Candlestick Forecasting for Investments: Applications, Models and Properties 9780367703370, 9781003145769

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Candlestick Forecasting for Investments

Candlestick charts are often used in speculative markets to describe and forecast asset price movements. This book is the first of its kind to investigate candlestick charts and their statistical properties. It provides an empirical evaluation of candlestick forecasting. The book proposes a novel technique to obtain the statistical properties of candlestick charts. The technique, which is known as the range decomposition technique, shows how security price is approximately logged into two ranges, i.e. technical range and Parkinson range. Through decomposition-based modeling techniques and empirical datasets, the book investigates the power of, and establishes the statistical foundation of, candlestick forecasting. Haibin Xie is Associate Professor at the School of Banking and Finance, University of International Business and Economics. Kuikui Fan is affiliated with the School of Statistics, Capital University of Economics and Business. Shouyang Wang is Professor at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences.

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Candlestick Forecasting for Investments Applications, Models and Properties

Haibin Xie, Kuikui Fan and Shouyang Wang

First published 2021 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 52 Vanderbilt Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2021 Haibin Xie, Kuikui Fan and Shouyang Wang The right of Haibin Xie, Kuikui Fan and Shouyang Wang to be identified as the authors of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents 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 catalog record for this book has been requested Library of Congress Cataloging-in-Publication Data A catalog record has been requested for this book ISBN: 978-0-367-70337-0 (hbk) ISBN: 978-1-003-14576-9 (ebk) Typeset in Galliard by Apex CoVantage, LLC

Contents

List of figures List of tables About the authors Acknowledgements Preface

viii x xii xiii xiv

PART I

Introduction and outline 1 Introduction 1.1 1.2 1.3 1.4

1 3

Technical analysis before the 1970s 3 Technical analysis during 1990s–2000s 5 Recent advances in technical analysis 9 Summary 10

2 Outline of this book

11

PART II

Candlestick

13

3 Basic concepts

15

4 Statistical properties

19

4.1 4.2 4.3 4.4

Propositions 20 Simulations 21 Empirical evidence 22 Summary 23

vi Contents PART III

Statistical models

25

5 DVAR model

27

5.1 5.2 5.3 5.4 5.5

The model 28 Statistical foundation 29 Simulations 32 Empirical results 33 Summary 33

6 Shadows in DVAR 6.1 6.2 6.3 6.4

35

Simulations 35 Theoretical explanation 38 Empirical evidence 43 Summary 44

PART IV

Applications 7 Market volatility timing

45 47

7.1 7.2 7.3 7.4

Introduction 47 GARCH@CARR model 48 Economic value of volatility timing 49 Empirical results 51 7.4.1 The data 51 7.4.2 In-sample volatility timing 52 7.4.3 Out-of-sample volatility timing 54 7.5 Summary 58 8 Technical range forecasting 8.1 Introduction 62 8.2 Econometric methods 63 8.2.1 The model 63 8.2.2 Out-of-sample forecast evaluation 64 8.3 An empirical study 65 8.3.1 The data 65 8.3.2 In-sample estimation 65 8.3.3 Out-of-sample forecast 66 8.4 Summary 68

62

Contents 9 Technical range spillover

vii 69

9.1 Introduction 69 9.2 Econometric method 70 9.3 An empirical study: DAX and CAC40 71 9.3.1 The data 71 9.3.2 Estimation 73 9.4 Summary 74 10 Stock return forecasting: U.S. S&P500

76

10.1 Introduction 77 10.2 Econometric methods 78 10.2.1 The model 78 10.2.2 Out-of-sample evaluation 79 10.3 Statistical evidence 79 10.3.1 The data 79 10.3.2 In-sample estimation 80 10.3.3 Out-of-sample forecast 82 10.4 Economic evidence 84 10.5 More details 87 10.6 Summary 89 11 Oil price forecasting: WTI crude oil

91

11.1 Introduction 91 11.2 Econometric method 92 11.2.1 DVAR model 92 11.2.2 Forecast evaluation 93 11.3 Empirical results 94 11.3.1 The data 94 11.3.2 In-sample model estimation 96 11.3.3 Out-of-sample performance 96 11.4 Summary 98 PART V

Conclusions and future studies

101

12 Main conclusions

103

13 Future studies

104

Bibliography Index

106 115

Figures

3.1 3.2 5.1 5.2 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 7.5 8.1 9.1 9.2 9.3 10.1 10.2

