR statistics

Downside VaR

90% 95% 90% 95%

1.3883 1.8062 1.5033 1.9212

0.5910 0.7609 0.5937 0.7637

127 71 126 64

0.0883 0.0493 0.0876 0.0445

2.2870 0.0133 2.5714 0.9588

Upside VaR

Table 5.6 VaR estimation for the spot market based on GARCH model GARCH model

Confidence level

Mean

S.D.

Failure time

Failure rate

LR statistics

Downside VaR

90% 95% 90% 95%

1.3858 1.6391 1.5064 1.7610

0.7625 0.9260 0.7187 0.9240

123 62 130 67

0.0855 0.0431 0.0903 0.0466

3.5296 0.5165 1.5368 0.3665

Upside VaR

speculators grows. As a result, the risk grows and the reaction to the market uncertainty will become stronger. Therefore, though the mechanism for buying and selling is symmetrical in the futures market, the impacts of good news and bad news on the volatility of the market are still asymmetric. That is, the impacts of different information on the market are asymmetric and good news has stronger impacts than bad news. Perhaps it is due to the rising price tendency in the copper market after 2004. Based on formula (3.5) and (3.6), we calculate the Upside VaR and Downside VaR in the copper futures market and spot market. Tables 5.5 and 5.6 report some summary statistics for the estimated VaR in both the futures and spot markets. It can be seen from Tables 5.5 and 5.6 that the VaR of the futures market is larger than that of the spot market, indicating that the former has a larger risk. For the futures market, Kupiec’s (1995) LR tests of Downside VaR and Upside VaR are insignificant at conventional significance levels, suggesting the adequacy of the VaR model for the futures market. A similar conclusion is reached for the VaR model in the spot market, although in terms of the spot market, at the 10 per cent significance level, there exists some variance for the Downside VaR. Time series of Downside VaR and Upside VaR at the 95 per cent confidence level are given in Figure 5.3 and 5.4. The correlation coefficients between the rate of return of copper futures price and spot price are calculated and the results are presented as Table 5.7. According to Table 5.7, there exists a strong relationship between the futures market and the spot market.

8 7 6 5 4 3 2 1 0 250

500

750

CUFDOWNVAR

1000

1250

CUSDOWNVAR

Figure 5.3 Time series of Downside VaR at the 95 per cent confidence level 8 7 6 5 4 3 2 1 0 250

500

CUFUPVAR

750

1000

1250

CUSUPVAR

Figure 5.4 Time series of Upside VaR at the 95 per cent confidence level Note: CUFDOWNVAR denotes Downside VaR for copper futures, CUSDOWNVAR denotes Downside VaR for copper spot; CUFUPVAR denotes Upside VaR for copper futures, CUSUPVAR denotes Upside VaR for copper spot. Table 5.7 The correlation coefficient between the futures market and the spot market Correlation coefficient of rate of return

Correlation coefficient of conditional variance

95% Downside VaR correlation coefficient

95% Upside VaR correlation coefficient

0.7483

0.9209

0.9974

0.9938

Information spillover effects between markets

97

Table 5.8 The linear-Granger causality test Null hypothesis

Sample size

CUS does not 1440 linear Granger cause CUF CUF does not 1440 linear Granger cause CUS

F statistics and the probability of accepting the null hypothesis M=1

M=2

M=3

M=5

M=10

24.4340 (0.0000)*

11.8682 (0.0000)*

11.4969 (0.0000)*

6.6758 (0.0000)*

3.5580 (0.0000)*

37.5664 (0.0000)*

39.9672 (0.0000)*

25.7405 (0.0000)*

16.4553 (0.0000)*

9.1082 (0.0000)*

Note: CUF demotes rate of return for copper futures price, CUS denotes rate of return for copper spot price, numbers in () are probability of accepting the null hypothesis. M denotes the lag order. Lag orders of 1 to 10 are considered, with all p-values being .0000. For space we do not list them.

Thus it is necessary to study the information spillover effects between the two markets.

5.4 Empirical results for information spillover between futures market and spot market 5.4.1

Linear Granger causality test

Table 5.8 reports the results of the F-test for Granger causality. The empirical results show that at the 1 per cent significance level, there is a two-way linear Granger causality between the prices of the futures market and the spot market, with the impacts of futures market on the spot market being stronger than the impacts of the spot market on the futures market. Also, the results are not sensitive to changes in lag length, indicating high reliability of the conclusion. 5.4.2

Granger causality test in mean, volatility and risk

Next, Tables 5.9 to 5.14 report results on Granger causality in mean, volatility and risk at different risk level. It can be seen from Table 5.9 that, at the 1 per cent significance level, there exist significant two-way Granger causality in mean between the rate of returns of the copper futures prices and spot prices. Further tests suggest that the direction of spillover in mean is from the futures market to the spot market. This finding is consistent with Hua and Zhong’s (2005) conclusion based on regression analysis. Therefore, using historical information in two markets to predict the future price tendency is better than using historical information in one market only, which indicates that copper futures market in China is a deriving force for price movements.

Table 5.9 Test for Granger causality in mean Mean spillover M=5

M=10

M=20

M=30

M=40

M=50

CUF š CUS 331.3119 235.5847 167.4821 137.6718 119.8097 107.4254 (0.0000)* (0.0000)* (0.0000)* (0.0000)* (0.0000)* (0.0000)* CUF › CUS 0.9149 0.5294 0.6716 1.0380 1.1918 1.1833 (0.1801) (0.3709) (0.2509) (0.1496) (0.1167) (0.1183) CUF  CUS 13.4083 10.4109 7.7150 6.7770 6.1291 5.5023 (0.0000)* (0.0000)* (0.0000)* (0.0000)* (0.0000)* (0.0000)*

Table 5.10 Test for Granger causality in volatility Volatility spillover

M=5

M=10

M=20

M=30

CUF š CUS 248.3928 176.8413 124.8379 101.6055 (0.0000)* (0.0000)* (0.0000)* (0.0000)* CUF › CUS 3.3893 2.9500 1.7065 1.1566 (0.0004)* (0.0016)* (0.0440)* (0.1237) CUF  CUS 4.4462 3.7219 2.2456 1.3441 (0.0000)* (0.0000)* (0.0124)* (0.0895)

M=40

M=50

88.0006 78.7081 (0.0000)* (0.0000)* 0.9622 0.7464 (0.1680) (0.2277) 1.0173 0.8507 (0.1545) (0.1975)

Table 5.11 Test for Granger causality in downside risk at 10 per cent risk level 10% downside risk spillover

M=5

M=10

CUF š CUS 165.2339 117.3496 (0.0000)* (0.0000)* CUF › CUS 1.5269 0.8785 (0.0634)* (0.1898) CUF  CUS 4.0979 3.3837 (0.0000)* (0.0004)*

M=20

M=30

M=40

M=50

82.9010 (0.0000)* 0.2365 (0.4065) 2.3625 (0.0091)*

67.6323 (0.0000)* 0.2278 (0.4099) 1.6433 (0.0502)*

58.6700 52.5827 (0.0000)* (0.0000)* 0.3548 0.3469 (0.3614) (0.3643) 1.2757 1.1504 (0.1010) (0.1250)

Table 5.12 Test for Granger causality in downside risk at 5 per cent risk level 5% downside risk spillover

M=5

CUF š CUS 132.9721 (0.0000)* CUF › CUS 3.4565 (0.0003)* CUF  CUS 10.8574 (0.0000)*

M=10

M=20

M=30

M=40

M=50

95.0302 (0.0000)* 2.2911 (0.0110)* 8.3337 (0.0000)*

67.0208 (0.0000)* 1.4590 (0.0723) 5.3491 (0.0000)*

54.9261 (0.0000)* 1.6074 (0.0540) 4.0649 (0.0000)*

47.6370 (0.0000)* 1.3980 (0.0811) 3.4891 (0.0002)*

42.7005 (0.0000)* 1.2348 (0.1085) 3.1690 (0.0008)*

Information spillover effects between markets

99

Table 5.13 Test for Granger causality in upside risk at 10 per cent risk level 10% upside risk spillover

M=5

CUF š CUS

149.4089 (0.0000)* CUF › CUS 1.9193 (0.0275)* CUF  CUS 13.2910 (0.0000)*

M=10

M=20

M=30

M=40

M=50

106.7192 (0.0000)* 1.9448 (0.0259)* 9.3442 (0.0000)*

75.3339 62.3504 54.3737 48.9543 (0.0000)* (0.0000)* (0.0000)* (0.0000)* 1.4126 1.8204 1.9698 1.9247 (0.0789)* (0.0343)* (0.0244)* (0.0271)* 6.1928 5.1673 4.4799 4.1885 (0.0000)* (0.0000)* (0.0000)* (0.0000)*

Table 5.14 Test for Granger causality in upside risk at 5 per cent risk level 5% upside risk spillover

M=5

M=10

107.5952 CUF š CUS 149.1132 (0.0000)* (0.0000)* CUF › CUS 4.0829 5.2624 (0.0000)* (0.0000)* CUF  CUS 1.1170 1.2636 (0.1320) (0.1032)

M=20

M=30

77.3050 63.8822 (0.0000)* (0.0000)* 4.0182 3.4630 (0.0000)* (0.0003)* 2.0464 2.4028 (0.0204)* (0.0081)*

M=40

M=50

55.9579 50.8875 (0.0000)* (0.0000)* 3.2485 3.0315 (0.0006)* (0.0012)* 2.6033 3.2579 (0.0046)* (0.0006)*

Note: ‘š’ denotes tests for two-way Granger causality between the two markets; ‘›’ and ‘’ denote tests for one-way Granger causality; numbers in () are corresponding p-values. Also, risk spillovers with lag order M=15,25,35,45 are examined, which are not listed considering the length limitation.

According to Table 5.10, at the 1 per cent significance level, there exists twoway Granger in volatility between the copper futures and copper spot prices, and the spillover effects from the futures price to the spot price is stronger, which is consistent with the earlier findings in Liu and Wang (2006) on the volatility spillover effects between China’s agricultural product futures market and spot market. The results in Tables 5.11 to 5.14 show, at the 10 per cent and 5 per cent significance levels, the two-way Granger causality in downside risk and upside risk between the return rate of copper futures price and spot price are significant. Thus, when an event with large market volatility happens in one market, a similar event will almost necessarily happen in the other market. To test the direction of risk spillover, we further study the test statistics for one-way Granger causality in downside risk and upside risk. It can be seen that, one-way Granger causality in downside risk and upside risk are both significant, indicating that there are risk spillovers from the futures market to the spot market as well as from the spot market to the futures market. And test statistics suggest that the spillover effects from the futures market to the

100

Information spillover effects between markets

Table 5.15 The information spillover effect between the futures market and the spot market Spillover direction

Linear Mean Volatility 10% spill- spill- spillover downside over over risk spillover

CUF š CUS * CUF › CUS * CUF  CUS *

* *

* * *

* * *

5% downside risk spillover

10% upside risk spillover

5% upside risk spillover

* * *

* * *

* * *

Note: * Denotes significant spillover.

spot market is much stronger. That indicates that futures market in China has a dominant role in the copper markets and is of efficiency. The summary for information spillovers between the futures market and the spot market are shown as Table 5.15. We observe that tests for Granger causality in mean, volatility and risk are proved to have better power than test for linear Granger causality, as the kernel weight function can enhance good power when many lags have been employed. The results of the above tests suggest that there is a two-way Granger causality between the copper futures market and the copper spot market, and a one-way Granger causality test shows that information spillover from the futures market to the spot market is stronger than information spillover from the spot market to the futures market. The reasons are not difficult to understand. First, the futures market serves as a driving force for price discovery. On one hand, the open, fair, efficient and competitive trading mechanism in the futures market attracts a large number of traders, consequently, information such as commodity supply and demand can be reflected into price more accurately and promptly. As a result, new information is reflected first in changed futures prices and the price tendency can be reflected in advance. On the other hand, the futures price can better reflect investors’ expectations for price tendency with contracts trade on a future date. Second, lower costs associated with the futures market provide much greater leverage than the spot market, and market participants can trade more effectively with relatively small amount of cash. In other words, it is a better functioning, more efficient market. In summary, short selling mechanism, low cost, high liquidity and leverage in the futures ensures that the futures market can update prices more frequently than the spot market and hence exhibit dominance in information transmission.

Information spillover effects between markets

5.5

101

Conclusion

This chapter quantifies the potential risk in the Chinese copper market by applying a parametric approach based on GARCH-type models. Also, we study the information and risk spillover effects between the copper futures market and the copper spot market. The main findings are: First, although the trading mechanism in the futures market is symmetric, the impacts of good news and bad news on the market volatility are asymmetric. The market’s reactions to good news are stronger than to bad news. Second, the Downside VaR estimation and the Upside VaR estimation pass backtests at conventional significance levels, suggesting the adequacy of the estimated VaR models. Third, we find that the prices of copper futures market and copper spot market change in the same direction, and the variation ranges are very close. This indicates the two markets share the same long run tendency and investment capitals between the two markets have flexible two-way liquidity. Obviously, investors in the two markets have similar strategy and risk attitudes and their reactions to new information tend to be similar. Forth, significant conditional heteroskedasticity and volatility clustering are found in both futures and spot return series, which can be fitted adequately by a GARCH-type model. Moreover, there exist two-way Granger causality between the copper futures market and spot market, and information and risk spillovers from the futures market to the spot market are much stronger, showing the dominant role the futures market plays in terms of information spillover effects. Empirical results suggest that test statistic based on the kernel function gives better power than the F-test statistic based on regressions. In summary, the empirical results suggest that accurate estimation of the risk in China’s futures market can be realized by employing appropriate econometric models. And as there is a interactive relationship between the risk in the futures market and the risk in the spot market, it is important to strengthen the supervision and regulation over the futures market, introduce more futures contracts and enforce market rules. In addition, to further enhance the development of the futures market in emerging markets like China can help ensure the stability and efficient resource allocation of the spot market as the futures price can help ‘regulate’ supply and demand in advance.

6

How well can autoregressive duration models capture the price durations dynamics of foreign exchanges?

In this chapter, using the omnibus tests developed in Hong, Li and Zhao (2004); Egorov, Hong and Li (2004), we examine whether commonly used ACD models can well forecast the probability density of foreign exchange price durations. The foreign exchange market is one of the most important financial markets in the world, with trading taking place 24 hours a day around the globe and trillions of dollars of different currencies transacted each day. Transactions in the foreign exchange market determine the rates at which currencies are exchanged, which in turn determine the costs of purchasing foreign goods and assets. Density forecasts for the price durations of exchange rates are particularly useful for many outstanding issues in international economics and finance. Price durations measure how long it takes for the price of an asset to move beyond a certain threshold. A trader might be interested in knowing this time interval as it could influence the speed with which he places an order. Price duration models are essentially a volatility model or more precisely the inverse of a volatility model (Engle and Russell 1997, 1998; Giot 2000) and thus play an important role in intraday risk management. Density forecasts for price durations are crucial for the prediction of the price change intensity, which is important for valuing currency options and other currency derivatives. For example, Prigent, Renault and Scaillet (2001) offer an option pricing framework in incomplete markets by using a log ACD model to capture the full dynamics of a price duration process. In Section 6.1, we describe the evaluation procedures for the out-of-sample performance of an ACD model. A class of separate inference tests is also discussed, which can reveal useful information about where an ACD model is likely to be misspecified. In Section 6.2, we review a variety of ACD models and discuss their relative merits. Section 6.3 describes the data and estimation results. Section 6.4 reports the in-sample and out-of-sample performances of the ACD models. Section 6.5 concludes.

6.1

Nonparametric density forecast evaluation

Density forecasts have become a standard practice in many economic and financial applications. For example, modern risk control techniques often

Autoregressive duration models

103

involve some form of density forecasts. In macroeconomics, monetary authorities in the US and UK (the Federal Reserve Bank of Philadelphia and the Bank of England) have been conducting quarterly surveys on density forecasts for inflation and output growth to help set their policy instruments (e.g. inflation target). There is also a growing literature on extracting density forecasts from options prices to obtain useful information on otherwise unobservable market expectations (e.g. Fackler and King 1990; Jackwerth and Rubinstein 1996; Soderlind and Svensson 1997; Ait-Sahalia and Lo 1998; Rosenberg and Engle 2002). One of the most important issues in density forecasts is the evaluation of the quality of density forecasts (Diebold, Gunther and Tay 1998; Granger 1999). Suboptimal density forecasts for important macroeconomic variables, for example, may lead to suboptimal policy decisions (e.g. inappropriate level and timing in interest rate setting), which could have adverse consequence on resource allocations of an economy. In finance, suboptimal density forecasts may lead to misleading calculation of Value at Risk in risk management, and to large errors in derivatives pricing and hedging. 6.1.1

Nonparametric omnibus evaluation test

Evaluation of density forecasts is not trivial, since the probability distribution is not observable even ex post. Suppose {X i ,  1, 2, } is a stationary time series with conditional density p0 ( x | I i 1 ) ; where I i 1  {X i 1 , X i 2 ,…} is the information available at time ti 1. In our application, X i will be the price duration of foreign exchange rates. For a given ACD model for X i , there is a model-implied conditional density u P ( X i b x | I i 1 ,R ) y p( x | I i 1 ,R ), ux where θෛΘ is an unknown finite-dimensional parameter vector and Θ is a parameter space. Suppose we have a random sample {X i }Ti 1 of size T; and we divide it into two subsets: an estimation sample {X i }iR1 of size R and a prediction sample {X i }Ti  R 1 of size n y T R. The former is used to estimate model parameter θ, and the latter is used to evaluate density forecasts. To evaluate density forecasts, we use the probability integral transform of X i with respect to the model density, which is defined as follows: Zi (R ) y °

Xi

d

p( x | I i 1 ,R ).

(6.1)

The transformed series {Zi (R )} can be called the ‘generalized residuals’ of model p( x | I i 1 , R ) . In an important work, Diebold, Gunther and Tay (1998) show that if model p( x | I i 1 ,R ) is correctly specified in the sense that there exists

104

Autoregressive duration models

some R 0 1 such that p( x | I i 1 ,R 0 ) coincides with the true conditional density p0 ( x | I i 1 ) , then the sequence {Zi (R 0 )} should be i.i.d. U[0,1]. This characterization provides a convenient approach to evaluation p( x | I i 1 , R ) . Intuitively, the U[0,1] distribution indicates proper specification of the stationary distribution of X i , and the i.i.d. property characterizes correct specification of its dynamics. If {Zi (R )} is not i.i.d. U[0,1] for all θෛΘ, then p( x | I i 1 , R ) is not optimal, and there exists room to improve p( x | I i 1 , R ) . Thus, p( x | I i 1 , R ) can be evaluated by checking whether its generalized residuals is i.i.d. U[0,1]. Most existing density forecast evaluation procedures examine the i.i.d. property and the uniformity property separately. While this is informative about possible sources of suboptimal density forecasts, it is preferable to use a single omnibus evaluation criteria that takes into account deviations from both i.i.d. and U[0,1] jointly when comparing different models. Otherwise it may be difficult to decide which model is better in capturing the full dynamics of X i if (e.g.) the generalized residuals of one model has less serial dependence but displays a more non-uniform distribution than the generalized residuals of another model. Hong and Li (2004) developed a portmanteau in-sample evaluation procedure for a conditional density model, and Egorov, Hong and Li (2004) extend it to an out-of-sample context. They consider the impact of parameter estimation uncertainty and the choice of relative sample sizes between estimation and prediction samples on the evaluation procedure, two issues ignored by most existing evaluation procedures for density forecasts. They measure the distance between a density model and the true density by comparing a kernel estimator gˆ j ( z1 , z2 ) for the joint density of the pair of generalized residuals {Zi (R ), Zi j (R )} with unity, the product of two U[0,1] densities, where integer j is a lag order. The kernel estimator of the joint density of {Zi (R ), Zi j (R )} is given by T

gˆ j ( z1 , z2 ) y ( n j ) 1

¤L

h

( z1 , Zˆ t )L h ( z2 , Zˆ t j ), j  0

(6.2)

tR j

where Zˆ i  Zi (Rˆ ),Rˆ is any R -consistent estimator for R 0 ,L h ( z1 , z2 ) is a boundary-modified kernel function given by: « 1 ¥ x y ´ 1 ®h K ¦§ h µ¶ / ° ( x / h ) K ( u )du, if x [ 0, h ) ® ® ¥ x y´ L h ( x, y ) y ¬h 1K ¦ , if x [ h,1 h ], § h µ¶ ® ® 1 ¥ x y ´ (1 x ) / h K ( u )du, if x  (1 h,1), ®h K ¦§ h µ¶ / ° 1 ­

(6.3)

Autoregressive duration models

105

and K  i is a prespecified symmetric probability density, and h y h( n ) is a bandwidth such that h m 0, nh m d as n m d. One example of K  i is the quartic kernel K ( u )  15 / 16(1 u 2 )2 1(| u |b 1), where 1 i is the indicator function. We will use this kernel in our application. In practice, the choice of h is more important than the choice of K  i . Like Scott (1992), we choose h  SˆZ n 1/ 6 , where SˆZ is the sample standard deviation of {Z i }Tt  R 1 . This simple bandwidth rule attains the optimal rate for bivariate kernel density estimation. The modified kernel in (6.2) automatically deals with the boundary bias problem associated with standard kernel estimation. The weighting functions in the denominators of the modified kernel K h ( x, y ) for x in the boundary regions [ 0, h ) ‡ (1 h,1] account for the asymmetric coverage and ensure that the kernel density estimator (6.3) is asymptotically unbiased uniformly over the support [0, 1] of Zi (R ) . Hong and Li (2004) propose an in-sample test based on a quadratic form between gˆ j ( z1 , z2 ) and 1, the product of two U[0,1] densities. This test has been extended to the out-of-sample context in Egorov, Hong and Li (2004): 1 1 Qˆ ( j ) y [( n j )h° ° [ gˆ j ( z1 , z2 ) 1]2 dz1dz2 Ah0 ] / V01/ 2 , 0 0

j  1, 2,…

(6.4)

where the nonstochastic centering and scaling factors 1

1

1

0

Ah0 y [( h 1 2 )° K 2 ( u )du  2° 1

1

1

1

°

b

1

K b2 ( u )dudb ]2 1,

V0 y 2[ ° [ ° K ( u  v )K (v )dv ]2 du ]2 , b

and K b ( i ) y K ( i ) / ° K (v )dv. Under suitable regularity conditions on the data

1

generating process {X i }, the model p( x | I i 1 , t,R ), the estimator RˆR, the kernel K  i , the bandwidth h, and the relative sizes n, R between estimation and prediction samples, Qˆ ( j ) m N ( 0,1) in distribution when p( x | I i 1 ,R ), is optimal. The use of the Qˆ ( j ) statistics with various lag orders can reveal the lag orders at which we have significant departures from i.i.d. U [0,1]. To avoid the difficulty when one model has a smaller Qˆ ( j ) at lag j1 but another model has a smaller Qˆ ( j ) at lag j2 x j1, Egorov, Hong and Li (2004) propose a portmanteau evaluation statistic

W ( p) 

1 p

p

¤ Q( j ).