10.3 10.4

A typical candlestick Black and white candlesticks Information sets and price changes Granger causality: an illustration Granger causality test: the histogram of p values when σ = 0.01 Granger causality test: the histogram of p values when σ = 0.05 Granger causality test: the histogram of p values when σ = 0.1 Shadows in DVAR: Granger causality Time series plots of Parkinson price range and excess return In-sample volatility forecasting: GARCH@CARR, GJR-GARCH and EGARCH Cumulative return using different in-sample volatility timing strategies: 1983.01–2016.12 Out-of-sample optimal allocation weight on risky Asset: 1997.01–2016.12 Cumulative portfolio return formulated using different out-of­ sample volatility timing strategies: 1997.01–2016.12 Out-of-sample technical range forecasting Time series plot of log closing prices, DAX and CAC40: 1994.01–2014.12 Time series plot of log technical range, DAX and CAC40: 1994.01–2014.12 Plots of variance decomposition of technical range, DAX and CAC40 Time series of cumulative squared forecast error: 1995.01– 2015.12 Cumulative squared forecast error for the historical mean bench­ mark forecasting model minus the cumulative squared forecast error for the competing model: 1995.01–2015.12 Dynamic weights allocated on equities over time: 1995.01– 2015.12 Dynamic cumulative portfolio returns formed by different trading strategies over time: 1995.01–2015.12

16 17 30 31 39 40 41 43 52 55 57 59 60 68 72 73 75 83

84 86 87

Figures ix 10.5 10.6

10.7 11.1 11.2

Time series of cumulative squared forecast error over business cycle Cumulative squared forecast error for the historical mean bench­ mark forecasting model minus the cumulative squared forecast error for the DVAR model over business cycle Dynamic cumulative portfolio return formed by different trading strategies over business cycle Time series of monthly WTI crude oil price over 1986.01– 2013.01 Out-of-sample forecasting comparison, ARMA v.s. DVAR over 2001.01–2013.01

88

88 89 95 98

Tables

4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2 6.1 6.2 6.3 6.4 7.1 7.2 7.3 8.1 8.2 8.3 8.4 8.5 9.1 9.2 10.1 10.2 10.3 10.4

Unit root test results on simulated technical range Co-integration rank test on simulation observations Unit root test results on closing price of S&P500, DAX and SSEC Unit root test results on technical range of S&P500, DAX and SSEC Co-integration rank test on S&P500 stock index Co-integration rank test on DAX stock index Granger causality tests between ΔRt and ΔWt: simulation results Granger causality tests between ΔRt and ΔWt: empirical results Granger causality tests: σ = 0.01 Granger causality tests: σ = 0.05 Granger causality tests: σ = 0.1 Empirical studies on S&P500 Summary statistics of the range, the excess return, the absolute excess return and the squared excess return Estimates of GARCH@CARR, GJR-GARCH and EGARCH Summary statistics of the portfolio returns formulated using different volatility models Summary statistics of closing price and technical range Vector error correction estimates on S&P500 stock index Out-of-sample MAE and RMSE for VECM, ARMA and MA Out-of-sample forecasting evaluation: DM test Out-of-sample forecasting evaluation: regression and encompassing regression Summary statistics of technical range Vector error correction estimates on DAX and CAC40 Summary statistics of stock return and other variables Estimates of the ARMA-GARCH-in-Mean model Estimates of the DVAR model: S&P500 Realized utilities and CER gains

21 22 23 23 23 24 32 33 36 37 37 44 52 53 56 65 66 67 67 67 72 74 80 81 81 86

Tables 11.1 11.2 11.3

Summary statistics of ΔRt, ΔWt and crude oil return: 1986.01– 2013.01 Estimates of the DVAR model: WTI oil price Out-of-sample prediction error comparison: DVAR v.s. ARMA

xi 95 96 98

About the authors

Haibin Xie is an associate professor of the School of Banking and Finance at the University of International Business and Economics in Beijing, China. His research interests include econometric modeling, financial risk management, technical analysis and financial market predictability, and financial market anomalies. He has published one book and more than 30 journal papers in the fields of empirical finance and time series forecasting. He obtained his Ph.D. in Management Science and Engineering at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences in 2012. Kuikui Fan is a post doctor of the School of Statistics at the Capital University of Economics and Business in Beijing, China. He received his Ph.D. in Economics from the Shanghai University of Finance and Economics in 2017. His research interests focus on financial market microstructure, financial risk management, quantitative and technical analysis. He has published about ten papers in the fields of finance. He used to be a practitioner in China’s financial industry, serving as a trader of foreign exchange and commodities in the headquarters of the Bank of Communications and China Minsheng Bank. Shouyang Wang received his Ph.D. in Operations Research from the Institute of Systems Science, Chinese Academy of Sciences in 1986. He is currently Bairen Distinguished Professor of Management Science at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Changjiang Distinguished Professor of Systems Management of the University of the Chinese Academy of Sciences. He is also the founding director of the Center for Forecasting Science of the Chinese Academy of Sciences and the dean of School of Economics and Management of the University of the Chinese Academy of Sciences. He also serves as an adjunct professor of over 30 universities around the world. He is the editor-in-chief, an area editor or a co-editor of 16 journals including Energy Economics. He has published 35 monographs and over 350 papers in leading journals. His current research interests include financial engineering, economic forecasting and financial risk management.

Acknowledgements

This book summarizes some joint works of Haibin Xie, Kuikui Fan and Shouyang Wang. This book also includes some joint results with Professors Guohua Zou, Mo Zhou and Xun Zhang. We are indebted to Yong Fang, Fengbin Lu, Xiujuan Zhao, Hui Bu, Xiuli Liu, Ge Zhang, Ke Cheng, Bin Li, Ziran Li, Ai Han, Qin Bao, Jue Wang, Prof. Yong Zhou of Shanghai University of Finance and Economics, Prof. Zudi Lu of Uni­ versity of Southampton, and Prof. Yongmiao Hong of Cornell University for their very helpful discussions and comments on our work. We are greatful to Prof. Weixuan Xu of the Institue of Policy and Management, Chinese Academy of Sciences, Prof. Haijun Huang of Beihang University, Prof. Yiming Wang of Peking University, Prof. Jichang Dong of the University of the Chinese Academy of Sciences, Prof. Mei Yu, Prof. Weixing Wu, and Prof. Zhijie Ding of the University of International Business and Economics for their valuable feedback regarding the draft of the book. Of course, all remaining errors are ours. This research is partly supported by The National Nature Science Foundation of China (Nos. 71401033 and 71988101) and Fundamental Research Funds for the Central Universities in the University of International Business and Eco­ nomics (No. 19YB26). We would like to thank The National Natural Science Foundation of China, The Ministry of Education, the University of International Business and Economics, the Center of Forecasting Science of Chinese Academy of Sciences, and the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. We would like to express our tremendous gratitude to all the people dedicated to this book. Haibin Xie, Kuikui Fan and Shouyang Wang June, 2020

Preface

For hundreds of years, candlestick charts have been widely used in speculative markets to describe and forecast the movements of asset prices, however their sta­ tistical properties, neither theoretical nor empirical, have ever been investigated in the academic community. One of the main obstacles is their highly subjective nature. The presence of geometric shapes in candlestick charts is often in the eyes of the beholder, which makes it difficult to be investigated in an objective way and poses a great challenge. This book takes up this challenge by proposing a novel technique, the range decomposition technique, to investigate the properties of the candlestick. The range decomposition technique shows that the security price can be approxi­ mately decomposed into a couple of ranges, technical range and Parkinson range. Under some mild assumptions together with the range decomposition technique, this book for the first time obtains the statistical properties of the can­ dlestick. Based on these properties, this book then empirically investigates the power of candlestick forecasting with both in-sample and out-of-sample evi­ dence. We believe that this book, to some extent, establishes the statistical foun­ dation of candlestick forecasting. This book is valuable to both academic researchers and investment practition­ ers. To most of the academic researchers, technical analysis is usually known as “voodoo finance” due to its lack of theoretical underpinnings. The statistical foundation of candlestick forecasting proposed in this book establishes its theo­ retical underpinnings, and thus provides academicians with objective economet­ ric tools to evaluate its forecasting power. To investment practitioners, the empirical evaluation of candlestick forecasting power performed in this book is valuable and offers good guidance for their future investment practice. This book is structured with five parts. Part I presents an introduction to the technical analysis in Chapter 1 and the outline of this book in Chapter 2. Part II covers the basic concepts of candlestick charts in Chapter 3 and its statistical properties in Chapter 4. Part III presents the statistical models. There are two chapters in this part. Chapter 5 proposes a decomposition-based vector autoregres­ sive (DVAR) model for return modeling. Chapter 6 shows how the upper shadow and lower shadow of the candlestick can be used to improve return forecasting under the framework of the DVAR model. Part IV presents the applications of