(6.5)

j 1

For any given lag truncation order p, W ( p ) m N ( 0,1) in distribution when p( x | I i 1 ) is optimal. This may be viewed as a generalization of the popular

106

Autoregressive duration models

Box–Pierce–Ljung autocorrelation-based portmanteau test from a linear time series context to a nonlinear time series context. It can check misspecification in a conditional density model of X i rather than only misspecification in a conditional mean model of X i . As long as model misspecification occurs such that Qˆ ( j ) m d at some lag j > 0, we will have W ( p ) m d in probability. Therefore, W ( p ) can be used as an omnibus procedure to evaluate density forecasts. Another important feature of W ( p ) is that any R -consistent parameter estimator RˆR suffices. It is not necessary to use an asymptotically most efficient estimator for R . This is convenient, because asymptotically most efficient estimators such as maximum likelihood estimation (MLE) or approximated MLE may be difficult to obtain. One could choose a suboptimal but convenient estimator in implementing the test. 6.1.2

Diagnostic tools

Diagnostics based on generalized model residuals When a density model is rejected by using Qˆ ( j ) or W ( p ), it would be interesting to explore possible reasons of the rejection. Diebold, Gunther and Tay (1998) illustrate how to use the histogram of {Z i } and autocorrelogram in the powers of {Z i } to reveal sources of model misspecification. Although informative, these graphical methods ignore the impact of parameter estimation uncertainty in RˆR on the asymptotic distribution of evaluation statistics, which generally exists even when R, n m d. Hong and Li (2004) provide a class of rigorous separate inference procedures that explicitly address the impact of parameter estimation uncertainty. This class of test statistics is defined as follows: n 1

n 1

n 2

2 ( j ) ¤ k 2 ( j / p )] / [ 2¤ k 4 ( j / p )]1/ 2 , M z ( m, l )  [ ¤ k 2 ( j / p )( n j )Sml j 1

j 1

j 1

(6.6) where m, l  0, Sml ( j ) is the sample cross-correlation between Ztm and Ztl j . It is asymptotically N (0,1) under correct model specification. Different choices of orders ( m, l ) examine various dynamic aspects of the underlying process {Zt }. For example, the choice of ( m, l ) = (1,1), (2,2), (3,3), (4,4) is sensitive to autocorrelations in level, volatility, skewness and kurtosis of {Zt } respectively (see Diebold, Gunther and Tay 1998 for related discussion). Moreover, M z (1, 2 ) and M z ( 2,1) can check ARCH-in-Mean and leverage effects respectively, which were not previously investigated in the density forecast evaluation literature. Diagnostics based on standardized residuals In ACD modeling, we have X i  Z i (R ) Fi , where X i is a duration process, Z i (R ) is a model for the conditional expected duration E ( X i | I i 1 ) and εi is

Autoregressive duration models

107

the innovation. Different insight can be obtained by examining the standardized residual Fˆi  X i / Z i (RˆR ). For example, to see whether it is necessary to model higher order conditional moments of X i , one can check whether {Fi } is i.i.d. To this end, we follow Hong and Lee (2003) and consider T j ( u, v ) y cov( eiuFi, eivFi | j| ), the covariance function between eiuFi and e ivFi | j| . Straightforward algebra yields T j ( u, v )  K j ( u, v ) K ( u )K (v ), where K j ( u, v )  E ( eiuFi  ivFi | j| ) and K( u )  E ( eiuFi ) are the joint and marginal characteristic functions of ( Fi , F i | j | ) respectively. Define the empirical measure

Tˆ j ( u, v )  Kˆ j ( u, v ) Kˆ j ( u, 0 )Kˆ j ( 0, v ),

j  0, p1, p2,…

n

Where Kˆ j ( u, v )  ( n | j |) 1 ¤ i  | j |1 e i(uFˆ i  vFˆ i | j| ) . We consider the following class of test statistics n 1

( m ,l )

( u, v ) |2 ]dW1 ( u )dW2 (v )

( m ,l )

(1, l )¤ k 4 ( j / p )]1/ 2 ,

M F ( m, l )  {° [ ¤ k 2 ( j / p )( n j ) | T j j 1

( m ,l )

C0

n 1

¤k

2

( j / p )} / [ D 0

j 1

n 2 j 1

(6.7)

( m ,l )

where the integer m, l r 0, T j ( u, v )  u m  l T j ( u, v ) / u m ul ,W1 ( i ) and W2 ( i ) are positive and nondecreasing weighting functions, k( i ) is a kernel function1, and the centering and scaling factors Cˆ 0( m,l )  ° Tˆ (0m, m )( u, u )dW1 ( u )° Tˆ (0l ,l )(v, v )dW2 (v ), Dˆ 0( m,l )  2° | Tˆ (0m, m )( u, u a ) |2 dW1 ( u )dW1 ( u a )° | Tˆ (0l ,l )(v, v a ) |2 dW2 (v )dW2 (v a ). Most commonly used kernels weigh down higher order lags. This is expected to enhance the power of the tests in practice, because financial markets are more influenced by recent events than by remote past events. We will use the Bartlett kernel k ( u )  (1 | z |)1(| z |b 1) in our application. The lag order p can be chosen via suitable data-driven methods. Following Hong and Lee’s (2003) proof, we can show that for each given pair of nonnegative integers ( m, l ) , M F ( m, l ) m N ( 0,1) in distribution under correct ACD model specification, provided lag order p y p( n ) m d, p / n m 0. We note that parameter estimation uncertainty in RˆR has no impact on the asymptotic distribution of M F ( m, l ). The choice of ( m, l ) and {W1 ( i ),W2 ( i )} provide much flexibility in capturing various serial dependence of {Fi }. For example, put ( m, l ) = (0,0),

108

Autoregressive duration models

W1 ( i )  W2 ( i )  W0 ( i ), where W0 : m  is nondecreasing and weighs sets symmetric about 0 equally.2 Then we obtain n 1

( 0 ,0 )

M F ( 0, 0 )  {° [ ¤ k 2 ( j / p )( n j ) | T j ( u, v ) |2 ]dW0 ( u )dW0 (v ) C 0 j 1

( 0 ,0 )

/ [D0

n 1

¤k

2

( j / p )}

j 1

n 2

(1, l )¤ k 4 ( j / p )]1/ 2 , j 1

(6.8) where Cˆ 0( 0,0 )  [ ° Tˆ 0 ( u, u )dW0 ( u )]2 and 2

Dˆ 0( 0,0 )  2 ¨ª ° | Tˆ (00,0 ) ( u, u a ) |2 dW0 ( u )dW0 ( u a )·¹ . This checks generic serial dependence, which is useful in judging whether it is necessary to model higher order conditional moments of duration. Suppose {Fi } is found to be serially dependent. It would be then interesting to examine the pattern of serial dependence in {Fi }. To end this, we put l= 0, and use a Dirac E ( i ) function for W1 ( i ), with W2 ( i )  W0 ( i ).3 Then n 1

( m,0 )

M F ( m, 0 )  {° [ ¤ k 2 ( j / p )( n j ) | T j

( m,0 )

( 0, v ) |2 ]dW0 (v ) C 0

n 1

¤k

2

( j / p )}

j 1

j 1

( m,0 )

/ [D0

n 2

¤k

4

( j / p )]1/ 2 ,

j 1

(6.9) where Cˆ 0( m,0 )  Rˆ m ( 0 )° Tˆ 0 (v, v )dW0 (v ), Dˆ 0( m,0 )  2Rm2 ( 0 )° | Tˆ 0 (v, v a ) |2 dW0 (v )dW0 (v a ), and Rˆ m ( 0 ) is the sample variance of Fˆ im . Note that Tˆ (j m,0 ) ( 0, v ) is consistent for T (j m,0 ) ( 0, v ) y cov[( iF i )m , e iuFi j ]. The choice of m=1 checks the martingale hypothesis for {Fi }. In particular, it has power against alternatives that have zero autocorrelation but a nonzero mean conditional on the Fi j , such as some

Autoregressive duration models

109

bilinear and nonlinear moving average processes. This can reveal useful information whether a conditional mean duration model is adequate. Similarly, the choice of m = 2, 3, 4 can be used to test whether the conditional variance, skewness and kurtosis of Fi are time-varying.

6.2

ACD models

ACD models, first introduced by Engle and Russell (1998), are used to study the dynamics of arrival times between successive occurrences of trading events. Let X i  ti ti 1 be the time intervals between two market events. Examples include the time between successive transactions, the time until a price change occurs or until a prespecified number of shares or level of turnover has been traded.4 All ACD models can be embedded in the following framework: «X i  Z i F i , ® ¬Z i  E ( X i | I i 1 ) ®­Fi 1 ~ MDS with conditional pdf f ( i| I i 1 ).

(6.10)

By construction, the innovation Fi is nonnegative, with E ( Fi | I i 1 ) =1 and conditional pdf f ( i| I i 1 ). Because Fi  X i / Z i , Fi is also called as a standardized duration. The specification of an ACD model includes: (i) Z i , the conditional mean duration, and (ii) f ( i| I i 1 ), the conditional distribution of Fi . For most commonly used ACD models in the literature, {Fi } is assumed to be i.i.d. with marginal pdf f ( i ). This may be called a strong form ACD model, following the analogy of the strong form volatility model termed by Drost and Nijman (1993). For this class of ACD models, all past information enters the current duration through the conditional mean duration Z i . The dynamics of X i is completely captured by Z i . Often, the assumption of the i.i.d. innovation is too strong to capture financial duration dynamics (e.g. Drost and Werker 2004). Again, by analogy with Drost and Nijman (1993), the case in which the demeaned innovation Fi –1 is a martingale difference sequence but not i.i.d. may be called a weak form ACD model, which allows for higher order dependence in durations. The flexibility of an ACD model lies in the rich host of candidates for the conditional pdf of Fi as well as the functional form of the conditional mean Z i . For a strong form ACD model, the conditional pdf of Fi coincides with the marginal pdf of Fi . In this case, several innovation distributions have been used in practice: the standard exponential, Weibull, generalized Gamma and Burr distributions: (a) f ( Fi )  exp( Fi ), 1 1 (b) f ( Fi )  H [ ' (1  )]H Fi H 1 exp{ [ ' (1  )Fi ]H }, H H

H  0,

110

Autoregressive duration models

(c) f ( Fi ) 

HFi HM 1 ' ( M  1 / H ) HM F '( M  1 / H ) H [ ] exp{ [ i )] }, '( M ) '( M ) '( M )

(d) f ( Fi ) 

1 H Fi H 1 F ( M )1 H ' (1  M 1 )

1 ( ) [1  M ( i )H ] (1 M ) , c  , ' (1  H 1 )' ( M 1 H 1 ) c c c

M , H  0, H M0

where '( i ) is the Gamma function. Note that the generalized Gamma distribution reduces to the Weibull when M = 1, and to the standard exponential when M  H  1. When H  1, the Weibull distribution assigns a higher probability than the exponential distribution to extreme observations (very short and long durations). It also allows a non-flat hazard function, which is constant for the exponential distribution.5 However, the Weibull hazard function is monotone: increasing if H  1, and decreasing if H  1. A more flexible hazard function can be obtained with the generalized Gamma distribution (e.g. Lunde 2000). It exhibits a nonmonotonic hazard function in certain regions of the parameter space: a †-shaped hazard when MH  1 and H  1, and a U-shaped hazard when MH  1 and H  1. Another distribution that has a hump-shaped hazard function and that nests the Weibull distribution is the Burr distribution. It is used in Grammig and Maurer (2000) to account for the stylized fact that the hazard function of some financial durations may be increasing for small durations and decreasing for long durations. We will consider these four innovation distributions to see which best describes the price duration of foreign exchange rates. Another key ingredient in an ACD model is the conditional mean duration Z ( i ) . We consider six most popular duration models: LINACD, LOGACD, BCACD, EXPACD, TACD and MSACD models. The first four belong to the class of strong form ACD models and the last two belong to the class of weak form ACD models. For a meaningful comparison of alternative ACD models and for simplicity, we follow Dufour and Engle (2000b); Bauwens et al. (2003) to limit the dynamic structure of the ACD models to the first lag order only. 6.2.1

Strong form ACD models

Linear ACD models (LINACD) Engle and Russell (1998) assume that ψi is a linear function of past durations and conditional durations, namely,

Z i  X  B X i 1  CZ i 1 ,

(6.11)

where X  0, B r 0 and C r 0, ensuring Z i r 0. This is analogous to a GARCH(1,1) model. It can account for duration clustering, a salient feature of financial high-frequency data. However, it has two main limitations, as pointed out by Dufour and Engle (2000b). First, constraints on the parameters are needed to ensure that the linear model does not yield negative durations.

Autoregressive duration models

111

When additional explanatory variables are added linearly to the model of (6.11), Z i may become negative, which is not admissible. Second, empirical evidence suggests (e.g. Engle and Russell (1998)) that a nonlinear Z i may more accurately describe the dynamics of the conditional mean duration. Logarithmic ACD models (LOGACD) The limitations of LINACD have motivated Bauwens and Giot (2000) to introduce a LOGACD model: ln Z i  X  B ln X i 1  C ln Z i 1

(6.12)

 X  B ln Fi 1  C ’ ln Z i 1 where C a  B  C . While retaining the main characteristics of the LINACD model, this model is more flexible because no restrictions are required on the sign of its coefficients. Furthermore, for positive B , durations lower than the current conditional mean (so Fi  X i / Z i  1) have a negative effect, while long durations (Fi > 1) have a positive and marginally decreasing effect on the log of the expected duration. Thus the LOGACD model allows for nonlinear effects of short and long durations in the conditional mean, without requiring the estimation of additional parameters. However, it imposes a rigid adjustment process of the conditional mean to recent durations. For instance, because the logarithmic function asymptotically converges to minus infinity at zero, it is likely to have an overadjustment of the conditional mean after very short durations. Box–Cox ACD models (BCACD) To further improve upon the limitation of the LOGACD model, Dufour and Engle (2000b) proposed a Box–Cox transformation ACD model ln Z i  X a  B a( FiE 1 1) / E  C ln Z i 1  X  BFiE 1  C ln Z i 1 .

(6.13)

This includes the LOGACD model as a special case (E m 0). The choice of an appropriate shock impact specification is data driven in this model (i.e. E is estimated from data). Exponential ACD models (EXPACD) Dufour and Engle (2000b) also introduce a class of EXPACD models similar in spirit to Nelson’s (1991) EGARCH models. This allows for a piecewise linear news impact function kinked at the mean E ( Fi 1 )= 1: ln Z i  X  BFi 1  E | Fi 1 1 | C ln Z i 1 .

(6.14)

112

Autoregressive duration models

EXPACD models offer a captivating compromise between the need of greater flexibility and the burden of higher complexity. For standardized durations shorter than E ( Fi 1 ) = 1, it has a slope B E and an intercept X  E ; while for standardized durations longer than E ( Fi 1 ) = 1, the slope and intercept are B  E and X E respectively. It can capture asymmetric behaviors in price durations. 6.2.2

Weak form ACD models

Threshold ACD models (TACD) Zhang, Russell and Tsay (2001) propose a TACD model that allows the conditional expected duration Z i to be nonlinear in past information variables. The TACD model is a simple but powerful generalization of the LINACD model, allowing different subregimes to have different conditional means and innovation distributions. Put j  [ rj 1 , rj ], j  1,…, J, for a positive integer J, where the rj , with d  r0  r1  …  rJ  d, are thresholds. The process {X i } follows a J-regime threshold ACD model if, when the threshold variable Zi d  j , «Z i  X j  B j X i 1  C j Z i 1 , ¬F y X / Z ~ f ( F ;R ), i i i j ­ i

(6.15)

where the delay parameter d is a positive integer. We allow parameter R in the innovation distribution to be different across regimes. Exogenous variables (e.g. observable market characteristics, such as bid–ask spreads and volumes) as additional regressors and other functional forms of conditional mean (e.g. linear, logarithmic, Box–Cox and exponential forms) can also be used in (6.15). As documented in Zhang, Russell and Tsay (2001), there is a strong evidence that fast and slow transaction periods of NYSE stocks display different dynamics. We will examine whether this is also true of price durations of foreign exchanges. Here, we assume Zi d  X i 1 and focus on a two-regime TACD model analogous to Zhang et al. (2003): «Z i  X j  B j X i 1  C j Z i 1 , ¬ ( j) ­F i y X i / Z i ~ f ( F i ;R j ),

if X i 1 

j

, j  1, 2,

(6.16)

where the parameters in both conditional expected duration Z i and innovation density f ( i, i ) are allowed to vary across regimes. The innovation Fi has a discrete mixture distribution. For a given regime j, {Fi( j ) } is i.i.d. Also, {Fi( j ) } and {Fi( k ) } are independent for j x k . However, the conditional distribution of Fi given I i 1 is time-varying. In our empirical study, we will also employ a regime-adapted version of logarithmic forms for Z i .

Autoregressive duration models

113

Markov regime-switching ACD models (MSACD) The MSACD model, proposed by Hujer et al. (2002), allows the duration process X i to depend on a latent state variable Si that follows a Markov chain. This model nests many existing ACD models. It is closely related to Markov Switching autoregressive regression models popularized by Hamilton (1989) in econometrics. The introduction of the latent state variable Si can be justified in the light of recent market microstructure theories. For instance, the unobservable regime can be associated with the presence (or absence) of private information about an asset’s value that is initially available exclusively to a subset of informed traders and only eventually disseminates through the process of trading to the broader public of all market participants. Hujer et al. (2003b) fit a MSACD model to the Boeing stock data on NYSE and show that it can capture several specific characteristics of intertrade durations while other ACD models fail. In our application to price durations of foreign exchanges, we assume that there are two regimes and the conditional mean duration Z i depends on the latent state variable Si as follows: «ln Z i( Si )  X (Si )  B (Si ) ln X i 1  C (Si ) ln Z i 1 , ¬ ( Si ) ( Si ) ­F i y X i / Z i ~ f ( Fi ;R (Si )).

(6.17)

We refer to the regime in which Si =1 the first regime, and Si =2 the second regime. To avoid the computational intractability due to the dependence of the conditional mean Z i on the entire history of data, we follow Gray (1996) to average over all regime-specific conditional expectations according to

Z i  P (Si  1 | I i 1 )Z i(1)  P (Si  2 | I i 1 )Z i( 2 ) , where P (Si  j | I i 1 ) is the probability that Si is in state j given the filtration I i 1 . We assume a constant transition probability:P (Si  j | Si 1  l )  p jl , where j , l = 1, 2,. The associated conditional density for the price duration is given by f ( x | I i 1 )  P (Si  1 | I i 1 ) f ( x | Si  1, I i 1 )  P (Si  2 | I i 1 ) f ( x | Si  2, I i 1 ), where P (Si  j | I i 1 ), the ex ante probability that the data is generated from regime j at ti 1, can be obtained by a recursive procedure described in Hamilton (1994). By letting the parameters in the innovation distribution depend on the state variable Si , the MSACD models allow for time-varying higher order conditional moments of X i . TACD models are closely related to the MSACD models. Both of them belong to the class of discrete mixture models and allow the

114

Autoregressive duration models

innovations to have time-varying conditional higher order moments. However, their mechanisms of regime determination are different: TACD models allow switches between different regimes to be driven by observable lagged dependent variables. It is interesting to examine the relative performance of these two classes of models in capturing the full dynamics of price durations of foreign exchanges.

6.3

Data and estimation

We consider two intraday foreign exchange rates – Euro/Dollar and Yen/ Dollar, from July 1, 2000 to June 30, 2001. Euro and Yen are two most important currencies in the world after the US dollar. The launch of the Euro has been probably the most important event in the history of the international monetary and financial system since the end of the Bretton Woods system in the early 1970s. It has created the world’s second largest single currency area after the US dollar. In the foreign exchange market, Euro/Dollar is the busiest pair of currencies: it is estimated that 40 per cent of the trading is between this pair, which is twice as large as the Dollar/DM pair had, and twice as large as the Yen/Dollar pair has. The Japanese economy was in a prolonged recession over the last decade, and as a result, the Yen/Dollar rate might have a different dynamics from the Euro/Dollar rate. The data, obtained from Olsen & Associates, are indicative bid and ask quotes posted by banks. We choose the sample period between July 1, 2000 to June 30, 2001 to wait for the market to have stabilized after the introduction of Euro as a new currency in January 1, 1999 and to avoid the impact of ‘September 11’ Incident. The foreign exchange market operates around the clock 7 days a week and the typical rate of quote arrivals differs dramatically on weekends and weekdays and between business hours in different countries and in different time zones (e.g. Goodhart and Figliuoli 1991; Bollerslev and Domowitz 1993; Engle and Russell 1997). To utilize days with a common typical pattern, only data on Wednesdays are used. This subsample consists of 52 days, and 1,264,553 observations on Euro/Dollar and 623,687 observations on Yen/Dollar. Although we focus on price durations, it is useful to first examine quote arrival rates and quote durations. For our data, a typical weekday has almost 24,300 and 16,000 quote arrivals, and on average, a quote arrives every 3.5 s and 5.5 s for Euro/Dollar and Yen/Dollar respectively, where ‘s’ denotes the unit of second. Figure 6.1 plots the histogram of raw durations up to 30 s, showing that two exchange rates share a similar pattern. The majority of quotes (about 86 per cent for Euro/Dollar and 67 per cent per cent for Yen/ Dollar) arrive within 5 s of the previous quote. Clearly, the Euro/Dollar market is more active than the Yen/Dollar market. There is a strong seasonality in quote durations as a deterministic function of the time of the day. It is important to deseasonalize raw duration data. Following Engle and Russell (1997), we regress the duration on a pure time-

Autoregressive duration models 35

25

Frequency

Frequency

30

20 15 10 5 0

0

5

10

15

20

25

115

18 16 14 12 10 8 6 4 2 0

30

Duration (Seconds)

0

5

10

15

20

25

30

Duration (Seconds)

Figure 6.1 Histograms of raw quote durations

of-day to obtain a consistent estimator of the typical duration shape due to the time-of-day effect. Dividing durations by their estimated typical shape thus gives ‘seasonally adjusted’ durations. Figure 6.2 presents the predicted duration as a deterministic function of the time of day. This was obtained by regressing the observed duration on 96 time-of-day dummy variables, each per 15 minute time interval. Both Euro/Dollar and Yen/Dollar display the same pattern of seasonality as have been revealed in previous studies of quote frequency (Bollerslev and Domowitz 1993; Engle and Russell 1997). Trading activity picks up after midnight as the Asian Pacific markets such as Tokyo, Sydney, Singapore and Hong Kong open. The abrupt decline in arrivals between hours 3:00{4:00 GMT signals lunchtimes in these markets. We find most quote activity between 5:00 and 16:00 GMT in the afternoon Far Eastern trading session and during the overlap of the New York and European markets. During this period quotes arrive at a rate of about 1 trade every 2 s for Euro/Dollar and 4 s for Yen/Dollar on average. Activity declines after the New York market closes and before the Far Eastern markets open again. Price durations are the time needed to witness a given cumulative change in the price. They are usually defined on the mid-point of the bid–ask quote process. Such definition is one of the favorite ways used in the literature to thin the quote point process (e.g. Bauwens and Giot 2000, 2003; Engle and Russell 1997, 1998). It has several advantages. First, the bid–ask bounce can be avoided. In a dealer’s market, the bid–ask bounce can be annoying to work with, as it is a main feature of data but gives little information. Second, as the minimum amount of time for the price to increase or decrease by at least c, a pre-defined threshold, price durations are closely linked to the instantaneous volatility of the mid-quote price process. Therefore, we can investigate the determinants of price volatility by adding exogenous and lagged dependent variables to an ACD model of price durations. Thirdly, as documented in Engle and Russell (1997, 1998) for the foreign exchange

116

Autoregressive duration models

35 30 Seconds

Seconds

25 20 15 10 5 0 0

5

10

15

45 40 35 30 25 20 15 10 5 0 0

20

5

10

15

20

Hour of Day

Hour of Day

45

45

40

40

35

35

30

30

Frequency

Frequency

Figure 6.2 Expected quote durations conditional on time of day

25 20 15

25 20 15

10

10

5

5

0

0 0

0.0002 0.0004 0.0006 0.0008 0.001 Spread

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Spread

Figure 6.3 Histograms of observed spreads

market and the New York Stock Exchanges or in Biais, Hillion and Spatt (1995) for the Paris Bourse, the quotes process is often characterized by a short term transitory component that gives little information about the value of the asset; while significant movement of the mid-point often reflects some informative events. Define the ‘midprice’ at time ti as Pi  ( bidi  aski ) / 2, where bidi and aski are the current bid and ask prices associated with transaction time ti . A threshold c characterizes price changes. If c = 0, we would count every single movement in the midpoint as a price change. To better capture movements in the price at which transactions occur, we will choose a positive value of c. Figure 6.3 displays the histogram of the spreads. Most spreads are 0.0005 for Euro/Dollar and 0.05 for Yen/Dollar respectively, which both account for more than 40 per cent respectively. We thus set c = 0.0005 and 0.05 for Euro/ Dollar and Yen/Dollar respectively. These choices of c yield a sample size of 20,584 for Euro/Dollar and 15,818 for Yen/Dollar. Table 6.1 gives some descriptive statistics. For Euro/Dollar, the minimum price duration is 1 s, the maximum is 12,666 s (about 3.5 hours) and the average is nearly 214 s. For

Autoregressive duration models

117

Table 6.1 Summary statistics for price durations of Euro/Dollar and Yen/Dollar Euro/Dollar

Yen/Dollar

Raw duration

Deseasonalized duration

Raw duration

Deseasonalized duration

The whole sample Sample size Mean S.D. Minimum Median Maximum

20,584 213.89 500.64 1 57 12,666

20,584 1 1.7068 0.0009 0.3427 32.363

15,818 276.31 570.63 1 99 12697

15,818 1 1.6293 0.0012 0.4166 26.354

First half sample Sample size Mean S.D. Minimum Median Maximum

10,292 217.69 542.92 1 48 12,666

10,292 1.02 1.77 0.001 0.3094 21.862

7,909 331.51 896.96 1 115 12,697

7,909 1.20 1.9337 0.001 0.4966 26.354

Second half sample Sample size Mean S.D. Minimum Median Maximum

10,292 210.1 454.43 1 64 9,654

10,292 0.9816 1.6459 0.0009 0.3725 32.363

7,909 221.11 399.28 1 87 9,191

7,909 0.8013 1.2212 0.0014 0.3592 19.403

Note: Raw price durations are measured in seconds by the time interval between two bid–ask quotes during which a cumulative change in the mid-price of at least 0.0005 for Euro/Dollar and 0.05 for Yen/Dollar is observed. The quotes obtained from Olsen & Associates are intraday Euro/Dollar and Yen/Dollar exchange rates from July 1, 2000 to June 30, 2001 on Wednesdays. Deseasonalized price durations are produced from raw price durations by filtering out the timeof-day effects.

Yen/Dollar, the minimum price duration is 1s, the maximum is 12,697 s and the average is nearly 276 s, with the last two larger than those of Euro/Dollar. Figure 6.4 presents the histogram for the price durations, whose patterns differ a bit from those of raw quote durations. Most common price durations are 1s, accounting for 7.3 per cent for Euro/Dollar and 3.6 per cent for Yen/ Dollar respectively. Figure 6.5 presents the seasonality for price durations, which are calculated in the same manner as for quote durations. The two kinds of seasonality show similar patterns. The price durations are on average about once every 100 s for Euro/Dollar and 160 s for Yen/Dollar between 6:00 and 15:30 GMT. When the European market closes, price changes occur much less frequently. They become as infrequent as roughly once every 1,000 s and 830 s for Euro/Dollar

Autoregressive duration models

8

4

7

3.5

6

3

5

2.5

Frequency

Frequency

118

4 3

2 1.5

2

1

1

0.5

0

0 0

50

100

150

200

250

300

0

100

Duration (Seconds)

200

300

400

500

Duration (Seconds)

1800 1600 1400 1200 1000 800 600 400 200 0

900 800 700 Seconds

Seconds

Figure 6.4 Histograms of raw quote durations

600 500 400 300 200

0

5

10

15

Hour of Day

20

100

0

5

10

15

20

Hour of Day

Figure 6.5 Expected price durations conditional on time of day

and Yen/Dollar respectively around 22:30 GMT. However, Yen/Dollar price changes occur most frequently around 24:00 GMT when the Asian markets open again. We adjust price durations by filtering out the seasonality as follows: X i  Yi / g (ti ), where {Yi } is an original price duration, and g( i ) is the seasonal effect on price durations. The mean of the deseasonalized series X i is approximately unity. The standard deviations are 1.71 and 1.63 for the ‘seasonally adjusted’ Euro/Dollar and Yen/Dollar price durations respectively, indicating overdispersion. The adjusted price duration process in Figure 6.6 looks stationary. Prices tend to experience periods of rapid and slow movement respectively, displaying strong price duration clustering. There are a few jumps or outliers as well. Figure 6.7 shows that adjusted price durations have a decreasing

Autoregressive duration models 35

30

30

25 Seconds

Seconds

25 20 15 10

20 15 10 5

5 0

119

0 0

5,000

10,000

15,000

0

20,000

5000

10000

15000

7000

10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

6000 5000 Frequency

Frequency

Figure 6.6 Seasonally adjusted price duration

4000 3000 2000 1000 0

0

5

10

15

20

25

30

0

5

Seconds

10

15

20

25

Seconds

Figure 6.7 Histogram of seasonally adjusted price duration

pdf. A striking feature of price durations is the presence of autocorrelation even after removing the time-of-day effects. Figure 6.8 reports the first 50 sample autocorrelations of two adjusted price durations. The slow decaying autocorrelations indicate persistent price duration clustering. Ljung-Box test statistics with 15 lags are 34,719 and 30,558 for Euro/Dollar and Yen/Dollar respectively, implying rather significant autocorrelation in price durations. Following the usual practice in the literature, we focus on the adjusted series X i rather on the original series Yi . For out-of-sample evaluation, we divide the data into two equal halves. The first half (10,292 observations for Euro/ Dollar and 7,909 observations for Yen/Dollar) is used for estimation and the second half is used for out-of-sample evaluation. For both Euro/Dollar and Yen/Dollar, durations in two subsamples appear to have similar characteristics. We consider six classes ACD models: LINACD, LOGACD, BCACD, EXPACD, TACD and MSACD, combined with four innovation distributions – the exponential, Weibull, generalized Gamma and Burr distributions.6 This generates twenty four ACD models, all of which are estimated via MLE.

120

Autoregressive duration models

0.4

0.46

0.38

0.44 0.42

0.36

0.4

0.34

0.38

0.32

0.36 0.34

0.3

0.32

0.28

0.3

0.26

0.28 0

10

20

30

40

Lag order

50

0

10

20

30

40

50

Lag order

Figure 6.8 Sample autocorrelations of seasonally adjusted price duration

The optimization algorithm is the well-known BHHH with STEPBT for step length calculation and is implemented via the constrained optimization code in GAUSS Window Version 5.0. For TACD models, the MLE estimation is performed by a grid search over threshold values r and by maximizing the likelihood function given r. Parameter estimates of various ACD models are reported in Table 6.2.

6.4

Empirical evidence

We now use Hong and Li’s (2004) test to evaluate various ACD models for the price durations of foreign exchanges. The performance of each model is measured by the W ( p ) statistic, reported in Panels A and B of Table 6.3. For space, we only report W (5 ) , W (10 ) and W ( 20 ).7 6.4.1

In-sample performance

We first evaluate in-sample performance of ACD models. Based on parameter estimates in Table 6.2, we calculate in-sample generalized residuals {Zi (R )}tR1 in (6.1), where R is the size of in-sample observations. Although some models perform better than others, W ( p ) in Panel A of Table 6.3 overwhelmingly rejects all ACD models at any conventional significance level. In other words, none of the ACD models adequately captures the full dynamics of price durations for Euro/Dollar and Yen/Dollar. Our results differ from Bauwens, Giot, Grammig and Veredas (2004), who find that LOGACD models based on generalized Gamma or Burr innovations perform satisfactorily for some stock price durations.8 Among all ACD models, the MSACD model with the Burr distribution performs best for both Euro/Dollar and Yen/Dollar, with W (5 ) statistics around 30 and 20 respectively. This is in line with the results of Hujer et al. (2003b) that MSACD models have a better in-sample fit than other ACD models. With the same innovation distribution, LINACD, LOGACD,

γ

β

α

ω

Parameter

0.0418 (0.0044) 0.1699 (0.0091) 0.7997 (0.0115)

E

0.0562 (0.0072) 0.2551 (0.0174) 0.7139 (0.0190) 0.6239

W

Euro/Dollar

Panel A: Linear ACD model (LINACD)

0.0880 (0.0102) 0.4096 (0.0281) 0.6059 (0.0227) 0.2489

G 0.0653 (0.0080) 0.3482 (0.0254) 0.6560 (0.0207) 0.6968

B

exp( Fi ), « ® 1 1 ® H [ ' (1  )]H Fi H 1 exp{ [ ' (1  )Fi ]H }, H  0, H H ® ® f ( Fi )  ¬ HFi HM 1 ' ( M  1 / H ) HM Fi ' ( M  1 / H ) H ] }, M, H  0, ® ' ( M ) [ ' ( M ) )] exp{ [ '( M ) ®

1 ( M )1 H ' (1  M 1 ) Fi H (1 M 1 ) ® H Fi H 1 ,c  , H  M  0, ® c ( c ) [1  M( c ) ]

1 ' (1  H )' ( M 1 H 1 ) ­

0.0521 (0.0058) 0.1925 (0.0098) 0.7767 (0.0116)

E

if Fi ∼ B ( H , M ).

if Fi ∼ G ( H , M )

if Fi ∼ W ( H )

if Fi ∼ E

0.0616 (0.0098) 0.2466 (0.0177) 0.7244 (0.0202) 0.6799

W

Yen/Dollar

0.0792 (0.0129) 0.3167 (0.0260) 0.6634 (0.0266) 0.4710

G

0.0668 (0.0109) 0.2767 (0.0221) 0.6997 (0.0233) 0.7095

B

Table 6.2 Parameter estimates of ACD models for Euro/Dollar and Yen/Dollar price durations This table reports maximum likelihood estimates of linear, logarithmic, Box–Cox, Exponential, Threshold and Markov Switching models for Euro/Dollar and Yen/Dollar exchange rate price durations. For each model, four commonly used innovation distributions – exponential, Weibull, generalized Gamma and Burr distributions denoted by ‘E, W, G’ and ‘B’ respectively are considered. The whole sample are deseasonalized price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/ Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation. The numbers in the parentheses are standard errors for the estimates. The density function for innovation distribution is as follows:

–8943.24

E (0.0186) 5.2979 (0.7491) –6292.28

(0.0047)

–6452.94

G

W

Euro/Dollar

0.1334 (0.0057) 0.1150 (0.0046) 0.7749 (0.0118)

E

Log–9050.77 likelihood

λ

γ

β

α

ω

Parameter

–6480.52

0.1712 (0.0099) 0.1512 (0.0079) 0.7266 (0.0184) 0.6208 (0.0047)

W

Euro/Dollar

Panel B: Log ACD model (LOGACD)

0.2652 (0.0159) 0.1983 (0.0093) 0.6596 (0.0201) 0.2207 (0.0181) 6.6539 (1.0456) –6289.94

G 0.2379 (0.0158) 0.1876 (0.0095) 0.6823 (0.0196) 0.7060 (0.0104) 0.2300 (0.0260) –6422.23

B

E

0.1294 (0.0067) 0.1105 (0.0052) 0.7812 (0.0137)

E

–8310.07

–8401.26

(0.0100) 0.1963 (0.0247) –6407.41

B

X i  Z i Fi « ® The specified linear ACD model is ¬ Z i  X  BX i 1  CZ i 1 ®F ∼ i.i.d .E or W ( H ) or G ( H , M ) orB ( H , M ). ­ i

Log-likelihood

λ

Parameter

Table 6.2 (cont.)

–7145.65

0.1652 (0.0117) 0.1436 (0.0091) 0.7215 (0.0238) 0.6748 (0.0059)

W

Yen/Dollar

–7116.01

(0.0059)

W

Yen/Dollar

0.2244 (0.0166) 0.1840 (0.0112) 0.6485 (0.0275) 0.4138 (0.0245) 2.3356 (0.2456) –7098.11

G

(0.0261) 1.8703 (0.1795) –7087.59

G

0.1989 (0.0155) 0.1679 (0.0110) 0.6795 (0.0268) 0.7220 (0.0111) 0.1171 (0.0240) –7130.44

B

(0.0105) 0.0733 (0.0218) –7109.39

B

–0.2434 (0.0167) 0.2999 (0.0223) 0.9238 (0.0067) 0.5569 (0.0283)

E

Log–8859.20 likelihood

λ

γ

δ

β

α

ω

Parameter

–6405.91

–0.4107 (0.0388) 0.5090 (0.0490) 0.8967 (0.0109) 0.4402 (0.0358) 0.6262 (0.0047)

W

Euro/Dollar

Panel C: Box–Cox ACD model (BCACD)

–0.6600 (0.0689) 0.8598 (0.0831) 0.8659 (0.0125) 0.3378 (0.0325) 0.2423 (0.0185) 5.6292 (0.8158) –6236.09

G –0.5636 (0.0557) 0.7251 (0.0700) 0.8857 (0.0113) 0.3768 (0.0324) 0.7045 (0.0102) 0.2089 (0.0249) –6354.23

B

X i  Z i Fi « ® ln Z i  X  B ln X i 1  C ln Z i 1 The specified log ACD model is ¬ ®F ∼ i.i.d .E or W ( H ) or G ( H , M ) or B( H , M ). ­ i

–8241.34

–0.2611 (0.0169) 0.3277 (0.0228) 0.9198 (0.0070) 0.5604 (0.0310)

E

–7073.12

–0.3602 (0.0343) 0.4536 (0.0446) 0.9008 (0.0120) 0.4910 (0.0406) 0.6828 (0.0060)

W

Yen/Dollar

–0.4891 (0.0541) 0.6284 (0.0687) 0.8764 (0.0148) 0.4224 (0.0399) 0.4563 (0.0258) 1.9959 (0.1967) –7039.38

G

–0.4222 (0.0446) 0.5372 (0.0575) 0.8909 (0.0133) 0.4572 (0.0402) 0.7192 (0.0108) 0.0894 (0.0224) –7063.47

B

Loglikelihood

λ

γ

δ

β

α

ω

Parameter

–8867.44

–0.0806 (0.0052) 0.2061 (0.0107) 0.9149 (0.0074) –0.1309 (0.0112)

E

–6413.34

–0.0876 (0.0087) 0.2920 (0.0068) 0.8840 (0.0092) –0.2154 (0.0024) 0.6261 (0.0047)

W

Euro/Dollar

–0.0531 (0.0118) 0.4138 (0.0204) 0.8444 (0.0135) –0.3363 (0.0228) 0.2463 (0.0181) 5.4455 (0.7586) –6248.41

G –0.0681 (0.0107) 0.3752 (0.0207) 0.8674 (0.0126) –0.2972 (0.0225) 0.7003 (0.0100) 0.2010 (0.0241) –6366.08

B –0.0889 (0.0062) 0.2208 (0.0100) 0.9170 (0.0072) –0.1321 (0.0109)

E

–8237.14

X i  Z i Fi « ® ln Z i  X  BFiE 1  CZ i 1 The specified Box–Cox ACD model is ¬ ®F ∼ i.i.d .E or W ( H ) or G ( H , M ) orB ( H , M ). ­ i Panel D: Exponential ACD model (EXPACD)

Table 6.2 (cont.)

–7070.80

–0.0965 (0.0097) 0.2773 (0.0167) 0.8956 (0.0122) –0.1830 (0.0183) 0.6829 (0.0060)

W

Yen/Dollar

–0.0836 (0.0227) 0.3461 (0.0234) 0.8744 (0.0140) –0.2438 (0.0234) 0.4462 (0.0183) 2.0764 (0.1497) –7038.39

G

–0.0964 (0.0103) 0.3088 (0.0196) 0.8866 (0.0133) –0.2114 (0.0211) 0.7180 (0.0106) 0.0861 (0.0219) –7061.67

B

0.0184 (0.0039) 0.2783 (0.0207) 0.8069 (0.0128) 0.4013 (0.0414) 0.0986 (0.0105) 0.6268 (0.0341)

E

Log–8804.91 likelihood

λ2

λ1

γ2

γ1

β2

α2

ω2

β1

α1

ω1

Parameter

–6304.44

0.0364 (0.0080) 0.2134 (0.0877) 0.7376 (0.0210) 0.2450 (0.0272) 0.1288 (0.0139) 0.7280 (0.0276) 0.5734 (0.0060) 0.7001 (0.0076)

W

Euro/Dollar

4.3446 (0.5833) 5.3646 (0.7043) –6195.43

0.0478 (0.0090) 0.6391 (0.0692) 0.6438 (0.0230) 0.4562 (0.0582) 0.1251 (0.0206) 0.6964 (0.0412) 0.2684 (0.0190)

G

0.3559 (0.0235) 0.0857 (0.0217) –6292.29

–0.3070 (0.1024) –0.0181 (0.0253) 0.8893 (0.0234) 0.1601 (0.0124) 0.2227 (0.0154) 0.6725 (0.0258) 0.6971 (0.0092)

B 0.0171 (0.0037) 0.2786 (0.0191) 0.7940 (0.0099) 0.6003 (0.0566) 0.0771 (0.0116) 0.6059 (0.0400)

E

–8186.21

X i  Z i Fi « ® The specified exponential ACD model is ¬ ln Z i  X  BFi 1  E Fi 1 1  C ln Z i 1 ®F ∼ i.i.d .E or W ( H ) or G ( H , M ) orB ( H , M ). ­ i Panel E: Threshold ACD model (TACD)

–6950.79

0.0129 (0.0090) 0.3846 (0.0951) 0.7678 (0.0179) 0.2711 (0.0334) 0.1387 (0.0151) 0.7008 (0.0307) 0.5989 (0.0077) 0.7785 (0.0091)

W

Yen/Dollar

1.4781 (0.1352) 2.1655 (0.1981) –6948.84

0.0187 (0.0106) 0.4425 (0.1048) 0.7390 (0.0216) 0.2974 (0.0375) 0.1527 (0.0177) 0.6830 (0.0338) 0.4867 (0.0255)

G

0.4269 (0.0278) 0.0102 (0.0187) –6944.52

(0.0297) (0.0448) 0.0375 (0.0138) 0.9158 (0.0165) 0.1311 (0.0143) 0.2100 (0.0189) 0.6504 (0.0339) 0.7564 (0.0108)

B

Table 6.2 (cont.)

p22

p11

Parameter

W

0.5752 (0.0244) 0.8600 (0.0086)

E

0.6116 (0.0129) 0.7514 (0.0096)

Euro/Dollar

Panel F: Regime-switching ACD model (RSACD)

0.6558 (0.0403) 0.3544 (0.0380)

G

0.7140 (0.0120) 0.7552 (0.0100)

B

0.4443 (0.0161) 0.5827 (0.0285)

E

0.6140 (0.0333) 0.9008 (0.0084)

W

Yen/Dollar

0.6123 (0.0409) 0.3528 (0.0285)

G

0.6867 (0.0165) 0.8493 (0.0095)

B

Estimation is performed by a grid search over threshold values r and by maximizing the likelihood function conditional on given r . The first threshold values are for Euro/Dollar and threshold values in parentheses are for Yen/Dollar.

®« X1  B1 ln X i 1  C1 ln Z i 1 , Fi ∼ B  H 1 , M1 if X i 1 b 0.074  0.278 X i  Z i Fi , ln Z i  ¬ . ®­X 2  B 2 ln X i 1  C2 ln Z i 1 , Fi ∼ B  H 1 , M 2 if X i 1  0.074  0.278

with Burr error is

®« X1  B1X i 1  C1Z i 1 , Fi ∼ G  H 1 , M1 if X i 1 b 0.755  0.377 X i  Z i Fi , Z i  ¬ ; ®­X 2  B 2 X i 1  C2 Z i 1 , Fi ∼ G  H 1 , M 2 if X i 1  0.755  0.377

with generalized gamma error is

®« X1  B1X i 1  C1Z i 1 , Fi ∼ W  H 1 if X i 1 b 0.309  0.377 X i  Z i Fi , Z i  ¬ ; ­®X 2  B 2 X i 1  C2 Z i 1 , Fi ∼ W  H 2 if X i 1  0.309  0.377

with Weibull error is

®« X1  B1X i 1  C1Z i 1 , Fi ∼ E if X i 1 b 1.217 1.478 X i  Z i Fi , Z i  ¬ ; ­®X 2  B 2 X i 1  C2 Z i 1 , Fi ∼ E if X i 1  1.217 1.478

The 2-regime threshold ACD model with standard exponential error is

–2.4913 (0.0594) 0.0348 (0.0176) 0.5091 (0.0623) 0.5118 (0.0123) 0.0498 (0.0072) 0.5343 (0.0334)

E

W

–6027.42

–3.1667 (0.0876) –0.0367 (0.0212) 0.2541 (0.0632) 0.3491 (0.0138) 0.0792 (0.0085) 0.5789 (0.0362) 1.1374 (0.0360) 0.7616 (0.0105)

G

1.9161 (0.2078) 2.8640 (0.4258) –6123.99

0.2437 (0.0347) 0.0165 (0.0119) 0.9164 (0.0234) –0.2970 (0.0726) 0.6630 (0.0340) 0.2824 (0.0677) 0.4534 (0.0306) 1.2917 (0.0280) 0.4644 (0.0393) –5797.49

0.1061 (0.0502) 0.0176 (0.0108) 0.9090 (0.0253) 0.0596 (0.0241) 0.0601 (0.0076) 0.9226 (0.0197) 1.3211 (0.0276)

B

with constant transition probability p11  P Si  1 | Si 1  1 , p22  P Si  2 | Si 1  2 .

« X i  Z i Fi ® S ¬ ln Z i  i  X Si  B Si ln X i 1  C Si ln Z i 1 , Si  1, 2 ®F Si  X / Z Si ~ E or W ( H ) or G ( H , M ) or B ( H , M ). i i st 1 st 1 st ­ i

The 2-regime-switching ACD model is

Log–6181.97 likelihood

λ2

λ1

γ2

γ1

β2

α2

ω2

β1

α1

ω1

Parameter

Euro/Dollar

–7001.39

–1.1389 (0.0828) 0.4859 (0.0296) 0.5476 (0.0812) 0.5344 (0.0210) 0.0623 (0.0095) 0.8168 (0.0294)

E

–6935.41

–2.5765 (0.1175) 0.0881 (0.0342) 0.0346 (0.0891) 0.3656 (0.0192) 0.0911 (0.0095) 0.4611 (0.0480) 1.0298 (0.0443) 0.8076 (0.0126)

W

Yen/Dollar

1.2412 (0.1212) 1.4539 (0.1248) –6864.71

0.2345 (0.0315) –0.0004 (0.0099) 0.9702 (0.0221) –0.2973 (0.0586) 0.6979 (0.0257) 0.0796 (0.0654) 0.6750 (0.0351)

G

0.9874 (0.0561) 0.3593 (0.0365) –6761.33

0.0821 (0.0637) 0.0119 (0.0135) 0.9733 (0.0333) –0.1517 (0.0327) 0.0765 (0.0078) 0.8949 (0.0172) 1.1860 (0.0308)

B

W(10) 5081.27 372.83 191.76 299.53 5309.79 425.80 198.33 333.36 5037.01 395.77 187.74 318.05 5004.18 385.01 203.58 314.37 4860.84 264.49 177.77 271.68 113.73 82.47

W(5)

3813.51 278.63 147.68 224.73 3933.91 319.33 155.43 252.51 3638.72 287.89 145.29 234.31 3734.41 287.13 157.18 235.82 3643.36 198.24 136.93 202.55 88.07 63.78

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E EXPACD-W EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W

Euro/Dollar

6923.74 503.40 251.17 402.06 7322.43 577.57 258.70 449.23 6930.56 547.85 245.00 440.41 6849.53 522.15 267.08 423.49 6645.82 356.30 233.75 368.56 150.30 108.33

W(20) 1495.65 86.24 80.78 86.20 1555.12 92.81 75.24 87.97 1428.47 81.79 73.95 80.69 1432.69 84.36 78.90 84.47 1259.82 59.35 57.79 62.63 84.27 26.14

W(5)

Yen/Dollar

1960.61 103.31 96.29 103.52 2051.11 109.19 87.45 103.09 1883.91 97.66 88.11 96.52 1889.77 100.75 94.26 100.98 1744.29 72.65 70.61 80.92 112.54 33.27

W(10)

2663.42 130.88 121.90 131.20 2812.81 139.41 109.84 130.75 2573.63 124.98 113.30 123.89 2581.31 129.74 121.42 130.09 2427.03 95.88 93.56 107.41 156.54 42.13

W(20)

Table 6.3 Nonparametric portmanteau density evaluation statistics for in-sample and out-of-sample performance of ACD models Panel A: in-sample performance

113.60 29.47

149.06 36.34

196.24 44.78

35.38 16.99

44.09 20.84

58.92 27.15

W(10) 4039.90 312.96 221.57 275.83 4211.61 332.78 206.67 271.68 3984.67 305.99 210.12 262.46 4021.55

W(5)

3049.10 235.98 169.83 209.22 3145.42 255.00 165.90 212.47 2998.12 231.68 163.13 201.23 3025.14

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E

Euro/Dollar

Panel B: out-of-sample performance

5567.52 425.11 298.76 375.10 5851.44 451.05 273.05 365.42 5509.08 416.97 281.21 355.50 5556.89

W(20) 1447.52 96.72 83.75 90.71 1708.47 133.64 97.95 112.32 1433.75 102.11 84.58 92.46 1451.46

W(5)

Yen/Dollar

1901.04 117.18 99.38 108.60 2276.17 168.80 120.68 140.37 1895.51 126.22 103.32 113.38 1918.08

W(10)

2539.42 150.61 125.43 138.25 3089.67 224.70 157.02 184.64 2555.39 164.81 132.53 146.07 2586.03

W(20)

Note: The table reports the evaluation statistics W  p for the in-sample density forecasting performance of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr error distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation. The W  p statistics are asymptotically one sided N ( 0,1) distribution and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

RSACD-G RSACD-B

316.55 227.04 277.35 3973.34 229.52 204.62 241.43 156.22 103.70 175.02 39.68

239.88 175.90 212.56 2995.36 176.27 157.86 183.80 121.13 81.16 132.12 33.47

EXPACD-W EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W RSACD-G RSACD-B

431.20 305.39 375.33 5485.29 312.14 274.33 327.71 207.69 136.47 239.88 49.50

W(20) 105.70 90.55 97.06 1294.17 102.59 78.05 83.63 106.29 146.46 52.08 34.58

W(5)

Yen/Dollar

130.70 110.11 118.44 1775.72 135.81 99.67 107.79 144.92 200.07 67.85 42.49

W(10)

171.59 142.17 153.82 2499.07 185.91 133.66 145.58 201.26 276.79 91.41 54.61

W(20)

Note: The table reports the evaluation statistics W  p for the out-of-sample density forecasting performance of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr error distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. The W  p statistics are asymptotically one sided N ( 0,1) distribution and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

W(10)

W(5)

Euro/Dollar

Model

Table 6.3 (cont.)

Autoregressive duration models

131

BCACD, EXPACD and even TACD models in some cases perform rather similarly, although the TACD models are more sophisticated. This implies that nonlinear ACD models for Z i do not always outperform the LINACD model. In other words, a linear model for Z i performs as well as commonly used nonlinear ACD models for Z i in capturing the full dynamics of price durations of foreign exchanges. Among four innovation distributions, the exponential distribution always fits poorly while the generalized Gamma distribution performs best (the Burr innovation performs best for the MSACD model). For the ACD models with the exponential innovations (except the MSACD model), W (5 ) statistics are extremely large – over 3,000 for Euro/ Dollar and over 1,000 for Yen/Dollar. W (10 ) and W ( 20 ) tell the same story as W (5 ) . The W ( p ) statistics imply that sophisticated specifications of the conditional mean duration Z i do not help much in capturing the full dynamics of price durations. However, the specification of the innovation distribution is important: either the generalized Gamma or Burr distribution always performs better. Moreover, it is rather important to relax the i.i.d. assumption for the innovation and consider higher order conditional moments of X i . We find that the relative rankings among all ACD models are generally similar for two foreign exchange rates. The W ( p ) statistics for Yen/Dollar are only about half that of Euro/Dollar, indicating that these models can better at describing the price duration dynamics of Yen/Dollar. This may be due to the fact that Euro/ Dollar is more active than Yen/Dollar and its price durations display more time-varying clustering and dispersion. As a consequence, it is more difficult to capture the price duration dynamics for Euro/Dollar than for Yen/Dollar. Below, we investigate possible sources for the failure of ACD models by separately examining the uniform distribution and the i.i.d. properties of the generalized residuals of each model. Figures 6.9 and 6.10 display the histogram of the generalized residuals. Consistent with the W ( p ) statistics, we find that the marginal density of the generalized residuals of MSACD models is much closer to the uniform distribution than other ACD models. In particular, the distribution of the generalized residuals of the RSACD model with the Burr innovations is the closest to the uniform distribution. Given the same innovation distribution, the generalized residuals of LINACD, LOGACD, BCACD, EXPACD and TACD models have similar marginal distributional shapes. The density estimates of the generalized residuals of all ACD models (except RSACD) with the exponential innovation always exhibits a U shape, with pronounced peaks at both ends, especially at the left end. This indicates that the exponential innovation distribution underpredicts the tails of price durations, particularly extremely short price durations. In contrast, the generalized residuals of the ACD models with Weibull, generalized Gamma and Burr distributions usually show a similar †-shape density, implying that these models turn to overpredict ultra-long and ultra-short price durations. Both Euro/Dollar and Yen/Dollar ACD models often tell the same story. We now examine the performance of each model in capturing various specific dynamics of price durations. First, we check a pattern of serial

132

Autoregressive duration models

5

5

E W G B

4.5 4

4

LACD-E

3.5

3.5

3 LACD-B

LOGACD-E

3

LACD-W

2.5 2

E W G B

4.5

LOGACD-W

2.5

LACD-G

LOGACD-G

2

1.5

LOGACD-B

1.5

1

1

0.5

0.5

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

Linear ACD 5

0.8

5

E W G B

4.5 4 3.5

0.6

1

Log ACD

E W G B

4.5 4

EXACD-E

3.5

BCACD-E

3

3

BCACD-W

2.5

EXACD-W

2.5 BCACD-B

2

BCACD-G

2

1.5

EXACD-B

EXACD-G

0.2

0.6

1.5

1

1

0.5

0.5

0

0 0

0.2

0.4

0.6

0.8

0

1

0.4

0.8

1

Exponential ACD

Box–Cox ACD 5

1.6

E W G B

4.5 4

E W G B

1.4 1.2

3.5

TACD-E 1

TACD-W

3 2.5

0.8

TACD-G TACD-B

2

0.6

1.5

RSACD-W

0.4

1

RSACD-G RSACD-B RSACD-E

0.2

0.5

0

0 0

0.2

0.4

0.6

Threshold ACD

0.8

1

0

0.2

0.4

0.6

0.8

1

Regime-switching ACD

Figure 6.9 Histogram of in-sample generalized residuals of Euro/Dollar

dependence of the generalized residuals {Zi (RR )} of each ACD model for two exchange rates. Panels A and B of Table 6.4 report the diagnostic tests M z ( m, l ) for the in-sample generalized residuals. Almost all M z ( m, l ) statistics are rather significant at the 5 per cent level and most of them are significant at the 1 per cent level, indicating that there exists neglected dynamic structure in price durations for all ACD models. With the same innovation distribution, LOGACD, BCACD and EXPACD often exhibit similar performances

133

Autoregressive duration models 4.5

4.5

E W G B

4 3.5

E W G B

4 3.5

LACD-E

3

LOGACD-E

3

2.5 LACD-G

LACD-B

2

2.5

LACD-W

2

1.5

LOGACD-W LOGACD-G

LOGACD-B

1.5

1

1

0.5

0.5

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

4.5

0.8

4.5

E W G B

4

1

E W G B

4

3.5

3.5 BCACD-E

3

0.6

Log ACD

Linear ACD

EXACD-E

3

2.5

2.5

2

BCACD-G

BCACD-W

BCACD-B

2

1.5

EXACD-W

EXACD-B EXACD-G

1.5

1

1

0.5

0.5

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

4.5

0.8

1.8

E W G B

4 3.5

0.6

1

Exponential ACD

Box–Cox ACD

TACD-E

RSACD-W

1.4

3

E W G B

RSACD-G

1.6

RSACD-E

RSACD-B 1.2

2.5 2

TACD-W

TACD-B

TACD-G

1

1.5

0.8

1 0.6

0.5 0

0.4 0

0.2

0.4

0.6

Threshold ACD

0.8

1

0

0.2

0.4

0.6

0.8

1

Regime-switching ACD

Figure 6.10 Histogram of in-sample generalized residuals of Yen/Dollar

and perform a little better than LINACD. Weak form ACD models perform better than strong form ACD models. Among four innovation distributions, the exponential distribution always delivers a larger M z ( m, l ), with M z ( 0, 0 ) around 100. This may be due to the fact that the exponential distribution cannot capture overdispersion well in price durations. The generalized Gamma and Burr distributions always perform better in capturing price duration dynamics. In particular, the generalized Gamma distribution performs best in most cases. The models – TACD models with Burr innovations, RSACD models with generalized Gamma innovations and RSACD models with Burr