Preface xv candlestick charts in return and volatility forecasting. In this part, we show how the properties (in Part II) and models (in Part III) can be used in market volatility timing in Chapter 7, in technical range forecasting in Chapter 8, in range spillover testing in Chapter 9, in return forecasting in Chapter 10, and in oil price forecast­ ing in Chapter 11. Part V concludes in Chapter 12 and suggests future studies in Chapter 13. More details on the contents of each chapter follows. Chapter 1 makes an overview of the technical analysis. In this chapter, we present comprehensively the academic research results on technical analysis. This chapter serves as an introduction to the advances of academic research on technical analysis. Chapter 2 presents the outline of this book. The outline helps the readers to quickly understand the organization of this book. Chapter 3 presents the basic concepts and theoretical properties of candlestick charts. This chapter covers the drawing of the candlestick and the components of the candlestick, and presents the concepts of technical range, real body, upper shadow and lower shadow. As the forecasting power of candlestick charts is the focus of this book, we believe that a brief introduction to the candlestick would be a great help to those who are not familiar with candlestick forecasting. Chapter 4 presents the theoretical properties of candlestick charts. Three inter­ esting properties are obtained under some very mild assumptions. Both simula­ tions and empirical evidence are used to confirm these properties. Chapters 5 through 6 establish the statistical models of candlestick forecasting. Chapter 5 shows that the return series on an asset can be decomposed into a couple of components, and proposes to model the asset return with a decompo­ sition-based vector autoregressive (DVAR) model. Theoretical evidence, simula­ tions and empirical studies are used to show that the DVAR model is a reasonable one for return modeling. Compared with the classical univariate time series mod­ eling technique, the DVAR model uses both high, low prices and closing price, and thus is more efficient in using history price information. The DVAR model presents a new framework for return modeling. Chapter 6 investigates the role of upper shadow and lower shadow in the DVAR model. This chapter shows that both upper shadow and lower shadow are informative for improving DVAR forecasting performance. Theoretical expla­ nations, simulations and empirical evidence are provided to confirm this finding. This chapter serves as a supplement to Chapter 5, as it claims that upper shadow and lower shadow should be taken into consideration when using the DVAR model. Chapters 7 through 11 show with empirical applications that the statistical properties of the candlestick can be used to improve the forecasting performance of the candlestick. Chapter 7 demonstrates that the statistical properties of the candlestick can be used in market volatility timing trading strategy. An empirical study is performed on the monthly S&P500 stock index. The results show that this trading strategy is economically valuable.

xvi Preface Chapter 8 shows that technical range forecasting can be improved if the statis­ tical properties of technical range are considered. An empirical study is performed on the monthly S&P500 index, and the results demonstrate the outperformance of our model over both the ARMA model and the simple moving average model with significance for both in-sample and out-of-sample forecasting. Chapter 9 shows that the statistical properties of technical range can be used to investigate the information spillover across financial markets. An empirical study is performed on German DAX index and French CAC40 index, and the results demonstrate significant range spillover from the German stock market to the French stock market before 2007, while not vice versa. Chapter 10 investigates the performance of the DVAR model in return fore­ casting. An empirical study is performed on the monthly S&P500 stock index. We find the DVAR model outperforms not only the historical mean model but also the classic ARMA-GARCH-in-Mean model for both in-sample and out-of-sample forecasting. Moreover, we find the statistical outperformance of the DVAR model can bring investors with significant economic utility gain. Chapter 11 investigates the performance of the DVAR model on oil pricing. The results show significant outperformance of DVAR over the ARMA model for both in-sample and out-of-sample forecasting. Chapters 12–13 conclude the findings of this book and propose possible appli­ cations and extensions of these findings for future studies. This book is intended for researchers interested in candlestick charts and their theoretical underpinnings in forecasting. This book is also intended for invest­ ment practitioners to improve their understanding of the candlestick charts in practice. Of course, the contents of this book are also valuable for improving risk management. Anyway, this book can be useful to academic researchers, investment practi­ tioners, and all market participants.

Part I

Introduction and outline

1

Introduction

In this introductory chapter, we review and comment on academic studies on technical analysis.

1.1 Technical analysis before the 1970s Despite its popularity among practitioners, academic literature concerning tech­ nical analysis before the 1970s almost unanimously concluded that technical analysis is of no economic value in practice. Started by analyzing the weekly forecasting results of well-known professional agencies, such as financial services and fire insurance companies, in the period January 1928 through June 1932, Cowles (1933) found no statistically signifi­ cant forecasting performance. Furthermore, Cowles (1933) considered a 26­ year record forecasting of William Peter Hamilton in the period December 1903 until his death in December 1929. During this period Hamilton wrote 255 editorials in the Wall Street Journal and presented forecasts for the stock market based on the Dow Theory. Cowles (1933) found that Hamilton failed to beat the simple continuous investment in the DJIA or the DJIA after correct­ ing for brokerage charges and cash dividends. On 90 occasions Hamilton announced changes in the outlook for the market. Cowles (1933) found only half of his announcements were successful, not better than predicting purely by guess. Cowles (1944) repeated the analysis for 11 forecasting companies for the longer period January 1928 through July 1943, and still no significant evi­ dence of forecasting power of analysts was discovered. While Cowles (1933, 1944) focused on testing analysts’ advice, other academ­ ics shifted their attention to the behavior of time series of speculative prices. Working (1934), Kendall (1953) and Roberts (1959) found for series of specu­ lative prices, such as American commodity prices of wheat and cotton, British indices of industrial share prices and the DJIA, that successive price changes, as measured by auto-correlation, were linearly independent, and that these series behaved much like random walks. Since the dependence in price changes can be highly nonlinear and too com­ plicated to be captured by the standard linear statistical tools, such as autocorrelations, Alexander (1961) began defining filters to reveal possible trends