M z (1, 0 )

111.90 70.18 29.42 39.85 105.00 54.93 19.70 25.98 92.02 49.36 16.54 23.36 91.86 54.48 21.36 27.67 95.81 22.84 13.62 8.07 44.23 47.42 9.73 13.57

M z ( 0, 0 )

112.20 70.50 29.73 40.14 105.40 55.25 19.96 26.23 92.35 49.68 16.80 23.63 92.18 54.79 21.65 27.95 96.15 22.99 13.79 8.18 44.25 47.48 9.88 13.60

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E EXPACD-W EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W RSACD-G RSACD-B

90.56 35.69 34.47 37.62 84.87 23.12 23.21 24.93 90.25 34.40 32.70 35.85 88.96 32.20 30.66 34.01 91.54 6.13 6.31 7.97 3.27 9.41 22.73 1.53

M z ( 2, 0 ) 103.50 34.96 9.85 18.59 88.63 22.08 10.56 12.98 82.31 21.81 9.52 13.44 82.88 23.73 7.11 12.39 86.40 9.66 6.40 6.25 33.24 32.60 6.50 8.83

M z (3, 0 ) 85.35 23.74 25.93 27.49 76.11 13.93 18.13 17.63 84.21 23.22 25.94 26.83 82.66 21.02 23.60 24.51 85.73 22.75 2.40 1.98 4.95 7.51 30.15 3.73

M z ( 4, 0 ) 111.80 70.04 29.30 39.74 105.00 54.81 19.57 25.86 91.94 49.28 16.49 23.32 91.83 54.44 21.34 27.66 95.73 22.81 13.60 8.02 44.14 47.29 9.62 13.39

M z (1,1) 39.30 16.43 16.77 17.69 1.59 24.08 30.98 28.96 22.37 5.73 5.51 5.17 17.83 5.89 6.21 5.91 27.35 8.22 5.39 19.77 45.14 56.41 28.13 43.86

M z (1, 2 ) 90.50 35.71 34.52 37.66 84.74 23.04 23.14 24.86 90.19 34.40 32.70 35.86 88.87 32.18 30.65 34.00 91.48 6.05 6.24 7.94 3.18 9.30 22.72 1.41

M z ( 2,1) 43.43 12.57 11.12 12.47 52.30 26.60 25.60 26.67 43.28 15.81 15.86 16.40 45.49 17.54 16.23 17.54 43.34 19.72 24.89 14.80 20.04 19.47 18.59 27.54

M z ( 2, 2 )

Table 6.4 Separate diagnostic statistics for in-sample and out-of-sample generalized residuals of ACD models Panel A: in-sample generalized residuals of Euro/Dollar

96.57 24.65 5.59 12.47 76.74 15.42 6.97 8.78 75.31 14.81 6.42 8.97 80.34 19.48 4.30 9.19 82.16 7.80 3.38 2.67 33.82 32.27 4.89 8.44

M z (3, 3)

41.10 12.92 13.75 14.51 47.66 27.69 28.01 28.83 41.16 16.32 18.13 18.32 43.35 18.10 18.92 19.89 42.44 20.07 25.99 16.20 20.18 19.31 21.15 29.97

M z ( 4, 4 )

M z (1, 0 )

78.43 70.77 49.09 59.76 88.83 63.49 36.28 45.34 62.56 52.29 32.54 40.63 61.77 53.13

M z ( 0, 0 )

78.69 71.10 49.45 60.10 89.10 63.84 36.67 45.71 62.85 52.64 32.93 40.99 62.06 53.48

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E EXPACD-W

93.45 72.49 78.25 75.31 88.26 62.21 71.33 68.08 94.59 76.53 82.95 80.27 94.34 77.36

M z ( 2, 0 ) 84.42 52.68 33.69 43.81 89.07 38.35 17.23 24.70 67.74 35.84 19.89 27.06 67.04 36.63

M z (3, 0 )

Panel B: in-sample generalized residuals of Yen/Dollar

94.01 62.99 69.50 66.29 83.92 50.83 62.05 57.60 94.19 64.92 72.38 69.21 94.31 65.78

M z ( 4, 0 ) 78.27 70.60 48.92 59.60 88.76 63.38 36.18 45.24 62.45 52.19 32.46 40.54 61.71 53.07

M z (1,1) 38.52 16.43 17.19 16.83 7.35 10.80 12.14 11.74 22.27 4.68 2.80 3.81 17.59 4.23

M z (1, 2 )

93.41 72.48 78.25 75.30 88.12 62.06 71.18 67.93 94.54 76.49 82.90 80.23 94.28 77.29

M z ( 2,1)

37.04 12.74 11.49 12.40 57.36 36.59 34.11 35.72 39.96 17.76 17.94 18.18 43.15 20.41

M z ( 2, 2 )

77.48 36.05 21.71 29.68 79.72 26.64 10.37 16.18 61.73 24.00 12.57 17.89 65.28 28.28

M z (3, 3)

40.84 12.15 11.62 12.20 61.35 30.74 29.25 30.55 43.24 14.98 15.58 15.64 46.77 17.19

M z ( 4, 4 )

Note: This table reports M z ( m, l ) diagnostic test statistics for the serial dependence of generalized residuals [Zi ] of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. M z ( 0, 0 ) represents statistics on i.i.d. test. M z (l , 0 ), l  1, 2, 3, 4 represent test statistics on martingale, ARCH effect, conditional skewness and conditional heterokurtosis respectively. And M z (1,1), M z ( 2, 2 ), M z (3, 3) , M z ( 4, 4 ) are very sensitive to autocorrelations in mean, variance, skewness and kurtosis of the generalized residuals respectively. We only show results for preliminary lag truncation order p  20 , the results for other lag order is similar. The M z ( m, l ) tests are asymptotically one sided N ( 0,1) test and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

36.58 42.63 65.20 22.24 24.36 10.23 17.95 40.34 15.55 4.23

36.97 42.99 65.46 22.28 24.45 10.36 18.06 40.37 15.61 4.27

EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W RSACD-G RSACD-B

83.77 81.14 92.91 1.93 6.43 30.49 13.62 12.00 7.59 10.74

M z ( 2, 0 ) 22.93 28.62 72.84 15.15 13.90 6.93 9.41 32.88 9.04 4.29

M z (3, 0 ) 73.04 69.96 91.34 2.00 0.39 12.47 14.32 9.09 10.91 9.57

M z ( 4, 0 ) 36.53 42.58 65.11 22.21 24.32 10.23 17.88 40.27 15.51 4.15

M z (1,1) 4.05 3.99 22.64 3.38 3.19 1.57 8.32 28.28 3.94 21.67

M z (1, 2 ) 83.70 81.07 92.87 1.91 6.42 30.47 13.63 11.92 7.56 10.70

M z ( 2,1) 19.57 20.41 41.12 10.90 12.39 12.57 11.19 14.64 16.52 10.63

M z ( 2, 2 ) 17.44 22.30 70.19 12.43 11.46 6.81 6.35 30.20 6.63 5.38

M z (3, 3) 17.17 17.61 45.50 9.20 10.13 10.83 10.24 13.00 18.62 11.41

M z ( 4, 4 )

M z (1, 0 )

47.19 38.48 20.21

M z ( 0, 0 )

47.59 38.87 20.60

Model

LINACD-E LINACD-W LINACD-G

123.30 93.48 91.52

M z ( 2, 0 ) 55.82 25.56 14.21

M z (3, 0 )

Panel C: out-of-sample generalized residuals of Euro/Dollar

119.30 75.59 78.06

M z ( 4, 0 )

47.09 38.38 20.11

M z (1,1)

34.36 22.02 24.31

M z (1, 2 )

123.20 93.47 91.52

M z ( 2,1)

56.38 10.92 6.90

M z ( 2, 2 )

56.08 21.73 13.89

M z (3, 3)

55.38 13.62 12.75

M z ( 4, 4 )

Note: This table reports M z ( m, l ) diagnostic test statistics for the serial dependence of generalized residuals [Zi ] of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. M z ( 0, 0 ) represents statistics on i.i.d. test. M z (l , 0 ), l  1, 2, 3, 4 represent test statistics on martingale, ARCH effect, conditional skewness and conditional heterokurtosis respectively. And M z (1,1), M z ( 2, 2 ), M z (3, 3), M z ( 4, 4 ) are very sensitive to autocorrelations in mean, variance, skewness and kurtosis of the generalized residuals respectively. We only show results for preliminary lag truncation order p  20 , the results for other lag order is similar. The M z ( m, l ) tests are asymptotically one sided N ( 0,1) test and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

M z (1, 0 )

M z ( 0, 0 )

Model

Table 6.4 (cont.)

23.28 46.96 32.46 18.36 20.13 35.58 27.26 14.76 16.44 35.85 30.10 17.05 18.04 38.61 12.57 12.68 10.30 40.79 42.37 13.52 24.63

22.92 46.55 32.11 18.06 19.84 35.15 26.88 14.41 16.11 35.41 29.71 16.68 17.68 38.17 12.51 12.49 10.15 40.67 42.20 13.47 24.45

94.11 120.40 83.40 81.77 83.88 125.10 97.34 94.42 97.58 125.20 97.05 95.07 98.42 127.10 12.56 50.14 52.08 35.69 57.36 11.22 36.86

18.01 48.00 24.10 27.01 25.91 41.99 21.51 22.01 22.13 41.62 21.10 17.09 18.88 43.62 12.94 18.23 16.25 34.25 35.76 16.46 15.14

78.68 115.40 65.12 68.91 68.27 121.30 78.30 80.60 81.23 121.30 77.96 80.73 81.45 124.80 5.62 31.63 30.64 18.62 50.93 11.21 30.51

22.83 46.51 32.03 17.98 19.76 35.10 26.84 14.40 16.10 35.41 29.70 16.67 17.68 38.13 12.50 12.47 10.13 40.63 42.15 13.44 24.35

23.49 11.42 20.25 23.15 23.41 20.40 8.28 5.84 6.45 16.03 8.01 8.60 7.80 23.66 9.56 10.00 11.66 30.35 36.79 17.43 24.94

94.10 120.20 83.24 81.62 83.73 125.00 97.28 94.36 97.52 125.10 96.95 94.97 98.33 127.10 12.54 50.09 52.03 35.57 57.21 11.14 36.82

9.19 81.18 37.34 30.31 34.13 63.99 18.64 15.51 17.62 67.53 20.84 14.85 18.64 65.88 14.78 22.31 12.84 28.98 31.97 23.29 25.49

17.00 44.34 20.70 23.60 23.17 42.95 19.22 20.49 20.63 45.87 20.31 14.55 16.99 46.97 12.83 15.30 13.85 36.25 35.81 18.49 16.20

13.97 77.00 35.70 33.39 34.90 62.02 20.37 20.90 21.35 65.03 21.91 20.30 22.19 66.24 13.80 27.02 13.18 25.63 33.61 26.80 27.29

Note: This table reports M z ( m, l ) diagnostic test statistics for the serial dependence of generalized residuals [Zi ] of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. M z ( 0, 0 ) represents statistics on i.i.d. test. M z (l , 0 ), l  1, 2, 3, 4 represent test statistics on martingale, ARCH effect, conditional skewness and conditional heterokurtosis respectively. And M z (1,1), M z ( 2, 2 ) , M z (3, 3), M z ( 4, 4 ) are very sensitive to autocorrelations in mean, variance, skewness and kurtosis of the generalized residuals respectively. We only show results for preliminary lag truncation order p  20, the results for other lag order is similar. The M z ( m, l ) tests are asymptotically one sided N ( 0,1) test and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E EXPACD-W EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W RSACD-G RSACD-B

M z (1, 0 )

70.01 67.57 47.42 56.88 72.27 51.46 26.91 34.75 53.45 47.49 29.02 36.31 52.94 49.18 33.75 39.06 40.74 10.43 12.58 3.78

M z ( 0, 0 )

70.29 67.90 47.79 57.23 72.53 51.80 27.28 35.11 53.74 47.83 29.40 36.66 53.24 49.53 34.13 39.42 41.06 10.44 12.63 3.91

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E EXPACD-W EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B

105.10 79.29 83.62 80.93 98.09 74.76 80.94 78.22 100.20 81.17 85.59 83.17 100.50 83.32 87.76 85.53 97.96 2.51 7.97 35.36

M z ( 2, 0 ) 81.70 52.02 35.30 43.80 83.60 37.43 18.43 24.86 63.77 37.00 24.12 29.76 63.67 38.55 27.26 31.76 51.75 12.23 13.13 5.52

M z (3, 0 )

Table 6.4 (cont.) Panel D: out-of-sample generalized residuals of Yen/Dollar

101.00 65.41 70.55 67.60 93.81 59.54 66.06 63.35 96.56 66.98 71.83 69.51 97.11 68.90 73.80 71.54 95.79 -0.37 -0.25 15.63

M z ( 4, 0 ) 69.90 67.42 47.27 56.74 72.27 51.38 26.82 34.66 53.40 47.42 28.96 36.25 52.95 49.15 33.72 39.04 40.71 10.43 12.57 3.78

M z (1,1) 40.51 22.56 23.81 22.86 8.72 10.24 13.10 12.33 22.21 7.77 5.22 6.36 18.52 8.28 8.03 7.75 24.22 5.39 5.70 4.26

M z (1, 2 ) 105.00 79.26 83.61 80.91 97.96 74.61 80.82 78.10 100.20 81.11 85.54 83.11 100.40 83.24 87.68 85.45 97.88 2.47 7.94 35.33

M z ( 2,1) 55.51 20.49 17.84 19.45 72.34 46.02 42.04 44.05 59.44 28.37 26.73 27.74 62.59 30.36 27.47 29.12 64.03 15.83 17.34 16.55

M z ( 2, 2 ) 83.87 40.35 25.97 33.52 78.73 26.15 11.00 16.10 66.05 28.87 17.01 22.36 71.57 35.55 24.38 29.14 58.31 9.86 10.11 4.95

M z (3, 3)

57.26 16.10 15.06 15.74 72.14 33.74 31.21 32.89 61.59 21.39 20.70 21.31 64.80 23.36 22.06 22.91 66.80 10.43 11.17 10.91

M z ( 4, 4 )

7.95 13.30 6.37 4.55

7.87 13.21 6.32 4.48

9.85 29.13 6.76 13.65

5.03 9.99 5.07 2.58

9.52 19.67 8.62 11.81

7.82 13.14 6.29 4.37

13.15 30.65 8.76 31.14

9.84 29.09 6.71 13.62

12.36 19.63 21.71 12.28

4.97 11.18 5.05 4.04

10.62 14.01 25.20 9.87

Note: This table reports M z ( m, l ) diagnostic test statistics for the serial dependence of generalized residuals [Zi ] of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. M z ( 0, 0 ) represents statistics on i.i.d. test. M z (l , 0 ), l  1, 2, 3, 4 represent test statistics on martingale, ARCH effect, conditional skewness and conditional heterokurtosis respectively. And M z (1,1), M z ( 2, 2 ), M z (3, 3) , M z ( 4, 4 ) are very sensitive to autocorrelations in mean, variance, skewness and kurtosis of the generalized residuals respectively. We only show results for preliminary lag truncation order p  20 , the results for other lag order is similar. The M z ( m, l ) tests are asymptotically one sided N ( 0,1) test and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

RSACD-E RSACD-W RSACD-G RSACD-B

140

Autoregressive duration models

innovations have relatively small M z ( m, l ) statistics, with M z ( 0, 0 ) around 10. For Euro/Dollar, the TACD model with Burr innovations delivers the smallest M z ( 0, 0 ), which is 8.18; for Yen/Dollar, the RSACD model with the Burr innovations delivers the lowest M z ( 0, 0 ), which is 4.27. These results show that relaxing the i.i.d. assumption for the innovation helps a lot in capturing the price duration dynamics. Next, we check the independence assumption for the innovations {Fi }. Table 6.5 reports the M F ( m, l ) statistics for estimated standardized model residuals {Fi  X i / Z i }. The M F ( 0, 0 ) statistics in Panels A and B of Table 6.5 are all larger than 15.0, indicating strong serial dependence in {Fi }. This apparently contradicts the independence assumption for the innovation. The M F (1, 0 ) and M F ( 2, 0 ) statistics are significant at the 5 per cent level for almost all ACD models, indicating that there exists neglected dynamic structure in both conditional mean and conditional dispersion of price durations. For Euro/Dollar, M F (3, 0 ) and M F ( 4, 0 ) test statistics are not significant (except very few cases) at the 5 per cent level. For Yen/Dollar, while the M F (3, 0 ) statistics are smaller than M F (1, 0 ) and M F ( 2, 0 ), they are significant at the 5 per cent level in most cases. In contrast, M F ( 4, 0 ) statistics are not significant at the 5 per cent level. These results imply that the price duration may have strong serial dependence in its first two conditional moments (i.e. conditional mean duration and conditional dispersion of durations), but it may have rather weak or nonexistent higher order serial dependence. Similar to Hujer et al. (2003b), our analysis shows that the MSACD models, particularly the RSACD model with Burr innovations, have the best insample performance. Diagnostic analysis shows that RSACD models better capture the stationary density of the price duration. The weak form ACD models, such as TACD and RSACD, are better than strong form ACD models in capturing serial dependence of the generalized residuals. Moreover, both the portmanteau test W ( p ) and various diagnostic tests M z ( m, l ) and M F ( m, l ) suggest that the ACD models with generalized Gamma or Burr innovations can better capture the in-sample price duration dynamics of foreign exchanges. 6.4.2

Out-of-sample density forecast performance

Next, we study out-of-sample performance of ACD models. We are interested in checking whether the ACD models that have the best in-sample performance also have the best out-of-sample performance. Using parameters estimated in Table 6.2, we obtain out-of-sample generalized residuals {Zt (R )}Tt  R 1 for Euro/Dollar and Yen/Dollar respectively. Interestingly, the out-of-sample performances of ACD models are similar to their in-sample ranking. The out-of-sample W ( p ) statistics in Panel B of Table 6.3 are significant at the 1 per cent level, indicating that none of the ACD models can adequately forecast the full dynamics of price durations. Again, the RSACD model with Burr

M F (1, 0 )

11.05 7.46 16.73 14.10 19.94 8.44 6.59 6.93 7.61 5.64 13.24 12.23 8.36 3.56 8.64 7.68 5.65 4.47 7.15 4.29 48.27 27.87 11.02

M F ( 0, 0 )

40.56 22.07 22.04 21.19 61.42 40.18 24.73 27.92 35.44 21.25 19.41 19.99 39.75 24.22 17.53 18.71 36.14 26.86 20.10 25.20 100.90 82.47 36.85

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E EXPACD-W EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W RSACD-G

1.43 3.54 6.13 4.88 1.02 3.06 5.81 5.14 1.97 4.19 6.55 5.57 1.56 3.57 5.96 5.03 2.21 4.65 5.12 2.48 5.07 3.40 2.24

M F ( 2, 0 ) –0.79 –0.37 0.20 –0.12 –0.08 0.74 1.90 1.52 –0.46 0.19 0.98 0.60 –0.41 0.26 1.12 0.73 –0.19 1.13 0.67 –0.11 2.59 2.08 –0.36

M F (3, 0 ) –1.28 –1.12 –0.90 –1.00 –0.80 –0.38 0.38 0.11 –1.12 –0.76 –0.26 –0.46 –1.05 –0.64 0.05 –0.23 –0.89 0.13 –0.28 –0.79 1.90 1.01 –1.03

M F ( 4, 0 ) 6.56 7.57 15.43 13.15 26.59 10.73 2.26 3.46 4.87 4.36 8.23 7.87 4.62 3.38 6.35 5.91 4.20 3.58 3.30 5.99 47.67 30.17 8.91

M F (1,1) –0.43 0.06 1.20 0.56 13.95 8.34 4.09 4.99 –0.50 –0.28 0.37 0.12 –0.54 0.22 1.43 1.04 0.65 1.64 –0.07 4.85 18.11 11.52 1.90

M F (1, 2 )

Table 6.5 Separate inference statistics for in-sample and out-of-sample residuals of ACD models Panel A: in-sample residuals of Euro/Dollar

0.78 2.27 3.93 3.04 0.50 0.06 1.41 1.08 1.02 2.39 4.43 3.62 1.13 2.49 5.84 4.45 0.57 2.78 3.39 1.74 2.34 1.59 0.50

M F ( 2,1) –1.25 –1.50 –1.01 –1.39 5.10 4.01 5.42 5.24 –1.45 –0.84 2.58 0.99 –1.49 0.16 7.67 4.60 –1.17 0.57 0.58 2.74 3.74 3.78 –0.57

M F ( 2, 2 ) –0.79 –1.19 –1.61 –1.46 –0.53 0.44 6.46 4.60 –1.12 –1.68 –0.60 –1.62 –1.22 –1.81 4.47 0.69 –1.01 –1.65 –1.70 –1.59 2.30 4.12 –1.26

M F (3, 3) –0.85 –0.73 –1.15 –0.98 –1.41 –1.75 2.71 0.65 –0.71 –1.21 –1.83 –1.61 –0.76 –1.45 0.29 –1.62 –0.71 –1.64 –1.60 –1.53 1.53 2.57 –0.85

M F ( 4, 4 )

127.40

167.80

RSACD-B

10.19

M F ( 2, 0 ) 0.60

M F (3, 0 ) –0.97

M F ( 4, 0 ) 123.90

M F (1,1) 32.04

M F (1, 2 ) 15.11

M F ( 2,1) 11.72

M F ( 2, 2 ) 1.28

M F (3, 3) –1.03

M F ( 4, 4 )

M F (1, 0 )

6.77 6.48 10.58 8.18 22.21 11.38 8.14 8.62 4.70 5.16 9.30

M F ( 0, 0 )

16.41 12.02 12.60 11.87 42.63 28.12 18.61 21.63 14.12 10.38 10.78

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G

2.19 5.78 8.78 7.45 3.23 4.44 6.51 5.71 3.50 7.08 9.42

M F ( 2, 0 )

Panel B: in-sample residuals of Yen/Dollar

0.58 2.32 3.08 2.91 1.61 2.36 2.86 2.69 1.77 3.03 3.19

M F (3, 0 ) –0.09 0.92 1.11 1.17 1.20 1.58 1.65 1.66 0.88 1.36 1.19

M F ( 4, 0 ) 5.23 5.98 9.54 7.57 32.67 16.14 5.28 8.73 3.91 3.65 5.77

M F (1,1) 0.99 1.68 2.07 1.95 31.36 21.85 12.49 15.97 0.31 –0.09 –0.29

M F (1, 2 )

2.19 4.68 6.35 5.69 2.96 0.79 0.38 0.39 2.24 4.26 5.38

M F ( 2,1)

–0.40 –0.15 –0.23 –0.13 8.97 4.29 0.97 2.03 –0.88 –0.81 –0.84

M F ( 2, 2 )

–1.10 –1.07 –1.03 –1.05 0.60 –0.35 –0.77 –0.65 –1.14 –1.05 –0.98

M F (3, 3)

–0.98 –0.95 –0.85 –0.92 –0.71 –0.74 –0.76 –0.75 –0.97 –0.85 –0.76

M F ( 4, 4 )

Note: This table reports M F ( m, l ) diagnostic test statistics for the serial dependence of generalized residuals [Fˆ i ] of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. M F ( 0, 0 ) represents statistics on i.i.d. test. M F (l , 0 ), l  1, 2, 3, 4 represent test statistics on martingale, ARCH effect, conditional skewness and conditional heterokurtosis respectively. And M F (1,1), M F ( 2, 2 ), M F (3, 3), M F ( 4, 4 ) are very sensitive to autocorrelations in mean, variance, skewness and kurtosis of the generalized residuals respectively. We only show results for preliminary lag truncation order p  20 , the results for other lag order is similar. The M F ( m, l ) tests are asymptotically one sided N ( 0,1) test and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

M F (1, 0 )

M F ( 0, 0 )

Model

Table 6.5 (cont.)

10.33 15.54 10.54 9.36 9.65 15.46 13.07 10.89 17.13 36.89 58.52 16.93 117.50

7.37 4.73 3.56 6.05 4.83 4.41 3.21 3.26 5.65 18.44 33.25 6.65 100.80

8.64 2.89 6.11 8.66 7.73 3.89 4.68 6.50 1.23 2.05 8.92 3.09 16.23

3.26 1.65 2.96 3.48 3.38 2.54 2.66 3.38 0.60 1.12 4.90 1.56 3.17

1.32 0.91 1.45 1.41 1.49 1.55 1.26 1.46 0.21 0.91 3.40 0.74 0.44

4.86 3.57 2.65 4.33 3.52 4.23 2.52 1.92 11.79 25.70 28.24 6.53 98.16

–0.19 –0.02 –0.11 0.13 0.01 2.92 0.76 0.18 13.44 21.56 13.61 2.24 29.62

5.08 1.72 3.54 4.89 4.41 1.34 2.28 3.34 0.27 1.61 1.07 1.54 17.85

–0.80 –0.94 –0.83 –0.81 –0.81 –0.89 –1.04 –1.03 3.72 4.82 2.33 –0.76 8.41

–1.01 –1.14 –1.08 –1.07 –1.07 –1.23 –1.27 –1.16 –0.08 –0.21 –0.27 –1.24 –0.39

–0.80 –0.97 –0.88 –0.84 –0.86 –1.09 –1.08 –0.94 –0.95 –0.82 –0.82 –1.09 –1.07

Note: This table reports M F ( m, l ) diagnostic test statistics for the serial dependence of generalized residuals [Fˆ i ] of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. M F ( 0, 0 ) represents statistics on i.i.d. test. M F (l , 0 ), l  1, 2, 3, 4 represent test statistics on martingale, ARCH effect, conditional skewness and conditional heterokurtosis respectively. And M F (1,1), M F ( 2, 2 ), M F (3, 3), M F ( 4, 4 ) are very sensitive to autocorrelations in mean, variance, skewness and kurtosis of the generalized residuals respectively. We only show results for preliminary lag truncation order p  20 , the results for other lag order is similar. The M F ( m, l ) tests are asymptotically one sided N ( 0,1) test and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

BCACD-B EXPACD-E EXPACD-W EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W RSACD-G RSACD-B

M F (1, 0 )

6.54 19.86 39.34 34.15 12.54 13.73 19.74 18.82 6.36 18.68 33.45 31.28 5.05 13.28 26.45 23.51 4.21 11.14 22.08 10.98 47.10 31.58 7.97 81.67

M F ( 0, 0 )

13.45 19.77 36.66 31.15 25.27 21.62 22.00 22.12 12.24 18.38 29.83 27.77 12.29 15.25 24.48 21.86 11.83 14.42 21.45 14.29 67.11 52.64 13.37 90.29

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E EXPACD-W EXPACD-G EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W RSACD-G RSACD-B

9.38 18.11 21.17 21.21 7.41 10.52 11.64 11.83 10.20 16.27 17.20 18.02 8.36 13.63 16.25 16.18 8.27 13.07 15.16 10.82 10.25 6.85 10.16 15.08

M F ( 2, 0 )

Table 6.5 (cont.) Panel C: out-of-sample residuals of Euro/Dollar

5.15 7.74 6.64 7.36 4.14 4.74 4.48 4.76 5.07 6.03 4.86 5.56 4.21 5.37 5.03 5.36 2.98 4.72 4.33 3.67 3.73 2.63 4.94 9.49

M F (3, 0 ) 2.72 3.46 2.34 2.73 2.65 2.58 2.30 2.48 2.56 2.46 1.55 1.91 2.16 2.28 1.74 1.91 0.74 1.51 1.28 1.41 2.45 2.02 2.42 7.36

M F ( 4, 0 ) 5.20 18.47 32.78 29.26 11.50 4.84 2.53 2.65 3.91 11.73 18.19 18.13 2.35 7.60 14.13 12.75 1.74 5.45 9.34 1.02 50.33 31.90 2.83 83.74

M F (1,1) 1.16 4.66 6.04 5.83 21.11 17.12 12.62 13.97 –0.67 0.04 0.12 0.29 –0.80 –0.01 0.36 0.27 –0.92 –0.93 –0.90 2.07 24.57 20.97 –0.77 36.67

M F (1, 2 ) 9.06 14.70 15.07 15.59 0.91 1.21 1.95 1.77 8.01 10.53 9.42 10.44 6.35 8.59 8.95 9.22 5.79 8.29 8.12 3.42 8.32 2.80 6.03 18.17

M F ( 2,1) 1.34 1.69 0.81 1.12 4.09 2.08 0.18 0.61 –0.03 –0.24 –0.74 –0.53 –0.24 –0.34 –0.62 –0.50 –0.71 –0.52 –0.89 –1.38 1.64 0.05 –1.11 20.74

M F ( 2, 2 ) –0.84 –0.90 –0.91 –0.92 –0.44 –0.84 –1.12 –1.12 –0.93 –0.94 –0.98 –0.99 –0.91 –0.89 –0.86 –0.89 –0.89 –0.90 –0.86 –0.95 –1.01 –0.78 –1.03 3.78

M F (3, 3) –0.92 –0.85 –0.77 –0.80 –0.75 –0.85 –0.93 –0.98 –0.85 –0.79 –0.74 –0.78 –0.84 –0.78 –0.72 –0.74 –0.70 –0.75 –0.71 –0.71 –0.71 –0.82 –0.84 –0.90

M F ( 4, 4 )

M F (1, 0 )

7.02 5.37 8.02 6.42 23.54 10.63 6.02 6.87 6.28 5.76 9.15 7.50 7.07 4.92 6.46

M F ( 0, 0 )

15.47 10.44 10.57 10.08 40.25 24.55 15.53 18.15 13.56 9.55 10.22 9.59 14.97 9.73 8.76

Model

LINACD-E LINACD-W LINACD-G LINACD-B LOGACD-E LOGACD-W LOGACD-G LOGACD-B BCACD-E BCACD-W BCACD-G BCACD-B EXPACD-E EXPACD-W EXPACD-G

3.71 6.78 9.95 8.37 2.57 3.41 6.22 4.92 5.19 8.01 9.90 9.19 4.19 6.43 8.47

M F ( 2, 0 )

Panel D: out-of-sample residuals of Yen/Dollar

0.84 2.31 3.45 2.97 0.46 0.65 1.47 0.98 2.29 3.38 2.72 3.16 1.58 2.63 3.04

M F (3, 0 ) –0.36 0.32 0.75 0.64 –0.42 –0.60 –0.49 –0.62 0.94 1.31 0.21 0.75 0.51 1.05 0.95

M F ( 4, 0 ) 4.20 4.88 8.56 6.53 22.68 8.98 1.78 3.73 4.47 4.62 7.07 5.98 4.69 3.53 4.78

M F (1,1) 0.91 2.42 4.18 3.21 10.44 5.55 2.12 3.26 0.14 0.18 0.23 0.24 –0.01 –0.01 0.14

M F (1, 2 ) 3.31 6.12 8.79 7.45 0.41 0.54 2.44 1.51 3.75 5.84 7.10 6.63 2.79 4.36 5.73

M F ( 2,1)

0.08 0.74 1.28 0.99 –0.51 –1.30 –1.52 –1.50 –0.60 –0.45 –0.51 –0.48 –0.87 –0.75 –0.71

M F ( 2, 2 )

–1.13 –1.05 –1.05 –1.03 –1.14 –1.09 –1.16 –1.10 –1.11 –1.07 –0.98 –1.03 –1.13 –1.16 –1.15

M F (3, 3)

–1.00 –0.98 –0.99 –0.97 –0.73 –0.71 –0.71 –0.71 –0.93 –0.86 –0.70 –0.78 –0.90 –0.93 –0.88

M F ( 4, 4 )

Note: This table reports M F ( m, l ) diagnostic test statistics for the serial dependence of generalized residuals [Fˆ i ] of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. M F ( 0, 0 ) represents statistics on i.i.d. test. M F (l , 0 ), l  1, 2, 3, 4 represent test statistics on martingale, ARCH effect, conditional skewness and conditional heterokurtosis respectively. And M F (1,1), M F ( 2, 2 ), M F (3, 3) , M F ( 4, 4 ) are very sensitive to autocorrelations in mean, variance, skewness and kurtosis of the generalized residuals respectively. We only show results for preliminary lag truncation order p  20 , the results for other lag order is similar. The M F ( m, l ) tests are asymptotically one sided N ( 0,1) test and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

5.68 7.66 5.18 5.37 5.89 21.31 16.50 7.92 99.65

8.93 12.24 10.67 9.40 14.58 35.67 33.34 15.00 112.20

EXPACD-B TACD-E TACD-W TACD-G TACD-B RSACD-E RSACD-W RSACD-G RSACD-B

7.64 6.42 5.73 7.02 1.79 2.31 8.00 5.25 12.55

M F ( 2, 0 ) 2.95 2.54 2.46 2.97 0.68 0.68 5.29 2.24 3.20

M F (3, 0 ) 1.09 1.27 1.26 1.48 0.49 –0.07 3.35 1.13 0.32

M F ( 4, 0 ) 4.18 5.30 3.72 3.67 6.27 19.52 14.76 5.83 84.96

M F (1,1) 0.06 0.28 0.04 –0.27 2.98 7.66 7.01 0.59 17.18

M F (1, 2 ) 5.16 3.95 3.27 4.22 0.27 0.34 4.08 3.26 10.19

M F ( 2,1) –0.73 –0.81 –1.18 –1.16 –0.94 –0.65 –0.29 –0.89 12.35

M F ( 2, 2 ) –1.15 –1.06 –1.20 –1.27 –1.15 –1.40 –1.59 –1.08 8.73

M F (3, 3) –0.90 –0.85 –0.94 –1.01 –0.80 –1.02 –1.36 –0.86 0.63

M F ( 4, 4 )

Note: This table reports M F ( m, l ) diagnostic test statistics for the serial dependence of generalized residuals [Fˆ i ] of linear ACD, log ACD, Box–Cox ACD, Exponential ACD, threshold ACD and Markov regime-switching models based on standard exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. The whole sample are seasonally adjusted price durations from July, 1, 2000 to June 30, 2001 on Wednesdays, with a total of 20,584 and 15,818 observations for Euro/Dollar and Yen/Dollar respectively. The first half of the samples is used for estimation and the second half are used for forecasting. M F ( 0, 0 ) represents statistics on i.i.d. test. M F (l , 0 ), l  1, 2, 3, 4 represent test statistics on martingale, ARCH effect, conditional skewness and conditional heterokurtosis respectively. And M F (1,1), M F ( 2, 2 ), M F (3, 3), M F ( 4, 4 ) are very sensitive to autocorrelations in mean, variance, skewness and kurtosis of the generalized residuals respectively. We only show results for preliminary lag truncation order p  20, the results for other lag order is similar. The M F ( m, l ) tests are asymptotically one sided N ( 0,1) test and upper-tailed critical values should be used, which are 1.65 and 2.33 at the 5 per cent and 1 per cent levels, respectively.

M F (1, 0 )

M F ( 0, 0 )

Model

Table 6.5 (cont.)

Autoregressive duration models

147

innovations, which has the best in-sample fit, also has the best out-of-sample forecast, with W (5 ) equal to 29.5 and 17.0 for Euro/Dollar and Yen/Dollar respectively. With the same innovation distribution, LINACD, LOGACD, BCACD, EXPACD and TACD models perform rather similarly. This indicates that sophisticated nonlinear modeling for the conditional mean Z i helps little in improving out-of-sample density forecasts of price durations. However, the innovation distribution specification is important: the exponential distribution often has the worst performance, with overwhelming large W ( p ) statistics. The Weibull distribution helps a lot in improving density forecasts: the W ( p ) statistics decrease from over 1,000 for the exponential distribution to well below 500, given the same conditional mean specification Z i . Generalized Gamma and Burr innovations often have the best performances. This is similar to the results of Dufour and Engle (2000b) that the choice of the innovation distribution becomes critical when forecasting the trade duration density. We also find that it is important to relax the i.i.d. assumption for innovations. Taking into account higher order conditional dependence via regime shifts and threshold principles helps a lot in forecasting the full dynamics of price durations. To gauge possible sources of model misspecification, we next separately examine the uniform distribution and i.i.d. properties of out-of-sample generalized residuals. Figures 6.11 and 6.12 display histograms of out-of-sample generalized residuals of Euro/Dollar and Yen/Dollar. Consistent with the W ( p ) statistics, we find that the marginal densities of out-of-sample generalized residuals of RSACD models are much closer to the uniform distribution than those of other models. In particular, the density of the out-of-sample generalized residuals of the RSACD model with Burr innovations is the closet to the uniform distribution. Given the same innovation distribution, the density of the out-of-sample generalized residuals of LINACD, LOGACD, BCACD, EXPACD and TACD models perform rather similarly. Figure 10 shows that the density estimates of the out-of-sample generalized residuals of all ACD models (except RSACD) with exponential innovations exhibit a U shape, with pronounced peaks at two ends, especially at the left end. This pattern is similar to the in-sample pattern. However, this situation has been greatly improved in the RSACD model with exponential innovations. The ACD models with Weibull, generalized Gamma and Burr innovations all have an †-shape density for their out-of-sample generalized residuals: more realizations than predicted fall into the left and right ends. In most cases, the generalized Gamma distribution slightly outperforms the Burr distribution, which in turn outperforms the Weibull distribution. Although the RSACD model with Burr innovations is the closet to the uniform distribution, it still slightly underpredicts the very short durations and overpredicts the short durations. This implies that all ACD models cannot fully account for the tail of price durations. However, the RSACD models, which allow for different

148

Autoregressive duration models 6

6 E W G B

5 LACD-E

4

LOGCD-E

4

3

3

LACD-W LACD-B

2

E W G B

5

LOGCD-W

LACD-G

LOGCD-G

LOGCD-B

2

1

1

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

Linear ACD

0.6

0.8

1

Log ACD 6

6 E W G B

5

EXACD-E

BCACD-E

4

E W G B

5 4 3

3 BCACD-W

BCACD-G

EXACD-W

BCACD-B

2

2

1

1

EXACD-B

EXACD-G

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

Box–Cox ACD

0.4

0.6

0.8

1

Exponential ACD

6

1.8 E W G B

5

1.6 RSACD-W

1.4 TACD-E

4

E W G B

RSACD-E

1.2 1

3

TACD-W TACD-B

2

0.8

TACD-G

0.6 RSACD-B

0.4

1

RSACD-G

0.2 0

0 0

0.2

0.4

0.6

Threshold ACD

0.8

1

0

0.2

0.4

0.6

0.8

1

Regime-switching ACD

Figure 6.11 Histogram of out-of-sample generalized residuals of Euro/Dollar

regimes and higher order serial dependence, can better capture the fat tail of price durations. While the RSACD model with Burr innovations characterizes the marginal density of the out-of-sample generalized residuals well, it still has difficulty in capturing various aspects of price duration dynamics, as can be seen from the M z ( m, l ) and M F ( m, l ) statistics in Tables 6.4 and 6.5. Panels C and D of Table 6.4 reports M z ( m, l ) statistics for out-of-sample generalized residuals. All ACD models fail to satisfactorily capture serial dependence in the conditional mean, variance, skewness and kurtosis of their out-of-sample generalized residuals, with M z ( m, l ) statistics significant at

149

Autoregressive duration models 4.5

5

E W G B

4 3.5

E W G B

4.5 4

LACD-E

LOGACD-E

3.5

3

3

2.5

2.5

LACD-B LACD-W LACD-G

2

LOGACD-W

LOGACD-B LOGACD-G

2

1.5

1.5

1

1

0.5

0.5

0

0 0

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0

1

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4.5

0.8

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4 3.5

0.6

1

Log ACD

Linear ACD

E W G B

4 3.5

BCACD-E

3

EXACD-E

3

2.5

2.5

2

BCACD-W BCACD-G

BCACD-B

1.5

EXACD-B

EXACD-W

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Exponential ACD

Box–Cox ACD 1.8

4.5 E W G B

4

1.6

3.5

RSACD-E

E W G B

1.4

TACD-E

3

RSACD-W

1.2

2.5 TACD-W

2

TACD-G

1

TACD-B

0.8

1.5 1

0.6

0.5

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RSACD-B

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0.8

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0

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Regime-switching ACD

Figure 6.12 Histogram of out-of-sample forecasted generalized residuals of Yen/ Dollar

any conventional significance level. In general, the weak form ACD models (TACD and RSACD) with generalized Gamma and Burr innovations perform better, giving smaller M z ( 0, 0 ) statistics. There is a little difference from the in-sample case: the TACD (rather than the MSACD) model with Burr innovations has the smallest M z ( 0, 0 ) statistic for both Euro/Dollar and Yen/ Dollar. For Euro/Dollar, the RSACD model with Burr innovations has a larger M z ( 0, 0 ) than many other models (e.g. RSACD and BCACD models

150

Autoregressive duration models

with generalized Gamma innovations). This implies that relaxing the i.i.d. assumption for the innovation, allowing regime shifts and using generalized Gamma or Burr innovations can significantly improve forecasting the full dynamics of price durations of foreign exchanges. The M F ( m, l ) tests for the standardized residuals are reported in Panels C and D of Table 6.5. The empirical results are similar to that of the in-sample case: for both Euro/ Dollar and Yen/Dollar,M F ( 0, 0 ),M F (1, 0 ) and M F ( 2, 0 ) is rather large, while M F (3, 0 ) and M F ( 4, 0 ) are relatively small. In summary, our analysis shows that none of the commonly used ACD models can adequately capture the full dynamics of price durations of foreign exchanges, either in-sample or out-of-sample. This differs from Bauwens et al. (2003), who use different evaluation methods and find that LINACD and LOGACD models with generalized Gamma and Burr innovations perform satisfactorily for stock price durations. However, some ACD models outperform others. The RSACD models with Burr innovations have not only the best in-sample fit but also the best out-of-sample performance, which is consistent with the empirical results of Hujer et al. (2003b) for stock transaction durations. Generally speaking, each ACD model has similar in-sample and out-of-sample performances. In particular, the ACD models that have best in-sample fit usually have best out-of-sample density forecasts. It seems that sophisticated nonlinear specifications for conditional expected durations do not help much in capturing the full dynamics of price durations. However, the specification of the innovation distribution is important: the exponential distribution always fits poorly while generalized Gamma and Burr distributions perform much better. Moreover, relaxing the i.i.d. assumption for the innovation, allowing higher order dependence in price durations and taking into account possible regime shifts can help a lot in improving the performance of ACD models. Nevertheless, we seem to be a long way off finding an adequate ACD model for the full dynamics of price durations of foreign exchanges, which remains to be an important research topic in the literature.

6.5

Conclusion

In high-frequency financial econometrics, price duration dynamics is important due to its close links to market microstructure theory, options pricing and risk management. Applying Hong and Li’s (2004) nonparametric portmanteau test for time series conditional distributional models, we provide a relatively comprehensive empirical study on in-sample and out-of-sample performances of a wide variety of ACD models in capturing the full dynamics of price durations of two exchange rates – Euro/Dollar and Yen/Dollar. We find that none of the ACD models can adequately capture the price duration dynamics of Euro/Dollar and Yen/Dollar, either in-sample or outsample. However, some ACD models, particularly the Markov switching ACD model with Burr innovations, have not only the best in-sample fit, but also the best out-of-sample performance. We find that sophisticated models

Autoregressive duration models

151

for the conditional mean duration do not help much in capturing the full dynamics of price durations of foreign exchanges, but the specification of the innovation distribution is important: generalized Gamma or Burr distribution performs much better than Weibull and exponential distributions. The latter often performs poorest. Moreover, the conditional mean duration alone cannot fully capture the dynamics of price durations of foreign exchanges. It is important to relax the i.i.d assumption for the innovation, to model higher order conditional moments and to allow possible regime shifts in price durations. Our findings are similar for both Euro/Dollar and Yen/Dollar and for both in-sample and out-of-sample.

Notes 1 The kernel k( i ) here differs from the kernel K i used for probability density estimation. 2 A commonly used example is W0 ( i )  &( i ), the N(0,1) cumulative distribution function (cdf). 3 The Dirac delta function is defined as follows E( u )  0 if and only if u x 0 and

° E(u )du  1. 4 The price, volume and turnover duration processes can naturally be obtained from the trade duration series by dropping intervening observations from the sample, thus yielding a ‘thinned’ or ‘weighted’ duration process. 5 For a random variable X, its hazard function (or intensity function) is defined by h( x )  f ( x ) / S ( x ), where f ( i ) and S( i ) are the pdf and survival function of X, respectively. The survival function S ( x ) y P ( X  x )  1 P ( X b x ), x  0. 6 For TACD and MSACD, we assume that the innovations in a specific regime follows exponential, Weibull, generalized Gamma and Burr distributions respectively, with different parameters across different regimes. The marginal distribution of innovations is different from their conditional distribution in both TACD and MSACD models. 7 The results of W ( p ) ,for p = 1, 2, …, 30,are available from the authors upon request. 8 One possible reason is that the evaluation tools used are different. Another possibility is that the price durations of stocks have a different dynamics from the foreign exchange rates.

7

7.1

Intraday effect

Calendar Effect

High frequency data usually refers to financial data collected daily, by the hour, by the minute or even by the second. Ultra High Frequency data refers to tip-by-tip trading data. Models of high frequency financial data have developed quickly since the beginning of the 1990s, and they are now widely used in theoretical and empirical studies of microstructures of financial markets. Generally, because of continuity of impact of information on changes in prices, discrete models will cause loss of information; the lower the frequency of data, the more will be the loss of information. Low frequency data cannot reveal real-time dynamic characteristics of financial assets prices; high frequency data contain more and abundant intraday volatility information and, therefore, high frequency data is a more effective indicator of how information influences the markets. In research based on high frequency data, one important issue is description of the statistical laws and characteristics of the trading patterns. The ‘Calendar Effect’ is one of the most important findings in this field. An abnormal phenomenon in theories of microstructure of financial markets is the ‘Calendar Effect’, on yield and other financial variables, that demonstrates the characteristics of the markets; it is called the ‘Calendar Effect’ because frequencies vary, such as ‘Intramonth Effect’, ‘Intraweek Effect’ and ‘Intraday Effect’. For some mature markets, such as markets in the United States and Hong Kong, empirical analysis of high-frequency data has revealed some classical intraday characteristics. Admati and Pfleiderer (1988); Brock and Kleidon (1992) detected the ‘Calendar Effect’, and provided theoretical explanations of the intraday ‘U’ patterns. Hedvall (1995) compared these two findings. Andersen and Bollerslev (1997, 2001); Hasbrouck (1999), analyzed stock market data and found that the intraday volatility and volume had a ‘U’ pattern seasonality. They discovered that volatility is the highest in hours after opening in the morning, it goes down after that and rises gradually before the markets close in the afternoon. They also found the significant positive correlation between volatility and volume, which demonstrates the activity of the trades, and the asymmetric distribution of intraday trades,

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which probably contribute to the intraday seasonality. Ding and Lau (2001); Rahman, Lee and Aug (2002) used intraday trading data of some stocks to demonstrate the ‘U’ pattern, and found that volume, trading spreads and trading frequency also follow the pattern. Andersen, Bollerslev and Cai (2000) used Flexible Fourier Form Regression (FFFR) to analyze the Japanese stock market and found that the Japanese market showed double U-type movements, due to the Japanese market’s system of closing for a lunch break during the trading hours, which is different from the American market. Wood, Mginish and Reith (1985) also found the U-pattern in movement of volatility, which means volatility is high around opening and closing time, and low in the rest of the hours. Harris (1986) confirmed the existence of the phenomenon. Admati and Pfleiderer (1988) proved the rationality of the phenomenon through theoretical analysis. Jain and John (1988); Mcinish and Wood (1990, 1991, 1992) also analyzed the intraday price volatility, and found that the volatility was high around the morning hours immediately after opening and hours before the markets close in the afternoon; other variables, such as transaction frequency, volume and trading spread also appeared to be moving in the ‘U’ pattern. Engle and Russell (1997, 1998) studied the durations of trade, i.e. the intraday pattern of durations, and came to a similar conclusion – the movements followed an inverted ‘U’ pattern. Andersen and Bollerslev (1994) studied the relationship between the ‘Calendar Effect’ and volatility continuity, and proved that after eliminating the ‘Calendar Effect’ in the data, the reduction in continuity of the low frequency data is significant. Anderson and Bollerslev (1995) further studied the main factors that cause the ‘Calendar Effect’, in addition to time scale, i.e. holidays, midday rest and other factors. Regular dissemination of macroeconomic information also contributes to the ‘Calendar Effect’. The majority of scholars mentioned above focus their works on transaction variables, volatility and liquidity of securities markets. Chinese scholars particularly focus on estimation of volatility and liquidity of the stock market. Liu, et al. (2000) empirically studied and analyzed patterns of intraday movements of price and volume in Shanghai and Shenzhen A-share stock markets, using transaction data of five minutes periods. Guo and Du (2002) presented a synthesis of high-frequency data modeling. Fang and Wang (2004) analyzed the ‘U’-type characteristics of the Shanghai stock market, and established FFF models to estimate the intraday yield patterns. Qu and Wu (2002) studied the intraday pattern of volume and derived a conclusion similar to that of foreign researchers, i.e. the intraday volumes moved in an inverted ‘U’ pattern. However, the spread of price in the Chinese stock market moved in a ‘L’-type path, unlike the ‘U’-pattern of intraday movements in mature markets like the New York Stock Exchange. Wang and Li (2005) discovered the ‘L’-type movement of the volatility of absolute yield, and the ‘U’-pattern of transaction amounts, of the five-minute data of copper futures at the Shanghai commodity exchange. Other researches, which focus on the modeling of time intervals of transactions, lack detailed analysis of intraday characteristics.

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Studies mentioned above mostly focus on the stock markets, though some of the researches cover the foreign exchange markets also. Past studies have paid little attention to the internal structure and operational characteristics of the Chinese futures market. As a new booming market, the Chinese futures market is important to the international market; its microstructure has caused great concern. Compared with the developed markets and markets with market maker mechanisms, the microstructure of the Chinese futures market is quite different. This chapter uses high-frequency data of actively traded commodities futures in the Chinese market to empirically study the patterns of intraday changes of yield and volume, and explains the reason of the formation of the pattern, from the theories of microstructure of markets. Moreover, we analyze the relationship between price and volume, build up a Vector Autoregressive model, and empirically study the factors that influence changes in price volatility and factors that affect volume volatility. Finally, we derive the internal characteristics of the futures market. We hope to strengthen risk management and fill in the blanks in the study on futures market. This chapter is arranged as follows. Section 7.2 introduces the selection and presentation of the statistical characteristics of the yield sequences. Section 7.3 analyzes the intraday characteristics of yield and volume. Section 7.4 is the analysis of the dynamic correlation between the high frequency data series of yield, volume and open interest. Section 7.5 presents summarized comments.

7.2

Data

This chapter uses high frequency data, collected by the minute, of six actively traded kinds of commodities futures to study intraday movements of the Chinese futures market. All data, collected by the minute, come from Shihua Financial Information and Webstock of China. For each commodity, different futures contracts are traded at the same time, so we construct the continuous data series by uniting the most actively traded contracts. Data series include transaction time, price, volume and open interest of the most active contracts at that time. The data cover transactions between April 27 and September 28, though data of cotton go only up to August15 due to problems of the source. We first eliminate unreliable data from the trading records, such as significant mistakes and data pertaining to outside the trading hours and outside the continuous bidding hours transactions. Finally, we get 21,414 copper data, 21,405 natural rubber data, 23,043 soybean data, 22,493 corn data, 14,937 cotton data and 23,108 strong wheat data. Main results come from Matlab7.1.0 programming. Define yield Rt as 100 times of natural logarithm of the first order difference, i.e. Rt  100 * DLNP  100 * (ln( Pt ) ln( Pt 1 )).

(7.1)

0.0000 0.0000 0.0003 –0.0004 0.0002 –0.0003

21,312.00 21,303.00 22,938.00 22,390.00 14,864.00 23,003.00

0.0668 0.0902 0.0494 0.0570 0.0420 0.0586

S.D. 0.9061 1.5267 0.5539 0.4328 0.4668 0.5319

Maximum 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Median

Note: CU denotes copper, NR denotes natural rubber, WS denotes wheat strong.

CU NR Soybean Corn Cotton WS

Average

Sample size

Table 7.1 Statistical description of yield sequences

–1.0399 –2.6314 –0.6933 –0.5823 –0.5265 –0.5817

Minimum –0.2001 –1.1612 –0.0812 –0.1584 0.1119 –0.0475

Skewness

16.2782 56.7005 13.0782 7.1263 15.4591 8.0220

Kurtosis

156706 2564472 97100 15978 96170 24182

JB statistics

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Figure 7.1 Time series of yield

Then we eliminate the overnight yield, i.e. yield calculated from the difference between the opening price of the day and the closing price of the day before. We get yield sequences of copper with 21,312 data, natural rubber with 21,303 data, soybean with 22,938 data, corn with 22,390 data, cotton with 14,864 data and strong wheat with 23,003 data. Statistical description of the yield sequences is given in Table 7.1.