4 Introduction and outline in stock prices which may be masked by the jiggling of the market. A filter strat­ egy buys when price increases by x percent from a recent low and sells when price declines by x percent from a recent high. Thus filters can be used to identify local peaks and troughs according to the filter size. After applying several filters to the DJIA in the period 1897–1929 and the S&P Industrials in the period of 1929– 1959, Alexander (1961) concluded that in speculative markets a price move, once initiated, tended to persist. However he also noticed that commissions could reduce the profitability. Mandelbrot (1963, p.418) noted that there was a flaw in the computations of Alexander (1961), since he assumed that the trader could buy exactly at the low plus x percent and could sell exactly at the high minus x percent. However in real trading this would probably not be the case. In Alexander (1964) the computing mistake was corrected and allowance was made for transaction costs. The filter rules still reported considerable excess profits over the buy-and-hold strategy, but transaction costs wiped out all the profits. Fama (1965) tried to show with various tests that price changes were indepen­ dent and that history stock prices could not be used to make valuable future pre­ dictions. He applied serial correlation tests, runs tests and Alexander’s filter technique to daily data of 30 individual stocks quoted in the DJIA in the period January 1956 through September 1962. The serial correlation tests indi­ cated that the dependence in successive price changes was either extremely small or non-existent. Also the runs tests did not show a large degree of dependence. Profits of the filter techniques were calculated by trading blocks of 100 shares and were corrected for dividends and transaction costs. The results gave no prof­ itability. Moreover, even if there was some dependence, Fama (1965) argued that this dependence was too small to be profitably exploited because of transac­ tion costs. Fama and Blume (1966) further applied Alexander’s filters approach to the same data set as in Fama (1965). They found that the buy-and-hold strat­ egy could not consistently be outperformed. Levy (1971) first examined 32 possible forms of five point chart patterns, i.e. a pattern with two highs and three lows or two lows and three highs, which were claimed to represent channels, wedges, diamonds, symmetrical triangles, (reverse) head-and-shoulders, triple tops, and triple bottoms. Local extrema were determined with the help of Alexander’s (1961) filter techniques. After trading costs were taken into account it was concluded that none of the 32 pat­ terns gave any evidence of profitable forecasting ability in either bullish or bearish direction when applied to 548 NYSE securities in the period July 1964 through July 1969. In summary, early academic empirical studies concluded that successive price changes were independent and that trading strategies based on technical analysis were nonprofitable. These empirical findings combined with the theory of Paul Samuelson (1965), published in his influential paper ‘Proof that Properly Antic­ ipated Prices Fluctuate Randomly’, led to the efficient markets hypothesis (EMH). Eugene Fama (1970) reviewed the theoretical and empirical literature on the EMH and concluded that the evidence in support of the EMH was

Introduction

5

very extensive, and that contradictory evidence was sparse. Since then the EMH has been the central paradigm in financial economics. According to this hypoth­ esis it is not possible to exploit any information set to predict future price changes. Therefore, trading systems based on past information should not gen­ erate profits in excess of equilibrium expected profits or returns. It becomes com­ monly accepted in academic field that the study of past price trends and patterns is no more useful in predicting future price movements.

1.2 Technical analysis during 1990s–2000s Little work on technical analysis appeared during the 1970s and 1980s due to the dominance of efficient market hypothesis in financial paradigm. A critical problem concerning technical analysis is data snooping. Data snoop­ ing is the generic term of the danger that the best forecasting model found in a given data set by a certain specification search is just the result of chance instead of the result of truly superior forecasting power. Jensen and Benington (1969, p.470) argued: “Likewise given enough computer time, we are sure that we can find a mechanical trading rule which works on a table of random numbers-provided of course that we are allowed to test the same rule on the same table of numbers which we used to discover the rule.” Brock et al. (1992) claimed that they mitigated the problem of data snooping by (1) reporting the results of all tested trading strategies, (2) utilizing a very long data set, and (3) emphasizing the robustness of the results across various non-overlapping subperiods for statistical inference. They tested the forecastabil­ ity of a set of 26 simple technical trading rules by applying them to the closing prices of the DJIA in the period January 1897 through December 1986, nearly 90 years of data. The set of trading rules consists of moving average strate­ gies and support-and-resistance rules, very popular trading rules among technical trading practitioners. Brock et al. (1992) found that all trading rules reported significant profits above the buy-and-hold benchmark in all periods by using simple t-ratios as test statistics. Brock et al. (1992) found that the patterns uncov­ ered by their technical trading rules could not be explained by first order autocorrelation and by changing expected returns caused by changes in volatility. Therefore Brock et al. (1992) concluded that the conclusion reached in earlier studies that technical analysis was useless might have been premature. The strong results of Brock et al. (1992) resulted in renewed interest in aca­ demia for testing the predictability of technical trading rules in the 1990s. The following will mainly present the major results concerning the predictability of technical analysis. Levich and Thomas (1993) were the first to apply the bootstrap methodology, as introduced by Brock et al. (1992), to exchange rate data. Six filters and three moving averages were applied to the U.S. Dollar closing settlement prices of the BP, CD, DEM, JPY and SF futures contracts traded at the International Mone­ tary Market of the Chicago Mercantile Exchange over the period 1973.01– 1990.12. Consistent with Brock et al. (1992), they found that the simple