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It could be found that average yields of the minute by minute data of the six commodities are around 0, kurtosises are significantly larger than 3 and the yield series have significant large kurtosis and heavy tails. JB statistics also shows that yield series do not follow normal distribution. Yields of all the six commodities are graphically depicted in Figure 7.1.

7.3

Intraday trends of yield and volume

We first produce ONEDAY sequence, which means average of the daily data of all trading days, and we get the average yield, average absolute yield, average volume and average open interest. Because of the limited volatility of the daily data, it is reasonable to analyze the averaged sequences. This part focuses on analysis of intraday trends, and the reasons indicated by the microstructures of the financial markets, including the trading system and the psychology of investors. 7.3.1

Trends of intraday absolute yield

Because of the insignificant trend of the average yields, which is around zero, the average series is similar to a random sequence without seasonality. In order to detect the characteristics of price volatility, absolute yield should be considered. From Figure 7.2, it is clear that going by data collected by the minute, absolute yield of Chinese commodities futures is not a stationary process; generally, the yield moves on a slightly flat L-type path, from the opening to the closing, and this is different from the U-pattern of the Chinese stock market. The five-minute absolute yields of Shanghai and Shenzhen stock markets show a typical U-type feature, where volatility is high at the opening, declines gradually after that and rises again as the closing time approaches. It means that there is significant increase of volatility around the opening and closing times. In contrast, volatility of futures market does not rise near the closing; so it doesn’t have a U-type movement. Volatility of price around the opening is about three times the average of the day. Results of international and Chinese researchers support the existence of the intraday U-type feature. It is the microstructure of the financial markets, besides dissemination, absorption and assimilation of overnight information that directly lead to the movement patterns. However, Amihud and Mendelson (1986) believed that it is the call auction mechanism that causes the high volatility around the opening of the New York stock market. Meanwhile, from the graphics, we find that price volatility also rises slightly before the closing at noon and the reopening in the afternoon. Unlike the opening in mornings, there is no call auction before the afternoon reopening to explain the significant rise of price volatility at that time; therefore, we tend to believe that the high volatility comes from dissemination, absorption and

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Figure 7.2 Intraday pattern of absolute yield

assimilation of information that accumulated by the lunch-break, and not from the call auction mechanism. To explain the L-type characteristics of the intraday yield, there are many factors. The first is the arbitrage pressure that comes because of overnight information coming from movements in markets elsewhere. Secondly, institutional investors take advantage of private information and the call auction mechanism, which influences short-term volatility of price. Because of the asymmetric nature of the overnight information, it results in high volatility at the morning openings. It often takes about 20–30 minutes for ordinary

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investors to detect trends in behaviors of the more informed investors and to adjust their own investments. That is why price volatility reaches a stationary situation 20–30 minutes after the opening. 7.3.2

Trends of intraday volume

Figure 7.3 shows the trends of intraday volume. Generally, different from the ‘U’-pattern of foreign markets, the minute by minute volume data of Chinese futures shows ‘L’ type movement, which is high at the morning opening, then goes down gradually and finally reaches a stationary situation after about 30 minutes. Worthy of mention is the fact that only cotton has a ‘U’ shape. In the Shanghai market, after two rest sections (10:15 to 10:30 and 14:10 to 14:20) and the noon break (11:30 to 13:30), the trading volume has a slight but significant increase, similar to a new morning opening. As for Dalian and Zhengzhou markets, the volume increases slightly but not significantly after each rest section (10:15 to 10:30) and noon break (11:30 to 13:30). The movement of volume of futures correlates strongly to the mechanism of the exchange. The Chinese futures market follows a mark-to-market and marginal exchange mechanism; it encourages investors to speculate. Because of the limited ability to endure risks, often there is great pressure on investors because of the high volatility around the opening hour. Investors with more information tend to trade following information, so the volume at the early opening is large. The larger the volume, the more information the ordinary investors can detect from the behavior of institutional investors, and the larger is the possibility that they would adjust their investments, and the higher would be the price volatility. After about 30–40 minutes from the morning opening, information will be fully absorbed, and prices and volumes will stabilize. The high volatility of volume around the opening hours comes from the release of overnight information, which is similar to the results of Easley and O’Hara (1992); irrespective of whether the information of the market is positive or negative, volatility would go up. Information accumulates between the rest sections (noon break and naprest) also, though there is no call auction during these periods. We believe that release of information after the rests would increase price movement and volume, especially volume, which increases significantly around the reopening of the market after the noon breaks. Since breaks are not long, volume increases significantly, but not as much as in case of the morning openings. It is the highly asymmetric information of the Chinese futures market that results in high volatility of volume and price around the morning openings; this is coherent to behaviors of the stock markets. However, unlike the ‘U’-pattern movement of stock market, intraday yield and volume of futures markets demonstrates the ‘L’-pattern. The main reasons are: (1) The sources of overnight information are different. Because of limited correlation between Chinese and international stock markets, the

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Figure 7.3 Intraday pattern of volume

overnight information mainly comes from domestic news. A small number of traders take advantage of the overnight information, before the morning opening, using the call auction mechanism. In the process, they impact volatility in price and volume. In the case of the futures market, overnight information comes not only from domestic news but also from international markets such as London and Chicago. Thus, pressure

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161

coming from arbitrage opportunities offered by international futures markets results in adjustments of prices at and around the opening of Chinese futures markets. (2) The expectations of the next trading day are also different. Information that influences expectations of the next trading day of Chinese stock markets mainly comes from domestic news, but information influencing commodities futures markets comes mainly from international news. Investors tend to adjust their investments after the opening of the international markets, which offer much information that can help avoid risks. (3) The trading mechanisms are different. The ‘T+1’ trading mechanism of Chinese stock markets encourages investors to buy or sell 2–3 minutes before the afternoon closing, in order to avoid overnight risks. That is why volatility of stock markets is high before the closing. Unlike the stock markets, futures markets follow the ‘T+0’ trading mechanism. So investors could buy and sell many times as they wish in any trading time. So it is unnecessary to adjust investments before the closing of the markets on a given day.

7.4 Analysis of correlation among yield, volume and open interest The relationship between volume and yield is always important to analysis of microstructures of financial markets. In futures markets, the movement of price demonstrates the reaction of the markets to information, volatility is the degree of price fluctuation and volume indicates the diversity of acceptance of new information by investors. Volume of open interest reflects investors’ confidence in specific contracts. By analyzing the dynamic relationships between price, volume and open interest, we can understand the microstructure of futures markets, i.e. the way information impacts the market, and the speed at which prices react to information. What is more, it helps understanding of the effect of volume and open interest on price volatility and helps strengthen the management of futures markets. We first eliminate the intraday trends of yield, volume and open interest, and then build VAR Models. Through Granger causality tests, we analyze the models to conduct qualitative analysis of correlations among yield, volume and open interest. Under the framework of Vector Auto Regression (VAR) model analysis, we take advantage of variance decomposition and impulse response analysis to analyze correlations among variances, and the duration of their impact, quantitatively. VAR is used for forecasting of related time series; it helps analysis of the dynamic influence of random disturbances on the variances in the system. It has many advantages. It does not require any pre-restriction, and it does not require assumptions of variables being endogenous or exogenous, since

162

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it treats every variable as endogenous. VAR models should be stationary. Stability refers to the duration of the impulsive impact of an innovation, of a new equation of a VAR model, as time goes by. This section uses the OLS method to conduct ADF regression, and the results of ADF tests show that absolute yield and volume are stationary, but open interest isn’t stationary. Its first-order difference is stationary. Therefore, we consider Granger causality among absolute yield, volume and different open interests, and build the related VAR models. Selection of the lagged order k follows AIC and SIC criteria. That is, for copper k = 8, for natural rubber k = 9, for soybean k = 8, for corn k = 7, for cotton k = 4 and for strong wheat k = 6. 7.4.1

Test of Granger causality

About the relationship between price and volume, one theory suggested by Copeland (1976) is the Sequential Information Arrival Hypothesis. The hypothesis suggests that information transfers to the market gradually, and price volatility and volume changes take place during the process of information transfer. Volatility and volume go up as new information accumulates. According to the assumption, because of the step-by-step transfer of information, and adjustments made by investors in response, the market goes to a stable situation step-by-step, implying a two-way causality between absolute yield and volume. Therefore, information of the past volatility of absolute price helps forecast of future volume, while, information of volume in the past is a helpful indicator of the future absolute yield. Another hypothesis, first proposed by Clark (1973), is the Mixture Distribution Hypothesis (MDH). It says that yield and volume of financial assets are determined by a potential and invisible information flow. The arrival of the flow leads the movement of both volume and yield. The information flow changes the demand and supply situations, breaks the existing balance and directs the adjustment of price. Trading continues in the price adjustment process, and reflection of new information in price and volume is instantaneous. Regardless of the direction of how price fluctuates, volume will increase; so there is a positive correlation between absolute yield and volume. According to this hypothesis, there is no causality between absolute yield and volume in any direction. From Table 7.2 and Table 7.3, the tests reveal that there is a two-way Granger causality between the absolute yield and volume in the futures market. The two-way Granger causality exists between absolute yield and open interest, too. It comes from the instability of open interest series; we have to use the first-order difference series of open interest, which is strongly correlated to volume. This is quite different from the one-way Granger causality in the stock market. Due to the lack of a short selling mechanism of stock markets, volatility of absolute price increases when price falls, while volume does not increase significantly. Therefore, there is a natural conclusion that volume is the Granger cause of absolute yield, but the inverse does not

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Table 7.2 Granger causality test between volume and absolute yield

CU

NR

Soybean

Corn

Cotton

WS

Null hypothesis

F statistics and Null hypothesis the possibility to reject it

F statistics and the possibility to reject it

Volume doesn’t Granger cause absolute yield Volume doesn’t Granger cause absolute yield Volume doesn’t Granger cause absolute yield Volume doesn’t Granger cause absolute yield Volume doesn’t Granger cause absolute yield Volume doesn’t Granger cause absolute yield

53.9094

44.4803

(0.0000) 39.2471 (0.0000) 58.8187 (0.0000) 51.7488 (0.0000) 78.2439 (0.0000) 89.7499 (0.0000)

Absolute yield doesn’t Granger cause volume Absolute yield doesn’t Granger cause volume Absolute yield doesn’t Granger cause volume Absolute yield doesn’t Granger cause volume Absolute yield doesn’t Granger cause volume Absolute yield doesn’t Granger cause volume

(0.0000) 52.0915 (0.0000) 54.0642 (0.0000) 14.4673 (0.0006) 32.2993 (0.0000) 59.2945 (0.0000)

hold. This discussion shows that the mechanism of the commodities futures market decides the behavior of the market, which is quite different from that of the stock market. Meanwhile, it suggests that the Sequential Information Arrival Hypothesis is more likely to explain the information transfer pattern of Chinese commodities futures markets. Though there is a two-way Granger causality between volume and open interest, from the available statistics in Table 7.4, volume has a stronger influence on open interest than vice-versa, since the movement of open interest comes from the change of volume. Tauchen and Pitts (1983) believed that it was the unpredictable potential information flow that decides price volatility and volume. The impact of information flow affects price volatility and volume at the same time, i.e. their reaction to information is instantaneous, and volume increases as price volatility rises. Westerfield (1977) suggested a positive correlation between yield and volume in the stock market. Karpoff (1987); McCarthy and Najand (1993) found that yield of futures had no relationship with volume. They suggested that it is the lack of a short selling mechanism, or the high cost of shorting, in

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Table 7.3 Granger causality test between open interest and absolute yield

CU

NR

Soybean

Corn

Cotton

WS

Null hypothesis

F statistics and the possibility to reject it

Null hypothesis

F statistics and the possibility to reject it

Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield

12.0744

Absolute yield doesn’t Granger cause open interest Absolute yield doesn’t Granger cause open interest Absolute yield doesn’t Granger cause open interest Absolute yield doesn’t Granger cause open interest Absolute yield doesn’t Granger cause open interest Absolute yield doesn’t Granger cause open interest

24.6961

(0.0000) 7.0080 (0.0000) 25.1082 (0.0000) 21.8839 (0.0000) 46.4997 (0.0000) 47.5247 (0.0000)

(0.0128) 36.4145 (0.0000) 52.3226 (0.0002) 25.9594 (0.0000) 35.7556 (0.0000) 68.3696 (0.0000)

the stock market, that causes the difference. A rise in price results in increase of volume while a smaller volume follows lowering of the price. That is why the correlation between volume and price is positive. The cost of shorting is no more than that of a long position in the futures market; so there is no relationship between price and volume. Chinese researchers Hua and Zhong (2003) analyzed the daily data of the futures market and found that there was no correlation between yield and volume, while the absolute yield correlates to volume. We use high frequency data, collected by the minute, and discover the Granger causality between absolute yield and volume. More than the qualitative analysis, we build VAR models to measure the degree of causality between them, and in order to explain the results, variance decomposition and impulse response methods are used. 7.4.2

Variance decomposition

The variance decomposition method provides a description of the dynamic movement of the system; its main idea is to decompose each endogenous

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Table 7.4 Granger causality test between open interest and volume Null hypothesis

CU

NR

Soybean

Corn

Cotton

WS

Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield Open interest doesn’t Granger cause absolute yield

F statistics Null hypothesis and the possibility to reject it 2.9324 (0.0028) 2.1518 (0.0000) 4.6280 (0.0000) 13.8299 (0.0000) 2.9785 (0.0180) 4.6987 (0.0000)

Volume doesn’t Granger cause open interest Volume doesn’t Granger cause open interest Volume doesn’t Granger cause open interest Volume doesn’t Granger cause open interest Volume doesn’t Granger cause open interest Volume doesn’t Granger cause open interest

F statistics and the possibility to reject it 102.0770 (0.0000) 86.0026 (0.0000) 147.6970 (0.0000) 136.1600 (0.0000) 150.7080 (0.0000) 239.3400 (0.0000)

variable (total m) into components of the new information equations, according to its causes. Then we compare their contributions and know the importance of endogenous variables. The more the lagged orders are included, the more stationary is the influence of new information on each variable. The variance decomposition method is a way to analyze the relationships quantitatively. Decomposition results of all varieties of VAR model are listed in the Tables 7.5. When we treat absolute yield as an explanatory variable, its lagged orders explain the majority of the residual disturbance, about 90 per cent to 97 per cent, while volume contributes 3 per cent to 6 per cent and open interest has a little less than 1 per cent contribution to the explanation of absolute yield. It indicates that volume has a significant effect that explains the absolute yield volatility, while open interest does not. Explanatory variables reach a stationary situation in order 30. When volume is interpreted as an explanatory variable, we find that its lagged orders explain 70 per cent to 85 per cent of the residual disturbance.

100.0000 97.5201 96.5589 95.7718 95.6207 95.5910 95.5903 95.5902

0.0000 2.3691 3.1446 3.8787 4.0236 4.0525 4.0531 4.0532

0.0000 0.1109 0.2966 0.3495 0.3557 0.3565 0.3566 0.3566

15.9500 21.5759 22.4387 23.1984 23.3431 23.3712 23.3718 23.3719

CUABSRT

CUDOI

CUABSRT

CUVOL

CUVOL

CUABSRT

1 5 10 20 30 50 60 100

NR

100.0000 98.4307 97.7048 96.8865 96.6603 96.5853 96.5814 96.5800

0.0000 1.5309 2.1234 2.9035 3.1234 3.1966 3.2005 3.2018

0.0000 0.0384 0.1718 0.2100 0.2164 0.2181 0.2182 0.2182

23.2976 29.9262 29.8133 29.5858 29.4908 29.4585 29.4568 29.4562

NRABSRT

NRDOI

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NRVOL

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Panel B: decomposition results of VAR model of rubber

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CU

Panel A: decomposition results of VAR model of copper

Table 7.5 Variance decomposition

76.7024 70.0276 70.0002 70.0996 70.1630 70.1855 70.1867 70.1871

NRVOL

84.0500 78.3187 77.3058 76.4859 76.3311 76.3012 76.3005 76.3004

CUVOL

0.0000 0.0462 0.1866 0.3146 0.3461 0.3560 0.3566 0.3567

NRDOI

0.0000 0.1054 0.2555 0.3158 0.3258 0.3277 0.3277 0.3277

CUDOI

4.4086 6.2659 6.7812 7.1793 7.2676 7.2950 7.2964 7.2969

NRABSRT

NRDOI

3.9620 4.8963 5.3248 5.6890 5.7628 5.7774 5.7777 5.7777

CUABSRT

CUDOI

14.7006 17.8457 19.0227 20.0860 20.3431 20.4256 20.4299 20.4314

NRVO

25.2016 29.2452 31.0034 31.7630 31.8481 31.8604 31.8606 31.8607

CUVOL

80.8908 75.8885 74.1962 72.7346 72.3893 72.2795 72.2737 72.2718

NRDOI

70.8364 65.8585 63.6718 62.5480 62.3891 62.3622 62.3617 62.3616

CUDOI

100.0000 98.8088 97.5678 95.8111 95.0400 94.6049 94.5550 94.5225

0.0000 1.1731 2.3949 4.1346 4.8987 5.3299 5.3793 5.4116

0.0000 0.0182 0.0373 0.0543 0.0613 0.0652 0.0657 0.0659

DOI

1 5 10 20 30 50 60 100

CORN

100.0000 99.0130 97.9561 97.4487 97.3786 97.3677 97.3675 97.3675

0.0000 0.8932 1.8903 2.3844 2.4530 2.4636 2.4638 2.4638

0.0000 0.0937 0.1536 0.1669 0.1685 0.1687 0.1687 0.1687

DOI 9.1916 11.2808 11.6048 11.8044 11.8315 11.8356 11.8357 11.8357

ABSRT

VOL

ABSRT 90.8084 88.2803 87.8323 87.5669 87.5308 87.5252 87.5252 87.5251

VOL

CORN

CORN

CORN

CORN

CORN

CORNBEANVOL

84.7552 79.2539 79.9919 80.5916 80.8328 80.9591 80.9731 80.9822

VOL

CORNBEANABSRT

15.2448 20.6994 19.8562 19.1726 18.8994 18.7565 18.7407 18.7304

ABSRT

VOL

ABSRT

SOY

SOY

SOY

SOY

SOY

SOYBEANVOL

SOYBEANABSRT

Panel D: decomposition results of VAR model of corn

1 5 10 20 30 50 60 100

SOY

Panel C: decomposition results of VAR model of soybean

0.0000 0.4389 0.5629 0.6287 0.6377 0.6391 0.6391 0.6391

DOI

CORN

0.0000 0.0467 0.1519 0.2358 0.2678 0.2844 0.2863 0.2875

DOI

SOY

21.8998 25.9002 28.7802 31.5412 32.5888 33.1614 33.2264 33.2688

VOL

SOY

3.6958 4.7572 5.0924 5.2961 5.3251 5.3296 5.3297 5.3297

ABSRT

CORN

39.1948 42.5426 44.3375 45.1160 45.2197 45.2357 45.2360 45.2360

VOL

CORN

CORNBEANDOI

3.8942 6.1815 6.7918 7.2517 7.3975 7.4756 7.4845 7.4903

ABSRT

SOY

SOYBEANDOI

57.1094 52.7002 50.5701 49.5879 49.4552 49.4347 49.4343 49.4343

DOI

CORN

74.2060 67.9183 64.4279 61.2071 60.0137 59.3630 59.2891 59.2409

DOI

SOY

100.0000 97.6847 96.8996 96.7585 96.7553 96.7552 96.7552 96.7552

0.0000 2.3123 3.0963 3.2372 3.2404 3.2404 3.2404 3.2404

0.0000 0.0030 0.0041 0.0043 0.0043 0.0043 0.0043 0.0043

DOI

1 5 10 20 30 50 60 100

WS

100.0000 98.6626 97.1857 96.4957 96.3966 96.3802 96.3799 96.3799

0.0000 1.2995 2.7735 3.4597 3.5583 3.5745 3.5748 3.5748

0.0000 0.0378 0.0408 0.0446 0.0452 0.0453 0.0453 0.0453

13.9158 18.7328 18.1730 17.9949 17.9710 17.9671 17.9671 17.9671

WSABSRT

WSDOI

WSABSRT

WSVOL

WSVOL

WSABSRT

12.6174 16.4841 16.6361 16.6547 16.6551 16.6551 16.6551 16.6551

ABSRT

VOL

ABSRT

86.0842 81.1303 81.6524 81.8079 81.8289 81.8323 81.8323 81.8323

WSVOL

87.3826 83.4826 83.3255 83.3058 83.3054 83.3054 83.3054 83.3054

VOL

COTTON

COTTON

COTTON

COTTON

COTTON

COTTONVOL

COTTONABSRT

Panel F: decomposition results of VAR model of wheat

1 5 10 20 30 50 60 100

COTTON

Table 7.5 (cont.) Panel E: decomposition results of VAR model of cotton

0.0000 0.1368 0.1746 0.1972 0.2001 0.2006 0.2006 0.2006

WSDOI

0.0000 0.0333 0.0384 0.0395 0.0395 0.0395 0.0395 0.0395

DOI

COTTON

4.3933 6.7587 7.1504 7.3799 7.4127 7.4181 7.4182 7.4183

WSABSRT

WSDOI

5.1093 7.3349 7.7462 7.8073 7.8087 7.8087 7.8087 7.8087

ABSRT

COTTON

31.1658 35.5819 38.5046 39.7641 39.9420 39.9712 39.9717 39.9718

WSVOL

38.6818 43.9848 45.2146 45.4250 45.4297 45.4298 45.4298 45.4298

VOL

COTTON

COTTONDOI

64.4409 57.6594 54.3450 52.8560 52.6453 52.6107 52.6100 52.6099

WSDOI

56.2090 48.6803 47.0392 46.7677 46.7617 46.7615 46.7615 46.7615

DOI

COTTON

Intraday effect

169

Moreover, price volatility has a significant influence on volume, and absolute yield explains 15 per cent to 30 per cent of residual disturbance; open interest contributes little to the explanation of residual disturbance, less than 1 per cent. Explanatory variables reach a stationary situation in order 30. Treating open interest as an explanatory variable, we found that its lagged orders explain 45 per cent to 70 per cent of the residual disturbance; volume explains 25 per cent to 45 per cent; and absolute yield contributes 5 per cent to 10 per cent. Explanatory variables reach a stationary situation in order 30.Variance decomposition maps are as in Figure 7.4. Images more intuitively demonstrate explanation contributions of variables. In sum, we can see that although the response to new information from outside is real-time, considering their mutual relations, in three cases, explanatory variables are almost stable around order 30. It means that the markets take about 30 minutes to absorb new information. This also corresponds to the conclusion that volatility of volume and yield is high at the opening, and they turn to be stable after about 30 minutes. Meanwhile, price volatility influences volume more than vice-versa. Open interest has little influence on volume but volume contributes 40 per cent of the explanation of the residual disturbance of open interest. 7.4.3

Analysis of impulse response

The impulse response functions show the reaction of each variable affecting the unit impact in the system. It demonstrates that the change of one unit of disturbance, which comes from the accumulated impulse of deviation, equals the total influence of current and future variances. Adding an impact to a variable of one model will directly affect it and transmit it to other variables via the dynamic structure of the VAR model. Analysis of impulse response is described in Figure 7.5. Results of impulse response functions of the six commodities are similar; only some slight differences of degree are observed. Absolute yield has a fairly strong response immediately to arrival of its innovation of deviation. Volatility has a substantial increase initially, then it is weakened rapidly, after the first 2 orders, and is attenuated to around 1/8, after which slow decay begins, attenuating to 0 around order 15. The sequence has no response to innovation from other equations at the first order, where the influence from innovation of deviation of volume is potent at the second order (achieves the maximum positive response), and then slow decay starts, attenuating to zero around order 15. The influence from the open interest innovation of deviation diminishes, to around zero, which suggests that the influence of open interest on price volatility is small. Volume responds quite strongly, immediately on arrival of its innovation of deviation; it has a substantial increase initially, then it is weakened rapidly after the first two orders to around 1/2, and then the slow decay starts, which lasts longer than 20 orders. The sequence has a positive response, immediately,

170

Intraday effect Variance Decomposition of CURT

100

Variance Decomposition of NRRT 100

80

80

60

60

40

40

20

20

0

2

4

6

CURT

90 80 70 60 50 40 30 20 10 0

CUVOL

2

4

6

8 10 12 14 16 18 20 CUVOL

4

6

CURT

8 10 12 14 16 18 20 CUVOL

2

CUDOI

4

6

NRRT

8 10 12 14 16 18 20 NRVOL

NRDOI

Variance Decomposition of NRVOL 80 70 60 50 40 30 20 10 0

2

4

6

NRRT

CUDOI

Variance Decomposition of CUDOI

2

0

CUDOI

Variance Decomposition of CUVOL

CURT

80 70 60 50 40 30 20 10 0

8 10 12 14 16 18 20

90 80 70 60 50 40 30 20 10 0

8 10 12 14 16 18 20 NRVOL

NRDOI

Variance Decomposition of NRDOI

2 4

6

NRRT

8 10 12 14 16 18 20 NRVOL

NRDOI

Figure 7.4 Variance decomposition map

to the influence of the standard deviation of absolute yield, and achieves the maximum; then it weakens rapidly after the first six orders and then starts a slow decay, leading to almost total disappearance after 20 orders. The influence from the open interest is diminished, to around 0, which indicates that the influence of open interest on price volatility is small.

Variance Decomposition of SOYBEANRT

Variance Decomposition of CORNRT

100

100

80

80

60

60

40

40

20

20

0

0

2

4

6

8 10 12 14 16 18 20

2

4

SOYBEANRT SOYBEANVOL SOYBEANDOI

6

8 10 12 14 16 18 20

CORNRT CORNVOL CORNDOI

Variance Decomposition of SOYBEANVOL

Variance Decomposition of CORNVOL 100

90 80 70 60 50 40 30 20 10 0

80 60 40 20 0

2

4

6

8 10 12 14 16 18 20

2

4

SOYBEANRT SOYBEANVOL SOYBEANDOI Variance Decomposition of SOYBEANDOI 80 70 60 50 40 30 20 10 0

6

8 10 12 14 16 18 20

CORNRT CORNVOL CORNDOI Variance Decomposition of CORNDOI

60 50 40 30 20 10 0

2

4

6

8 10 12 14 16 18 20

SOYBEANRT SOYBEANVOL SOYBEANDOI

Figure 7.4 (cont.)

2

4

6

8 10 12 14 16 18 20

CORNRT CORNVOL CORNDOI

Variance Decomposition of COTTONRT

Variance Decomposition of WSRT

120

100

100

80

80

60

60 40

40

20

20 0

0

2

4

6

8 10 12 14 16 18 20

COTTONRT

2

COTTONVOL

4

6

WSRT

8 10 12 14 16 18 20 WSVOL

WSDOI

COTTONDOI Variance Decomposition of COTTONVOL

90 80 70 60 50 40 30 20 10 0 2

4

6

Variance Decomposition of WSVOL 90 80 70 60 50 40 30 20 10 0

8 10 12 14 16 18 20

COTTONRT

2

COTTONVOL

4

6

WSRT

8 10 12 14 16 18 20 WSVOL

WSDOI

COTTONDOI Variance Decomposition of COTTONDOI

Variance Decomposition of WSDOI

60

70

50

60 50

40

40

30

30

20

20

10

10

0

0

2

4

6

8 10 12 14 16 18 20

COTTONRT

COTTONVOL

COTTONDOI

Figure 7.4 (cont.)

2

4

6

WSRT

8 10 12 14 16 18 20 WSVOL

WSDOI

Response of NRRT to Cholesky One S.D. Innovations

Response of CURT to Cholesky One S.D. Innovations 1.0

1.2

0.8

1.0 0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 –0.2

–0.2 2

4

6

CURT

8 10 12 14 16 18 20 CUVOL

2

CUDOI

6

NRRT

Response of CUVOL to Cholesky One S.D. Innovations .7 .6 .5 .4 .3 .2 .1 .0 –.1

4

8 10 12 14 16 18 20 NRVOL

NRDOI

Response of NRVOL to Cholesky One S.D. Innovations .6 .5 .4 .3 .2 .1 .0 –.1

2

4

6

CURT

8 10 12 14 16 18 20 CUVOL

2

CUDOI

6

NRRT

Response of CUDOI to Cholesky One S.D. Innovations

8 10 12 14 16 18 20 NRVOL

NRDOI

Response of NRDOI to Cholesky One S.D. Innovations

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

–0.2

4

–0.2 2

4

6

CURT

8 10 12 14 16 18 20 CUVOL

CUDOI

Figure 7.5 Impulse response map

2

4

6

NRRT

8 10 12 14 16 18 20 NRVOL

NRDOI

Response of SOYBEANRT to Cholesky One S.D. Innovations 1.2

Response of CORNRT to Cholesky One S.D. Innovations 1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 –0.2

–0.2 2

4

6

8 10 12 14 16 18 20

SOYBEANRT

2

4

6

CORNRT

SOYBEANVOL

SOYBEANRT

CORNVOL

CORNDOI

SOYBEANDOI Response of SOYBEANVOL to Cholesky One S.D. Innovations .8 .7 .6 .5 .4 .3 .2 .1 .0 –.1 2 4 6 8 10 12 14 16 18 20

8 10 12 14 16 18 20

Response of CORNVOL to Cholesky One S.D. Innovations 1.2 1.0 0.8 0.6 0.4 0.2 0.0 –0.2 2

4

6

8 10 12 14 16 18 20

CORNRT

SOYBEANVOL

CORNVOL

CORNDOI

SOYBEANDOI Response of SOYBEANDOI to Cholesky One S.D. Innovations 1.2

Response of CORNDOI to Cholesky One S.D. Innovations 1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 –0.2

–0.2 2

4

6

8 10 12 14 16 18 20

SOYBEANRT

SOYBEANVOL

SOYBEANDOI

Figure 7.5 (cont.)

2

4

6

8 10 12 14 16 18 20

CORNRT

CORNVOL

CORNDOI

Response of WSRT to Cholesky One S.D.Innovations

Response of COTTONRT to Cholesky One S.D. Innovations 1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

–0.2 2

4

6

8 10 12 14 16 18 20

2

4

8 10 12 14 16 18 20

WSRT

COTTONRT COTTONVOL COTTONDOI Response of COTTONVOL to Cholesky One S.D. Innovations 1.2

1.0

1.0

0.8

0.8

6

WSVOL WSDOI

Response of WSVOL to Cholesky One S.D. Innovations

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

–0.2

–0.2 2

4

6

8 10 12 14 16 18 20

2

4

6

8 10 12 14 16 18 20

WSRT

COTTONRT COTTONVOL COTTONDOI Response of COTTONDOI to Cholesky One S.D. Innovations 1.2

1.2

1.0

1.0

WSVOL WSDOI

Response of WSDOI to Cholesky One S.D. Innovations

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

–0.2 2

4

6

8 10 12 14 16 18 20

COTTONRT COTTONVOL COTTONDOI

Figure 7.5 (cont.)

2

4

6

8 10 12 14 16 18 20

WSRT

WSVOL WSDOI

176

Intraday effect

Open interest has a strong response immediately on arrival of its innovation of deviation. It has a substantial increase in movement initially, then it weakens rapidly (after the first two orders, to around 1/10), and then starts the slow decay, attenuating to 0 at the 10th order. The sequence has positive and immediate response to the influence, which comes from the innovation of deviation of absolute yield, and it achieves the maximum level. It starts a slow decay after the 4th order, and almost disappears after order 16. The influence from open interest innovation of deviation also has immediate positive response to the influence of new information deviation on absolute yield; it is weakened rapidly after the first 5 orders but remains stronger than the response itself and almost disappears after 20 orders. This result matches the conclusion of Granger causality tests, i.e. volume has a great influence on open interest. It can be seen from the Figure 7.5 of impulse response that the behavior of the Chinese futures market follows the Sequential Information Arrival Hypothesis suggested by Copeland (1976). The transmission of market information is step-by-step. Asymmetric information has a longer lasting impact on volume and volatility. Absolute yield and volume are different forms of the effect of arrival of new information, and there is a common movement trend. While they affect each other mutually, this helps the reflection of impact of the new information on market. Large volume goes with high absolute yield. Information of volume contributes to the forecast of price volatility.

7.5

Conclusion

This chapter studies the minute by minute high-frequency data on yield, volume and open interest of six commodities futures in the three markets of China, and comes to the following conclusions: (1) Absolute intraday yield and volume movements are of the L-pattern. This is caused by the reaction of the market to release of information accumulated during the break hours, psychology of investors and the expectations of the markets. Meanwhile, it shows that intraday characteristics of market and the intraday movement patterns are not the only behavior of the market, which is also driven by the mechanism of continuous bidding under the market-making system. The automatic orders matching system of Chinese markets also has its own effects on intraday movement patterns. The results enrich the theory of the microstructure of the market. (2) The high volatility around the morning opening is because of release of information accumulated during non-trading hours, i.e. it is not caused by the call auction mechanism. (3) Among absolute yield, volume and open interest, there exist two-way Granger causalities in the futures market. This is different from the stock market, where volume is the Granger causal of absolute yield, and

Intraday effect

177

absolute yield doesn’t Granger cause volume since there is a shorting mechanism in the commodities futures market. (4) We give an empirical analysis of the dynamic relationships among absolute yield, volume and open interest. When the absolute yield is interpreted as an explanatory variable, volume explains about 10 per cent of the residual disturbance, and the impact of open interest on price volatility is very small. When volume is treated as an explanatory variable, absolute yield explains about 20 per cent of the residual disturbance. The impact of open interest on volume is very small. When open interest is the variable to be interpreted, volume explains about 25 per cent to 45 per cent of the residual disturbance, and the impact of absolute yield on open interest is around 5 per cent to 10 per cent. Explanatory variables are stable after around 20–30 minutes. (5) Absolute yield and volume have strong interactions and their mutual influence is positive. Information of volume contributes to the forecast of price volatility. Information dissemination by Chinese futures market follows the Sequential Information Arrival Hypothesis. Our conclusions are reached under the microstructure of the market, which follows the new mechanism of continuous bidding and automatic matching system. In spite of being a new market, when its microstructure is compared with those of matured foreign markets (such as New York, Chicago and the Nasdaq), which are directed by the price matching mechanism with marketmaker rules, the Chinese market has a certain degree of similarity with them, such as high price volatility at the opening. It also has some different and unique features, such as no high price volatility and large volume at the closing hours. Moreover, the different mark-to-market trading mechanism of futures, a marginal exchange mechanism and source of information contribute to the intraday characteristics of futures markets, which are different from those of the stock markets. Because change of volume has impact on price volatility to a certain extent, when investors forecast and analyze the trend of futures price, they should not only consider the change of price, but should also consider changes in volume, to improve the ability to understand the movement of the futures market and to detect market opportunities. Forecasting of yield volatility improves management of risks, too.

8

Conclusions and perspective studies

This book focuses on two key issues belongs to both time series econometrics and financial econometrics: the information spillover effect between financial markets and the autoregressive conditional duration (ACD) models tailored for high frequency data. We proposed both theoretical methodology and empirical application. Theoretically, we contribute to provide a new statistical methodology with comparative advantages for analyzing extreme comovements between two time series. Empirically, on one hand, we explore the extreme risk spillover between the Chinese stock market and the international market by the proposed test, in view of its specific characterizations. We also explore the information spillover between the futures market and spot market. On the other hand, we systematically investigate the density forecast performance of commonly used ACD models by applying new portmanteau tests, where often informal tests are used and not in an integrated framework; exchange rate data and futures data are adopted. Some new results are found. In Chapter 2, based on a new concept of Granger causality in risk which focuses on the comovements between the tails of the two distributions, a class of kernel-based statistical tests are proposed to test whether a large downside risk in one market will Granger-cause a large downside risk in another market. The proposed test checks a large number of lags but avoids suffering from loss of power due to the loss of a large number of degrees of freedom, thanks to the use of the downward weighting kernel function. This downward weighting is consistent with the stylized fact that financial markets are more influenced by more recent events than by remote past events, thus enhancing power of the proposed tests. It is proved that the proposed tests have a convenient asymptotically standard normal distribution under the null hypothesis of no Granger causality in risk and certain regular conditions. A simulation study shows that the proposed tests have reasonable levels and power against a variety of empirically plausible alternatives in finite samples, including the spillover from the dynamics in mean, variance, skewness and kurtosis respectively and further demonstrate it to have relative advantages compared with traditional regression based tests and are useful in investigating adverse large market comovements between financial markets such as financial contagions.

Conclusions

179

Simulation studies show that the procedures have reasonable level and good power against some empirically plausible alternatives in finite samples, no matter whether risk spillover arises from spillover in mean, in variance, or in skewness and kurtosis. In Chapter 3, we adopt the parametric approach, semi-parametric approach and nonparametric approach to estimate the conditional VaR and unconditional VaR of the copper futures market. In respect that good news and bad news have different impact on the market volatility, EGARCH and TGARCH models are adopted in parametric estimation, volatility, skewness and kurtosis are employed in semi-parametric estimation and nonparametric estimation is based on kernel function. Generally, financial markets consider Downside VaR only. However, Upside VaR, as a new concept, is being introduced for the first time in this chapter and both Downside VaR and Upside VaR are estimated with the preceding three approaches. Moreover, Kupiec’s backtest has been done to evaluate the efficiency of each approach. Parametric estimation and semi-parametric estimation fail the tests at some confidence levels, while nonparametric estimation passes the tests at all the three confidence levels. Our findings indicate that the asymmetric factors are all significant in both of the two GARCH models. Therefore, although the trading mechanism is symmetric in China’s futures market, good news and bad news have an asymmetric effect upon the volatility of futures market. The leverage effect is still significant and the market is more sensitive to good news than bad news. Moreover, combining parametric and nonparametric approach is an appropriate VaR estimation. In Chapter 4, we provide an empirical study on spillover of extreme downside market risk among Shares A, B and H in the Chinese stock market, between different stock markets in greater China, and between the Chinese stock market and other international capital markets. It is found that there exists strong risk spillover between Share A indices and Share B indices, and the occurrence of a large downside risk in Share B markets can help predict the occurrence of a similar future risk in Share A markets. There also exists strong risk spillover between Share A and Share H, and particularly between Share B and Share H. Share B, and particularly Share H, have significant risk spillover with the Asian and international stock markets. In contrast, although Share A has some risk spillover with Korean and Singapore stock markets, it has no risk spillover with leading international mature capital markets Japan, US and Germany. Our findings suggest that the market segmentation between Share A and Share B is effective in avoiding large adverse shocks from international capital markets. In terms of large adverse market movements, the Chinese stock market has some ties with the Asian stock markets, but its link with leading international capital markets is still weak. In Chapter 5, we employ the parametric approach based on TGARCH and GARCH to estimate the VaR of the copper futures market and spot market in China. Considering the short selling mechanism in the futures market, we introduce two notions: Upside VaR and extreme upside risk spillover for the

180

Conclusions

first time. And the Downside VaR and Upside VaR are examined by using the above approach. Also, we use Kupiec’s backtest to test the power of our approaches. In addition, we investigate information spillover effects between the futures market and the spot market by employing linear Granger causality test, Granger causality test in mean, volatility and risk. Moreover, this paper studies firstly the relationship between the futures market and the spot market by using test statistics based on kernel function. Empirical results indicate that there exist significant two-way spillover effects between the futures market and the spot market, and the spillover effects from the futures market to the spot market are much more significant. In Chapter 6, using a new omnibus density forecast evaluation procedure, we examine various commonly used ACD models in capturing the price duration dynamics of Euro/Dollar and Yen/Dollar exchange rates. The ACD models under investigation include Linear, Logarithmic, Box– Cox, Exponential, Threshold and Markov Switching ACD models with the Exponential, Weibull, generalized Gamma and Burr innovation distributions respectively. We find that none of the ACD models can adequately capture the full dynamics of foreign exchange rate price durations, either in-sample or out-of-sample. However, some ACD models, particularly the Markov switching ACD model with Burr innovations, have not only the best in-sample fit, but also the best out-of-sample performance. It is found that sophisticated nonlinear specifications for the conditional mean duration do not help much over linear ACD models in capturing the full dynamics of price durations, but the specification of the innovation distribution is important: the generalized Gamma or Burr distribution performs much better than the Weibull and exponential distributions. Moreover, it is important to relax the independence assumption for innovations and model higher order conditional moments of price durations. In Chapter 7, intraday effect is explored. It uses minute by minute data series to study patterns of intraday movement of yields and volumes and discovers the ‘L’ pattern of intraday absolute yield and volume. We apply the financial market microstructure theory, traders psychology and the trading mechanism to explain the different intraday patterns in commodities futures market and the stock market, which has a distinctive intraday movement pattern of ‘U’-type. Therefore, using the Granger causality test theories and Vector Autoregressive models (VAR), we study the factors that influence volatility of yields and the lagged orders. The results show that there is a two-way Granger causality among any two of the absolute yield, volume and open interest, and it is different from the empirical results of the stock market, in the sense that there is only a one-way Granger causal relationship from volume to absolute yield; the difference is accounted for by the different mechanism of the futures market. Using decomposition of variance of VAR models and the impulse response analysis, we analyze the dynamic relationship among the three factors. The empirical results tell that the influence of open interest on volatility

Conclusions

181

of absolute yield and volume is weak, and there is a strong correlation between volatility of absolute yield and volume. During the past thirty-five years, time-series econometrics has developed from infancy to relative maturity. And the financial econometrics is a very rapidly growing area of econometrics, which might be due to several reasons: one is that financial theories are very precise and very much amenable to testing, and another reason is that the data are very high quality, especially compared to the data we are used to in macro, labor and some of the other areas where there is a lot more concern about the data quality. In China, the research in these fields starts relatively late and there may be a big gap compared with the advanced research level. But it has been paid much attention to in recent years. Many universities open related courses and strengthen the communication with overseas academicians. All these have provided us not only good opportunities but also great challenges to do further research.

Appendix Mathematical proof

Proof of Theorem 2.1: Let f (X ) and f10 (X ) be defined in the same way as f (X ) and f10 (X ) in (2.11) and (2.12) with Rˆ y (Rˆ1 ,Rˆ2 ) a replaced by Rˆ 0 y (Rˆ 1 , Rˆ 2 ) a . d

Given D1T (M)  M ° k 4 (z)dz[1  o(1)] as M → ∞ under Assumption 6, it suf0

fices to show Theorems A.1 and A.2, below, under the conditions of Theorem 1. Theorem A.1 implies that parameter estimation uncertainty in Rˆ has no impact on the limit distribution of Q1 (M). The main technical challenge for the proof of Theorem A.1 is that the risk indicator Zlt (Rl ) y 1(Ylt  Vlt ) is not differentiable with respect to parameter Rl .



1

Theorem A.1: M 2 T [ L2 f ,f 10 L2 ( f,f 01 )] m p 0, where L2 ( i ) is defined as in (2.19). 1

Theorem A.2: [TL2 (f,f10 ) C1T (M)] / [ 2 D1T (M)] 2 md N ( 0,1). Proof of Theorem A.1: Throughout, let C(j) be defined as Cˆ (j) in (2.15) with Rˆ replaced by R 0 . We further replace the sample proportions Bˆ l and B l in Cˆ (j) and C(j) with α. Such a replacement does not affect the asymptotic distribution of Q1 (M). Putting T 2 y B (1 B ), we have T 1

T [L2 ( f , f 10 ) L2 (f,f10 )]  T 4T ¤ k 2 (j/ M)[C(j) C(j)]2 j 1 T 1

 2T 4T ¤ k 2 (j/ M)[C(j) C(j)]C (j)

(A.1)

j 1

y T 4TQ1  2T 4TQ 2 . We shall prove Theorem A.1 by showing Propositions A.1 and A.2 below. 1

Proposition A.1: M 2 TQˆ1 m p 0.

Mathematical proof

183

1

Proposition A.2: M 2 TQˆ 2 m p 0.

Proof of Proposition A.1: By straightforward algebra, we have for j > 0, ˆ 1 (j,Rˆ1 )  M ˆ 2 (j,Rˆ2 )  M ˆ 3 (j,Rˆ1 ,Rˆ2 ) Cˆ (j) C(j)  M

(A.2)

where T

ˆ 1 (j,R1 ) y T 1 M

¤ [Z

1t

(R1 ) Z1t (R10 )][Z 2 (t j) (R 20 ) B ]

t  j 1 T

ˆ 2 (j,R 2 ) y T 1 M

¤ [Z

1t

(R10 ) B ][Z 2 (t j) (R 2 ) Z2 (t j) (R 20 )]

t  j 1

ˆ 3 (j,R1 ,R 2 ) y T 1 M

T

¤ [Z

1t

(R1 ) Z1t (R10 )][Z 2 (t j) (R 2 ) Z2 (t j) (R 20 )].

t  j 1

By the definition of Qˆ1 in (A.1), we have ˆ (Rˆ )  Q ˆ (Rˆ )  Q ˆ (Rˆ ,Rˆ )] Qˆ1 b 3[Q 11 1 12 2 13 1 2

(A.3)

where T 1

ˆ 12 (j,R1 ) Qˆ11 (R1 ) y ¤ k 2 (j/ M) M j 1

T 1

ˆ 22 (j,R 2 ) Qˆ12 (R 2 ) y ¤ k 2 (j/ M) M j 1

T 1

ˆ 32 (j,R1 ,R 2 ). Qˆ13 (R1 ,R 2 ) y ¤ k 2 (j/ M) M j 1

Because the risk indicator Zlt (Rl ) is not differentiable with respect to Rl , we shall use the uniform convergence argument to show that Qˆ11 (Rˆ1 ), Qˆ12 (Rˆ2 ) and Qˆ13 (Rˆ1 ,Rˆ2 ) vanish in probability with suitable rates. Given Assumption 4, we have that for any given constant F  0, there exists $ 0 y $ 0 ( F )  d such 1 ¥

´ that P ¦ Rl Rl0  $ 0 T 2 µ  F for T suffciently large. Hence, it suffces to show § ¶ Lemmas A.1–A.3 below.

184

Mathematical proof

1 «

º Lemma A.1: Put 1l0 y ¬Rl  1l : Rl Rl0 b $ 0T 2 » for 0  $ 0  d , l  1, 2 ­ ¼

Then for any given constant $ 0  0 , supR1 110 TQ11 (R1 )  OP (1)

and

1

supR1 110 M 2 TQ11 (R1 ) m p 0 . Proof of Lemma A.1: This is one of the most involved proofs, due to the fact that the risk indicator Zlt (Rl ) is not differentiable with respect to Rl . Recalling Zlt (Rl ) y 1[Ylt  Vlt (Rl )] , we put Wlt (Rl ) y Zlt (Rl ) Zlt (Rl0 ) E[Zlt (Rl ) I t 1 ]  E[Zlt (Rl0 ) I t 1 ]  Zlt (Rl ) Zlt (Rl0 ) Flt [ Vlt (Rl )]  Flt [ Vlt (Rl0 )],

(A.4)

where Flt (y I t 1 ) is the conditional CDF of Ylt given I t 1. Given each Rl 1l , we have E [Wlt (Rl ) I t 1 ]  0 a.s., i.e. {Wlt (Rl )} is a m.d.s. with respect to I t 1 for each given Rl . Noting Zlt (Rl0 )  Zlt y 1(Ylt b Vlt ) a.s. under Assumption 3(i) ˆ 1 (j,R1 ) in (A.2), we have for j>0, and recalling the definition of M ˆ 1 (j, R1 )  T 1 M

T

¤

[Z1t (R1 ) Z1t (R10 )][Z 2 (t j) (R02 ) B ]

t  j 1 T

 T 1 T

¤W

1t

t  j 1 T

1

¤

(R1 )[Z 2 (t j) B ]

(A.5)

{F1t [ V1t (R1 )] F1t [ V1t (R10 )]}[Z 2 (t j) B ]

t  j 1

ˆ 11 (j, R1 )  M ˆ 12 (j, R1 ). yM ˆ 12 (j,R1 ) in (A.5). By a second order Taylor series expanWe first consider M sion, we have T

uF1t [ V1t (R10 )] [Z 2 (t j) B ] uR1 t  j 1 T 1 u 2 F1t [ V1t ( R1 )]  (R1 R10 )a T 1 ¤ [Z 2 (t j) B ](Rˆ 1 R10 ), 2 u R u R a 1 1 t  j 1 1 0 ˆ ˆ 122 (j, R1 )(R1 R10 ) y (R1 R1 ) a M121 (j, R1 )  (R1 R10 ) a M 2

ˆ 12 (j, R1 )  (R1 R10 ) a T 1 M

¤

(A.6)

T

where R1 lies between R1 and R10. Put '(j) y T 1

¤S

1t

t  j 1

S1t (R1 ) is defined in Assumption 5. Then

(R10 )[Z 2 (t j) B ], where

Mathematical proof

185

T 1

sup T ¤ k 2 (j/ M)[(R1 R10 )M121 (j, R1 )]2

R1 110

j 1

2

b 2T R1 R10

T 1

¤k

2

2

(j/ M) ' (j)  2T R1 R10

2

j 1

T 1

¤k

2

2

(j/ M) ' (j) ' (j) ,

j 1

 OP (1)  O P (M/ T) (A.7)

1

T 1

d

2

2

where supR1 110 R1 R10 b $ 0T 2 , ¤ j 1 k 2 (j/ M) ' (j) m ¤ h 1 ' (h)  O(1) given Assumptions 5(i) and 6, M → ∞,M/T → 0, and T 1

¤k

2

2

(j/ M) ' (j) ' (j)  OP (M/ T)

j 1

2

by Markov’s inequality and sup0  j T E ' (j) ' (j) b $T 1. Note that 2

sup0  j T E ' (j) ' (j) b $T 1 follows from Assumption 5 (e.g. Hannan (1970), p.209). For the second term in (A.6), we have T 1

sup T ¤ k 2 (j/ M) (R1 R10 ) aM122 (j, R1 )(R1 R10 )

R1 110

2

j 1

0 4 1

b sup R1 R R1 110

T 1

(A.8)

2

T ¤ k (j/ M) M 122 (j, R1 )  OP (M/ T), 2

j 1

1

given supR1 110 R1 R10 b $ 0T 2 , Assumption 2, and Markov’s inequality. It follows from (A.6)–(A.8), M → ∞,M/T → 0 that T 1

2

sup T ¤ k 2 (j/ M) M 12 (j, R1 )  OP (1).