6 Introduction and outline technical trading rules generated unusual profits (no corrections are made for transaction costs) and that a random walk model could not explain these profits. Lee and Mathur (1995) remarked that surveys in favor of technical trading if applied to exchange rate data, were mostly conducted on U.S. Dollar denomi­ nated currencies and conjectured that the positive results were likely due to the central bank intervention. Therefore they tested market efficiency of Euro­ pean foreign exchange markets by applying 45 different crossover movingaverage trading strategies to six European spot cross-rates (JPY/BP, DEM/ BP, JPY/DEM, SF/DEM and JPY/SF) over 1988.05–1993.12. After a correc­ tion for 0.1% transaction costs per trade, they found that moving-average trading rules were marginally profitable only for the JPY/DEM and JPY/SF cross rates. Further it was found that in periods during which central bank intervention was believed to have taken place, trading rules did not show to be profitable in the European cross rates. Finally Lee and Mathur (1995) applied a recursively optimizing test procedure with a rolling window for the purpose of testing out-of-sample forecasting power. Every year the best trading rule of the previous half-year was applied. Also this out-of-sample test procedure rejected the null hypothesis that moving averages had forecasting power. Bessembinder and Chan (1995) tested whether the trading rule set of Brock et al. (1992) had forecasting power when applied to the stock market indices of Japan, Hong Kong, South Korea, Malaysia, Thailand and Taiwan over 1975.01–1989.12. They found that the rules were most successful in the markets of Malaysia, Thailand and Taiwan if the break-even round-trip transac­ tion costs were set to be 1.57% on average. They concluded that excess profits over the buy-and-hold could be made, but emphasized the fact that the relative riskiness of the technical trading strategies was not controlled. For the UK stock market Hudson et al. (1996) tested the trading rule set of Brock et al. (1992) on daily data of the Financial Times Industrial Ordinary index over 1935.07–1994.01. They found that the trading rules on average generated an excess return of 0.8% per transaction over the buy-and-hold, but that the costs of implementing the strategy were at least 1% per transaction. Further results show that over the subperiod 1981–1994, the trading rules seemed to lose their forecasting power. Hence Hudson et al. (1996) concluded that although the technical trading rules examined did have predictive ability, their use would not allow investors to make excess returns in the presence of costly trading. Additionally Mills (1997) simultaneously found in the case of zero trans­ action costs with the bootstrap technique introduced by Brock et al. (1992) that the good results for the period 1935–1980 could not be explained by an AR­ ARCH model for the daily returns. Again, for the period after 1980 it was found that the trading rules did not generate statistically significant results. Bessembinder and Chan (1998) replicated the calculations of Brock et al. (1992) for the period 1926–1991 to assess the economic significance of the Brock et al. (1992) findings. Corrections were made for transaction costs and dividends and for non-synchronous trading. One-month treasury bills were

Introduction

7

used as proxy for the risk-free interest rate if no trading position was held in the market. It was computed that one-way break-even transaction costs were approx­ imately 0.39% for the full sample. Although Bessembinder and Chan (1998) con­ firmed the results of Brock et al. (1992), they concluded that there was little reason to view the evidence of Brock et al. (1992) as indicative of market inefficiency. Fernandez-Rodriguez et al. (2001) replicated the testing procedures of Brock et al. (1992) for daily data of the General Index of the Madrid Stock Exchange (IGBM) in the period January 1966 through October 1997. They found that, if transaction costs were not taken into consideration, technical trading rules were found to have forecastability in the Madrid Stock Exchange. Furthermore, the bootstrap results indicated that the forecasting power of the technical trading rules could not be explained by several null models for stock returns such as the AR(1), GARCH and GARCH-in-Mean models. Ratner and Leal (1999) applied ten moving-average trading rules to daily local index inflation corrected closing levels for Argentina (Bolsa Indices General), Brazil (Indices BOVESPA), Chile (Indices General de Precios), India (Bombay Sensitive), Korea (Seoul Composite Index), Malaysia (Kuala Lumpur Composite Index), Mexico (Indice de Precios y Cotaciones), the Philippines (Manila Com­ posite Index), Taiwan (Taipei Weighted Price Index) and Thailand (Bangkok S.E.T.) over 1982.01–1995.04. After correcting for transaction costs, the rules appeared to be significantly profitable only in Taiwan, Thailand and Mexico. Isakov and Hollistein (1999) tested simple technical trading rules on the Swiss Bank Corporation (SBC) General Index and on some of its individual stocks UBS, ABB, Nestle, Ciba-Geigy and Zurich in the period 1969–1997. They were the first who augmented moving-average trading strategies with momen­ tum indicators or oscillators, so called relative strength or stochastics. These oscillators were expected to indicate when an asset was overbought or oversold and were supposed to give appropriate signals when to step in or out of the market. Isakov and Hollistein (1999) found that the use of oscillators did not add to the performance of the moving averages. For the basic moving average strategies they found an average yearly excess return of 18% on the SBC index. However it was concluded that in the presence of trading costs the rules were only profitable if the costs were not higher than 0.3–0.7% per transaction. LeBaron (2000a) reviewed the paper of Brock et al. (1992) and tested whether the results found for the DJIA in the period 1897–1986 also held for the period after 1986. Two technical trading rules were applied to the data set, namely the 150-day single crossover moving average rule, because the research of Brock et al. (1992) pointed out that this rule performed consistently well over a couple of subperiods, and a 150-day momentum strategy. LeBaron (2000a) found that the results of Brock et al. (1992) changed dramatically in the period 1988–1999. The trading rules seemed to have lost their predictive ability. For the period 1897–1986 the results could not be explained by a random walk model for stock returns, but for the period 1988–1999, in contrast, it was concluded that the null of a random walk could not be rejected. LeBaron

8 Introduction and outline (2000b) tested a 30-week single crossover moving-average trading strategy on weekly data at the close of London markets on Wednesdays of the U.S. Dollar against the BP, DEM and JPY in the period June 1973 through May 1998. It was found that the strategy performed very well on all three exchange rates in the subperiod 1973–1989, yielding significant positive excess returns of 8, 6.8 and 10.2% yearly for the BP, DM and JPY respectively. However for the subpe­ riod 1990–1998 the results were not significant anymore. Coutts and Cheung (2000) applied the technical trading rule set of Brock et al. (1992) to daily data of the Hang Seng Index quoted at the Hong Kong Stock Exchange (HKSE) over 1985.10–1997.07. They found that although the trading range break-out rules had better results than the moving averages, they could not profitably be exploited after correcting for transaction costs. In contrast, Ming et al. (2000) found significant forecasting power for the strategies of Brock et al. (1992) when applied to the Kuala Lumpur Composite Index (KLCI) even after correction for transaction costs. Detry and Gregoire (2001) tested 10 moving-average trading rules of Brock et al. (1992) on the indices of all 15 countries in the European Union. They found that their results strongly supported the conclusion of Brock et al. (1992) for the predictive ability of moving average rules. Neftci (1991) showed that technical patterns could be fully characterized by using appropriate sequences of local minima and maxima. Hence it was concluded that any pattern can potentially be formalized. Osler and Chang (1995) were the first to evaluate the predictive power of head-and-shoulders patterns using a com­ puter-implemented algorithm in foreign exchange rates. The features of the head­ and-shoulders pattern were defined to be described by local minima and maxima that were found by applying Alexander’s (1961) filter techniques. The pattern rec­ ognition algorithm was applied to six currencies (JPY, DEM, CD, SF, FF and BP against the USD) in the period March 1973 to June 1994. Significance was tested with the bootstrap methodology described by Brock et al. (1992) under the null of a random walk and GARCH model. It was found that the head-and-shoulders pattern had significant predictive power for the DEM and the JPY, also after cor­ recting for transaction costs and interest rate differentials. Lo et al. (2000) developed a pattern recognition algorithm based on nonparametric kernel regression to detect (inverse) head-and-shoulders, broadening tops and bottoms, triangle tops and bottoms, rectangle tops and bottoms, and double tops and bottoms patterns that were the most difficult to quantify analyt­ ically. The pattern recognition algorithm was applied to hundreds of NYSE and NASDAQ quoted stocks in the period 1962–1996. It was found that technical patterns did provide incremental information, especially for NASDAQ stocks. Further it was found that the most common patterns were double tops and bottoms, and (inverted) head-and-shoulders. In summary, stimulated by the findings of Brock et al. (1992), voluminous lit­ erature investigated the forecasting power of technical analysis in the 1990s and 2000s, however conclusions on the economic value of technical analysis remained controversial.