R1 110

(A.9)

j 1

ˆ 11 (j,R1 ) in (A.5). Divide the cube 110 , which We now consider the first term M is centered at R10 with size 2 $ 0T

1 2

, into approximately LT y ( 2 $ 0 / FT )d1 1

cubes {11 (l ), l  1,...,LT } of size FT / T 2 , where FT y M

1 2

/ ln(T) m 0 and

186

Mathematical proof

d1 is the dimension of R1 . For 1 b l b LT , put R1at (l ) y infR 1 ( l ) F1t [ V1t (R1 )] 1 and R1bt (l ) y supR1 1 ( l ) F1t [ V1t (R1 )]. Note that R1ta (l ) and R1tb (l ) are measurable 1

I1(t 1) because V1t (R1 ) is a measurable function of I1t 1. Then, for

functions of

any R1  11 (l ) , we write T

ˆ 11 (j, R1 )  T 1 M

¤ {Z

1t

(R1 ) Z1t (R10 ) F1t [ V1t (R1 )]+F1t [ V1t (R10 )]}Z2 (t j)

t  j +1 T

1

¤ {Z

BT

1t

(R1 ) Z1t (R10 ) F1t [ V1t (R1 )]  F1t [ V1t (R10 )]}

t  j 1 T

b T 1 ¤ {Z1t [R1bt (l )] Z1t (R10 ) F1t [ V1t (R1at (l ))]  F1t [ V1t (R10 )]} t 1 T

T

1

T

1

¤ {Z

1t

[R1at (l )] Z1t (R10 ) F1t [ V1t (R1bt (l ))]  F1t [ V1t (R10 )]}

t 1 T

 2T

¤ {W

1t

t 1 T

1

[R1bt (l )] W1t [R1at (l )]}

¤ {F [ V 1t

1t

(R1bt (l ))] F1t [ V1t (R1at (l ))]}.

t 1

Similarly, we can obtain T

ˆ (j, R ) r T 1 {W [Rb (l )] W [Ra (l )]} M 11 1 1t 1t ¤ 1t 1t t 1

T

2T 1 ¤ {F1t [ V1t (R1bt (l ))] F1t [ V1t (R1at (l ))]}. t 1

It follows that T

max

0  j T

sup M 11  j , R1 b T 1 ¤ {W1t [R1bt (l )] W1t [ R1at ( l )]}

R1 11 l

t 1

T

 2 $ 0 FT T 2 T 1 ¤ sup 1

0 t 1 R1 11

uF1t ; V1t (R1 )= , uR1 (A.10)

1

given R1bt (l ) R1at (l ) b $ 0 FT / T 2 . Note that the second term in (A.10) does not depend on l. Therefore, we have

Mathematical proof T 1

sup T ¤ k 2 (j/ M) M 11  j , R1

R1 110

187

2

j 1

T 1

sup T ¤ k 2 (j/ M) M 11  j , R1

 max 1b l b LT

R1 11 ( l )

2

j 1

2 T 1

T

b 2 max T T 1 ¤ {W1t [R1bt (l )] W1t [R1at (l )]} 1b l b LT

t 1

T

¤k

2

(j/ M),

(A.11)

j 1 T 1

uF1t [ V1t (R1 )] 2 ] ¤ k 2 (j/ M) uR1 j 1

4T  $ 20 FT2 / T [T 1 ¤ sup

0 t 1 R1 11

 OP (M FT2 )  o P 1 given FT  M

1 2

/ ln T , where we have made use of the fact that

2 T ¥ ´ P ¦ max T T 1 ¤ {W1t ¨ªR1bt l ·¹ W1t ¨ªR1at l ·¹}  $ 20 FT2 µ t 1 § 1b l b LT ¶ 2 LT T ¥ ´ b ¤ P ¦ T 1 ¤ {W1t [R1bt (l )] W1t ¨ªR1at l ·¹}  $ 20 FT2 / T µ l 1 t 1 § ¶ LT

4

T

2

[

b ¤  $ 20 FT2 / T E T 1 ¤ W1t [R1bt (l )] W1t ¨ªR1at l ·¹ l 1

t 1

]

1 2 1 · ¨ ¥ ´ ¥ ´ d b 2 $ 0 / FT 1 ( $ 20 FT2 / T ) 2 ©T 2 ¦ $ 0 FT / T 2 µ  T 3 ¦ $ 0 FT / T 2 µ ¸ § ¶ § ¶ ¸¹ ©ª 3

d 2

d 3  O ( FT  1 / T  FT  1 / T 2 ) m 0

given M  cT V , FT  M

1 2

/ ln T , V 

2 , where the third inequality d1  2

follows from 4

T

ET

1

¤ {W

1t

2

[R (l )] W1t [R (l )]} b T b 1t

a 1t

t 1

2

1 1 ¥ ´ ¥ ´

3 ¦§ $ 0 FT / T 2 µ¶  T ¦§ $ 0 FT / T 2 µ¶

by Rosenthal’s inequality (e.g. Hall and Heyde (1980), p.23) the fact that {W1t [R1bt l ] W1t [R1at l ],Ft 1 } is a m.d.s. and the fact that E W1t [R1bt l ] W1t [R1at l ]

m

1

b $ 0 FT / T 2 for any m r 1 1

by the law of iterated expectation and R1bt l R1at l b $ 0 FT / T 2 . Note that a larger m does not imply a faster convergence rate due to the very nature of

188

Mathematical proof

the indicator function. The desired result then follows from (A.5), (A.9) and (A.11). Lemma A.2: For any given constant $ o  0, we have

1

supR2 102 TQ12 R2  OP 1 and supR2 102 M 2 TQ12 R2 m p 0. Proof of Lemma A.2: Similar to the proof of Lemma A.1. Lemma A.3: Put 1 0 y 110 ƒ 1 02 and R y R1 ,R 2 a . Then for any given constant $ 0  0,

1

supR10 TQ13 R1 , R2 m p 0 and supR10 M 2 TQ13 R1 , R2 m p 0. Proof of Lemma A.3: Recalling the definition of Mˆ 3  j , R1 , R2 as in (A.2) and Zlt Rl0  Zlt , we write ˆ 3  j , R1 , R2  T 1 M

T

¤ [Z R Z R ][Z  R Z  R ] 1t

1

0 1

1t

2 t j

2

2 t j

0 2

t  j 1 T

T 1 T

¤ W R ¨ªZ  R Z  1t

t  j 1 T

1

1

2 t j

2 t j

2

¤ {F [ V R ] F 1t

1t

1

1t

t  j 1

· ¹

¨ª V1t R10 ·¹} ¨ªZ2t j R2 Z2t j ·¹

ˆ 31  j , R1 , R2  M ˆ 32  j , R1 , R2 . yM (A.12) ˆ 31  j ,R1 ,R 2 in (A.12), following reasoning analogous to that for For M ˆ M11  j ,R1 in the proof of Lemma A.1, we can obtain T 1

2

sup T ¤ k 2  j / M M 31  j , R1 , R2 m p 0. R 1 0

(A.13)

j 1

ˆ 32  j ,R1 ,R 2 in (A.12), by the mean value theorem, we have For M ˆ 32  j , R1 , R2  R1 R10 a T 1 M

T

¤

t  j 1

 ¨Z

uF1t [ V1t R1 ] uR1

ª

2 t j

R2 Z2t j R02 ·¹

¥ 3 ´  OP ¦ T 4 µ § ¶ (A.14)

Mathematical proof

189

uniformly in R1 ,R 2 1 0 . Here, we have used the facts that R1 R10 b $ 0T and T

E sup T 1 R 1 0

¤

uF1t ¨ª V1t R1 ·¹ uR1

t  j 1

1 2

2

¨Z2t j R 2 Z2t j R 20 · ª ¹

« T uF1t ¨ª V1t R1 ·¹ ® b ¬T 1 ¤ E sup uR1 R1 110 t 1 ®­ b CT

2

2 º« º® ® ® 1 T 0 ¨ · T E sup Z R

Z R   »¬ ¤ 2t 2 2t 2 ¹ », ª R 2 1 20 t 1 ®¼ ­® ¼®

1 2

given Assumption 2 and 2

E sup ¨ªZ2t R 2 Z2t R 20 ·¹ b E {I ¨ª V2t R 2at  Y2t  V2t R 2bt ·¹} R 1 0 2

2

[

]

 E F2t ¨ª V2t R 2bt ·¹ F2t ¨ª V2t R 2at ·¹ b $ 0T given

R 2at R 2bt b $ 0T

1 2

,

R 2at y arg infR2 102 F2t ¨ª V2t R 2 ·¹

where

1 2

and

R 2bt y arg supR2 102 F2t ¨ª V2t R 2 ·¹ , and the equality follows by the law of iterated expectations and the fact that R 2ta and R 2tb are measurable functions of I 2t 1 . It follows that T 1 1 2 ¥ ´ supT ¤ k 2  j / M M 32  j , R1 , R2  OP ¦ M / T 2 µ  oP 1 . § ¶ R 1 02 j 1

(A.15)

Combining (A.12), (A.13) and (A.15) yields the desired result. Proof of Proposition A.2: Recalling the definition of Qˆ 2 in (A.1) and using (A.2), we can write









Qˆ 2  Qˆ 21 Rˆ1  Qˆ 22 Rˆ2  Qˆ 23 Rˆ1 ,Rˆ2 , where T 1

ˆ 1  j , R1 C  j , Qˆ 21 R1 y ¤ k 2  j / M M j 1

(A.16)

190

Mathematical proof T 1



ˆ 2  j ,R 2 C  j , Qˆ 22 Rˆ2 y ¤ k 2  j / M M j 1





T 1

ˆ 3  j , R1 , R2 C  j . Qˆ 23 Rˆ 1 , Rˆ 2 y ¤ k 2  j / M M j 0

Following reasoning analogous to that of Proposition A.1, it suffices to show Lemmas A.4–A.6:

1

Lemma A.4: For any given constant $ 0  0, supR1 110 M 2 TQ 21 R1 m p 0 . ˆ 1  j ,R1  M ˆ 11  j ,R1  M ˆ 12  j ,R1 in (A.5), Proof of Lemma A.4: Recalling M we have T 1

 yQˆ Rˆ  Qˆ Rˆ .



T 1



ˆ 11 j , Rˆ 1 C  j  ¤ k 2  j / M M ˆ 12 j , Rˆ 1 C  j Qˆ 21 Rˆ 1  ¤ k 2  j / M M j 1

211

1

212

j 1

1

(A.17) For the first term in (A.17), we have 1

1

¨ T 1 ·2 2 sup M T Q 211 R1 b ©T ¤ k 2  j / M M 11  j , R1 ¸ R1 110 ª j 1 ¹ y oP 1 OP 1  oP 1

¨ 1 T 1 2 ·2 © M T ¤ k  j / M C 2  j ¸ j 1 ª ¹

1 2

(A.18) by the Cauchy-Schwarz inequality, (A.11), and the fact that T 1

M 1T ¤ k 2  j / M C 2  j  OP 1 ,

(A.19)

j 1

2

which follows by Markov’s inequality, and E C  j b T 1 under H10 . For ˆ 12  j ,R1 can be decomposed as in the second term in (A.17), recalling that M (A.6), we have

Mathematical proof

191

T 1

TQˆ 212 R1  R1 R10 a T ¤ k 2  j / M '  j C  j j 1 T 1

 R1 R a T ¤ k 2  j / M ¨ª'  j '  j ·¹ C  j 0 1

j 1

T 1 1 ˆ 122 j , R1 C  j ,  R1 R10 a T ¤ k 2  j / M M 2 j 1





(A.20)

y T R1 R10 a Dˆ 1 R10  T R1 R10 a Dˆ 2 R10 1  T R1 R10 a Dˆ 3 R1 T R1 R10 2



where R lies between R1 and R10. For the first term in (A.20), we have T 1

Dˆ 1 R10 b T ¤ k 2  j / M '  j C  j  OP 1

(A.21)

j 1

2

by Markov’s inequality, Assumptions 5 and 6 and E C  j b T 1 under H10 . For the second term in (A.20), we have T 1 1 ¥ ´ Dˆ 2 R10 b T ¤ k 2  j / M '  j '  j C  j  OP ¦ M / T 2 µ § ¶ j 1

(A.22)

2

by Markov’s inequality, E '  j '  j b $T 1 given Assumption 5, and 2

E C  j b T 1 under H10 . Similarly, we have



T

Dˆ 3 R1 b T 1 ¤ sup

u 2 F1t ¨ª V1t R1 ·¹

T 1

uR1uR1a

j 1

0 t 1 R1 11

1 ¥ ´  OP ¦ M / T 2 µ § ¶

2

¤ k  j / M C  j 2

(A.23)

given Assumption 6, and E C  j b T 1. Collecting (A.20)–(A.23) and M / T m 0, we obtain

M

1 2

sup TQ 212 R1 m p 0.

R1 110

192

Mathematical proof

This completes the proof for Lemma A.4.

1

Lemma A.5: For any given constant $ 0  0 , supR2 102 M 2 TQ 22 R2 m p 0 . Proof of Lemma A.5: Similar to the proof of Lemma A.4.

1

Lemma A.6: For any given constant $ 0  0, sup( R1 ,R2 )10 M 2 TQ R1 , R2 m p 0. Proof of Lemma A.6: The result follows by the Cauchy-Schwarz inequality, Lemma A.3 and (A.19). Proof of Theorem A.2: The desired result follows from a modification of the proof of Hong (2001b, Theorem 1) by putting ut y Z1t B and Vt y Z2t B . Note that [ut ] is an i.i.d. sequence and ut is independent of [V s , s  t] under H10 . The difference between Theorem 1 of Hong (2001b) and the present case is that in the former, [ut ] and [Vt ] are mutually independent, while in the present case, we should allow for the possibility that Vt may depend on [us , s  t]. Given Assumption 3(ii), however, by going through all steps in the proof of Hong (2001b, Theorem 1), we can show that this does affect the asymptotic normality result of the proposed test statistic. In other words, the asymptotic normality of Q1 M holds under if [ut ] and [V s ] were mutually independent. Proof of Theorem 2.2: Recall that C1T M  O M and D1T M  2M ° k 4  z dz ¨ª1  o 1 ·¹ as M m d and M / T m 0, we have

M T Q M  [2° k  z dz] 1

2

4

1

1 2





L2 fˆ , fˆ10 ¨ª1  o 1 ·¹  o 1 .

Thus, it suffices to show Theorems A.3 and A.4 below.









Theorem A.3: L2 fˆ , fˆ10 L2 f , f10 m p 0. 2



0 1

Theorem A.4: L f , f

L f, f 2

0 1

m p 0.





Proof of Theorem A.3: Recall L2 fˆ , fˆ10  T 4Qˆ1  2T 4Qˆ 2 , where Qˆ1 and Qˆ 2 are as in (A.1). It suffices to show Qˆ1 m 0. The second term Qˆ 2 will also vanish in probability by the Cauchy-Schwarz inequality and Theorem A.4, which p





implies L2 f , f10  OP 1 given Assumption 7.









Next, recall Qˆ1 b 3 ¨ªQˆ11 Rˆ1  Qˆ12 Rˆ2  Qˆ13 Rˆ1 ,Rˆ2 ·¹ , where Qˆ11 R1 , Qˆ12 R 2 and Qˆ13 R1 ,R 2 are defined as in (A.3). We shall show that these three terms all vanish in probability under H1A .

Mathematical proof

193



We first consider Qˆ11 Rˆ1 . Given Assumption 4, we have that for any given





constant F  0, there exists $ 0 y $ 0  F such that P Rˆl Rl0  $ 0T 1 2  F for all T sufficiently large. Thus, it suffices to show Qˆ11 R1 m p 0 uniformly in ˆ ˆ R 1 0 , where 1 0 is as in Lemma A.1. By the definition of Q11 R1 in (A.3), 1

1



1

we have



T 1

sup Q11 R1 b max sup M 1  j , R1 ¤ k 2  j / M ,

R1 110

0  j T R 1 0 1 1

(A.24)

j 1

ˆ 1  j ,R1 is defined in (A.2). where M Put R1at y arg infR1 110 F1t ¨ª V1t R1 ·¹ and R1bt y arg supR1 110 F1t ¨ª V1t R1 ·¹ . Note that R1ta and R1tb are measurable functions of I1t 1 , because V1t R1 depends on I1t 1 and R1 . Then T

max sup M 1  j , R1 b T 1 ¤ sup Z1t R1 Z1t R10

0  j T R 1 0 1 1

0 t 1 R1 11 T

1

bT

¤

(A.25)

Z1t R1bt Z1t R1at  OP T 1 2 ,

t 1

¥ 1 ´ where the OP ¦ T 2 µ term follows from Markov’s inequality and the § ¶ fact that

[



E Z1t R1bt Z1t R1at  E F1t  V1t R1bt F1t Vt R1at b $ 0T

1 2

1

]

u E sup F1t ¨ª V1t R1 ·¹ 0 R1 11 uR

by the law of iterated expectations and R1bt R1at b $ 0T

1 2

. It follows from 1 ¥ ´ (A.24), (A.25), M 2 / T m 0 that supR1 110 Q11 R1  OP ¦ M / T 2 µ  oP 1 . § ¶ Similarly, we have supR2 120 Q12 R2 m p 0 and supR10 Q13 R m p 0 . This completes the proof.

194

Mathematical proof

Proof of Theorem A.4: See Hong (2001b, proof of Theorem 2). Proof of Theorem 2.3: The proof is similar to that of Theorem 2.1, so we omit it here. Part of the proof is more tedious than the proof of Theorem 1 because both positive and negative j’s should be considered, but the other part is simpler because under H 20 ,ut y Z1t B is independent of [V s  Z2t B , s b t 1] and Vt is independent of [us , s b t 1]. This is the reason why we do not need Assumption 3, which is required in Theorem 2.1. Proof of Theorem 2.4: The proof is similar to that of Theorem 2.2. Proof of Theorem 2.5: The proof is similar to that of Theorem 2.1. Note that under H30 , [ut ] and [V s ] are mutually independent, including ut and Vt . The proof is omitted. Proof of Theorem 2.6: The proof is similar to that of Theorem 2.2.

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Index

absolute yield, 8, 153, 157, 158, 162–165, 169, 170, 176, 177, 180, 181 ACD model, 2, 4, 5, 7, 8, 102, 103, 106, 107, 109–115, 119–128, 131–134, 140, 141, 147–151, 178, 180 asymmetric effect, 6, 40, 41, 51, 179 asymptotic theory, 6, 18 auction mechanism, 157, 158, 160, 176 autoregressive conditional duration, see ACD model backtest, 6, 7, 39, 44, 101, 179, 180 bandwidth, 43, 50, 105 BCACD, 7, 110, 111, 119, 123, 128, 129, 131–135, 137, 138, 141–145, 147–149 BHHH, 29, 120 bilateral, 22 Burr distribution, 7, 109, 110, 119–121, 131, 133, 147, 150, 151, 180 Calendar Effect, 152, 153 call auction, 157–160, 176 CAViaR, 15, 20, 38 centering factor, 90 comovement, 3, 5, 14, 15, 27, 35, 178 conditional mean, 4, 5, 7, 12, 87, 106, 109–113, 131, 140, 147, 148, 151, 180 conditional VaR, 2, 6, 40, 51, 179 conditional variance, 2, 12, 28, 40, 96, 109 confidence level, 6, 11, 37–39, 44, 45, 48, 49, 51, 95, 96, 179 copper, 6, 7, 39, 44–49, 51, 85, 86, 91–97, 99–101, 153, 154, 156, 162, 166, 179 cross-correlation function, 2, 16, 89 cross spectral density, 14, 20, 23, 25

Daniell kernel, 11, 22, 28–32, 34–36, 64 density forecast, 5, 102–104, 129, 130, 140, 147, 150, 178, 180 diagnostic statistic, 63, 134 downside risk, 3, 6, 9, 12, 13, 27, 35, 39, 83, 87–90, 98–100, 178, 179 Downside VaR, 6, 7, 38–41, 47–51, 88, 95, 96, 101, 179, 180 duration, 2, 4, 5, 7, 8, 102, 103, 106, 108– 121, 129–133, 135–137, 139, 140, 142, 143, 145–151, 153, 161, 162, 178, 180 econometrics, 9, 14, 86, 113, 150, 178, 181 EGARCH, 6, 40, 46–48, 111, 179 exchange rate, 7, 35, 38, 102, 103, 110, 114, 117, 121, 131, 132, 150, 151, 178, 180 EXPACD, 7, 110–112, 119, 124, 128–132, 134–138, 141, 143–147 exponential distribution, 7, 110, 131, 133, 147, 150, 151, 180 extreme downside risk, 3, 12, 13, 27, 35, 87 extreme downside risk spillover, 3, 87 extreme risk, 5, 9, 10, 35, 48, 52, 53, 178 extreme risk spillover, 5, 9, 10, 35, 52, 53, 178 extreme upside risk, 7, 86, 179 extreme upside risk spillover, 7, 86, 179 financial contagion, 1, 35, 53, 178 financial risk, 1, 2, 6, 9, 11, 37 finite sample, 6, 17, 26, 27, 35, 178, 179 foreign exchanges, 7, 102, 112–114, 120, 131, 140, 150, 151 futures market, 6–8, 37–40, 44, 46–51, 85, 86, 91–97, 99–101, 154, 157, 159–164, 176–180

208

Index

GARCH, 2–4, 6, 7, 12, 19, 20, 38, 40, 41, 46–48, 51, 60, 61, 63, 85, 86, 93–95, 101, 110, 111, 179 generalized Gamma distribution, 4, 7, 110, 131, 133, 147 generalized residual, 103, 104, 120, 131–140, 142, 143, 145–149 Granger causality, 2, 3, 6–8, 10, 11, 13–15, 17, 20–22, 24, 26–36, 55, 60, 64, 68, 72, 73, 78, 86–90, 97–101, 161–165, 176, 178, 180 Granger causality in downside risk, 88, 89, 98, 99 Granger causality in mean, 14, 17, 27–29, 33, 87, 88, 97, 98, 100 Granger causality in upside risk, 87–89, 99 Granger causality in volatility, 87, 88, 98 Granger-cause, 6, 13, 22, 24, 35, 64, 68, 72, 73, 86, 178 high frequency data, 8, 110, 152–154, 164, 176, 178 IGARCH, 12, 19 impulse response, 8, 161, 164, 169, 173, 176, 180 in sample, 7, 8, 102, 104, 105, 120, 128, 129, 132–135, 140–142, 147, 149–151, 180 indicator function, 14, 17, 88, 89, 105, 188 inference statistics, 141 information spillover, 2, 6, 7, 85, 86, 92, 97, 100, 101, 178, 180 innovation distribution, 4, 5, 7, 27, 35, 109, 110, 112, 113, 119–121, 131–133, 135–137, 139, 142, 143, 145–147, 150, 151, 180 intraday effect, 152, 180 intraday pattern, 8, 153, 158, 160, 180 investment, 1, 8, 10, 18, 52–55, 57, 64, 72, 73, 78, 82, 83, 101, 159, 161 jump process, 33 kernel function, 6, 7, 11, 19, 38, 42, 43, 49–51, 86, 90, 101, 104, 107, 178, 179 kurtosis, 6, 12, 14, 27, 28, 33, 35, 41, 45, 46, 49, 57–60, 93, 106, 109, 135–137, 139, 142, 143, 145, 146, 148, 155, 157, 178, 179

leverage effect, 40, 41, 47, 51, 60, 63, 106, 179 LINACD, 7, 110–112, 119–121, 128, 129, 131, 133–138, 141, 142, 144, 145, 147, 150 linear Granger causality test, 7, 86, 97 LOGACD, 7, 110, 111, 119, 120, 122, 128, 129, 131–135, 137, 138, 141, 142, 144, 145, 147, 149, 150 L-pattern, 176 market microstructure, 8, 113, 150 mean, 33, 35, 39, 41, 44–48, 57, 59–63, 73, 86–88, 91, 93, 95, 97, 98, 100, 106, 108–113, 117, 118, 131, 135–137, 139, 140, 142, 143, 145–148, 151, 178–180 mechanism, 1, 6–8, 39, 40, 47, 51, 86, 95, 100, 101, 114, 154, 157–163, 176, 177, 179, 180 microstructure, 2, 8, 113, 150, 152, 154, 157, 161, 176, 177, 180 Mixture Distribution Hypothesis, 162 MLE, 106, 119, 120 Monte Carlo simulation, 6, 38, 42 MSACD, 7, 8, 110, 113, 119, 120, 131, 140, 149 multivariate GARCH, 2, 3 nonlinear time series, 106 nonparametric, 5, 6, 15, 18, 38, 42, 43, 49–51, 102, 103, 128, 150 NYSE, 52, 54, 84, 112, 113 one-way, 8, 21, 30, 31, 34, 55, 64, 66, 68, 72, 73, 78, 88–90, 99, 100, 162, 180 open interest, 8, 154, 157, 161–165, 169, 170, 176, 177, 180 out-of-sample, 7, 8, 38, 102, 104, 105, 119, 128–130, 134, 136, 138, 140, 141, 144, 145, 147–151, 180 overnight information, 157–160 parametric, 5–7, 15, 38, 40, 43, 47, 49–51, 86, 101, 179 predict, 6, 13, 15, 23, 42, 64, 72, 83, 97, 131, 147, 179 price duration, 7, 8, 102, 103, 110, 112–121, 129–133, 135–137, 139, 140, 142, 143, 145–148, 150, 151, 180 QMLE, 4 quantile, 11, 20, 37–39

Index regulation, 9, 10, 44, 51, 52, 57, 101 return rate, 45, 58, 91, 92, 99 risk, 1–3, 5–7, 9–15, 18–40, 43, 44, 47, 48, 50–53, 55, 60, 63–66, 68–70, 72– 76, 78–80, 82, 83, 85–91, 95, 97–103, 150, 154, 159, 161, 177–180 risk management, 1, 6, 11, 18, 37, 38, 51, 83, 91, 102, 103, 150, 154 risk measure, 3, 6, 9, 35, 37 Risk Metrics, 12, 15, 20 risk spillover, 1, 3, 5–7, 9–11, 18, 24, 26, 27, 33, 35, 36, 52, 53, 55, 63–65, 68–70, 72–76, 78–80, 82, 83, 86–90, 98–101, 178, 179 scaling factor, 23, 90, 105, 107 seasonally adjusted, 115, 118–120, 129, 130, 135–137, 139, 142, 143, 145, 146 semi-parametric, 4, 6, 37, 38, 41, 48, 49, 51, 179 Sequential Information Arrival Hypothesis, 162, 163, 176, 177 SHFE, 45, 51, 85, 86, 91 short selling, 7, 39, 40, 47, 51, 86, 100, 162, 163, 179 significant level, 29–32, 34, 51 simulation, 4, 6, 15, 35, 38, 42, 50, 178, 179 skewness, 6, 12, 14, 27, 28, 33, 35, 41, 46, 49, 57–60, 93, 106, 109, 135–137, 139, 142, 143, 145, 146, 148, 155, 178, 179 spot market, 6, 7, 85, 86, 92–97, 99–101, 178–180 spread, 112, 116, 153 standardized residual, 3, 47, 49, 61, 63, 94, 106, 107, 150 stationary test, 45, 46, 93 stock market, 1, 3, 5–8, 10, 38, 52–57, 63, 65, 68–70, 72–76, 78–80, 82, 83, 152–154, 157, 159, 161–164, 176–180 stock price indices, 55, 56, 58, 60 strong form ACD, 109, 110, 133, 140 T+0 trading mechanism, 161 T+1 trading mechanism, 161

209

TACD, 7, 110, 112–114, 119, 120, 125, 128, 130–134, 136–138, 140, 141, 143, 144, 146–149, 151 TGARCH, 6, 7, 40, 46–48, 60, 61, 63, 86, 93–95, 179 time series, 2, 5, 11–14, 19, 20, 27, 45, 60, 86, 92, 95, 96, 103, 106, 150, 156, 161, 178, 181 trading mechanism, 6, 8, 51, 100, 101, 161, 177, 179, 180 two-sided, 3 two-way spillover, 7, 180 ‘U’ pattern, 153, 157 unconditional VaR, 6, 12, 51, 179 upside risk, 7, 86–89, 99, 100, 179 Upside VaR, 6, 7, 38–41, 47–51, 86, 89, 95, 96, 101, 179, 180 ‘U’-type, 153, 157 Value at Risk, 3, 11, 37, 103 VaR, 3, 6, 7, 9–15, 17–20, 35, 37–44, 47–51, 86, 88, 89, 93, 95, 96, 101, 179, 180 VAR, 3, 8, 161, 162, 164–169, 180 variance decomposition, 161, 164–166, 169–172 Vector Autoregressive Models, 154, 180, see also VAR volatility, 2–4, 6–9, 13, 33, 35, 37, 38, 40–45, 47–51, 56, 60, 63, 82, 85–88, 92, 95, 97–102, 106, 109, 115, 152–154, 157–163, 165, 169, 170, 176, 177, 179–181 volatility clustering, 45, 56, 60, 92, 101 volume, 8, 53, 64, 85, 112, 151–154, 157, 159–165, 169, 176, 177, 180, 181 weak form ACD, 109, 110, 112, 133, 140, 149 Weibull distribution, 4, 110, 147 yield, 8, 17, 35, 37, 39, 41, 43–46, 48–50, 107, 110, 116, 151–159, 161–165, 169, 170, 176, 177, 180, 181