Introduction

9

1.3 Recent advances in technical analysis Goyal and Welch (2008) showed that a long list of macroeconomic and financial predictors from the literature fail to deliver consistently superior out-of-sample forecasts of the U.S. equity premium relative to a simple forecast based on the historical average (constant expected equity premium model). Recent work on technical analysis mainly concerns whether technical indicators are informative predictors, as measured by out-of-sample R-square (Campbell and Thompson, 2008). The following will present the main results in academia. Neely et al. (2014) analyzed monthly out-of-sample forecasts of the U.S. equity risk premium based on popular technical indicators (moving average rule, momentum rule and on-balance volume) in comparison to that of a set of well-known macroeconomic variables, and found that technical indicators had statistically and economically significant out-of-sample forecasting power and frequently outperformed the macroeconomic variables. Furthermore, they found out-of-sample predictability was closely connected to the business cycle for both technical indicators and macroeconomic variables, although in a com­ plementary manner: technical indicators detected the typical decline in the equity risk premium near cyclical peaks, while macroeconomic variables more readily picked up the typical rise near cyclical troughs. It was concluded that uti­ lizing information from both technical indicators and macroeconomic variables substantially increased the out-of-sample gains relative to using either macroeco­ nomic variables or technical indicators alone. Huang et al. (2015) extended the traditional predictive regression model to a state-dependent one, in which a state variable was used to indicate an up- or down-market. They found that U.S. stock returns for one to 12 months could be predicted negatively in the up-market and positively in the down-market by a mean reversion indicator that was defined as the past year cumulative return of the market portfolio minus its long-term mean and standardized by its annu­ alized volatility, and this predictive pattern was found to be robust to crosssectional portfolios sorted by size, book-to-market ratio, industry, momentum, and long- and short-term reversals. Goh et al. (2013) studied the predictability of technical indicators (moving average rule and on-balance volume) for U.S. government bond risk premia. They found that technical indicators had economically and statistically significant forecasting power both in- and out-of-sample, and for both short- and long-term government bonds, and that technical indicators were more useful than eco­ nomic variables. Furthermore, they found that a forecasting model that combines information in technical indicators together with economic variables substantially outperformed forecasts based on models using economic variables only. Using the intraday data of the S&P500 ETF over 1993.02.01–2013.12.31, Gao et al. (2015) documented an intraday momentum pattern that the first half-hour return on the market predicted the last half-hour return on the market. The predictability was both statistically and economically significant, and was stronger on more volatile days, higher volume days, recession days

10

Introduction and outline

and some macroeconomic news release days. Moreover, they also found that the intraday momentum was also strong for ten other most actively traded ETFs. In summary, regarding out-of-sample predictability, recent work seems to confirm that technical analysis is informative for predicting future price changes.

1.4 Summary In financial practice technical analysis is not free from being criticized because of its highly subjective nature: the geometric shapes in historical price charts are often in the eyes of the beholder. It is said that there are probably as many methods of combining and interpreting the various techniques as there are char­ tists themselves. The attitude of many academics towards technical analysis is described by Malkiel (1996, p.139): “Obviously, I’m biased against the chartist. This is not only a personal predilection but a professional one as well. Technical analysis is anathema to the academic world. We love to pick on it. Our bullying tactics are prompted by two considerations: (1) after paying transaction costs, the method does not do better than a buy-and-hold strategy for investors, and (2) it’s easy to pick on. And while it may seem a bit unfair to pick on such a sorry target, just remember: It’s your money we are trying to save.” Due to its lack of theoretical underpinnings, the technical analysis, in some circles, is known as “voodoo finance” and chart reading is believed to share a pedestal with alchemy.

2

Outline of this book

This book is structured with five parts. Part I has two chapters. Chapter 1 pre­ sents a comprehensive review on technical analysis. Chapter 2 presents the outline of this book. Part II introduces the basic concepts and statistical proper­ ties of Candlestick. There are two chapters in Part II. Chapter 3 gives the basic concepts of candlestick. Chapter 4 presents the basic statistical properties of can­ dlestick charts. These properties establish the foundations of candlestick forecast­ ing. Part III presents the statistical models. There are two chapters in this part. Chapter 5 proposes a decomposition-based vector autoregressive (DVAR) model for predicting returns. Compared with the traditional return-based time series modeling technique, the DVAR model employs the high, low and closing prices, which makes it more efficient in information using. Chapter 6 shows, using both theoretical explanation and empirical evidence, that upper shadow and lower shadow are informative for predicting asset returns in DVAR model. Parts II and III are the cores of this book. Part IV presents the empirical applications of candlestick forecasting. Based on the statistical proper­ ties in Parts II and III, Part IV shows empirically how these statistical properties can be used in practice. There are five chapters in this part. Chapter 7 shows with an empirical example that the statistical properties of candlestick charts can be used in market volatility timing. Chapter 8 presents an empirical example to show how the statistical properties of candlestick can be used to improve range forecasting. Chapter 9 demonstrates with an empirical example that the statistical properties of candlestick charts can be used to investigate information spillover effect across financial markets. Chapters 10–11 show empirically that the statis­ tical properties of candlestick charts can be used to improve return forecasting. Part V concludes and proposes directions for future studies.

Part II

Candlestick

3

Basic concepts

Candlestick charts are thought to have been developed in the 18th century by Munehisa Homma, a Japanese rice trader of financial instruments (Morris, 2006). They were introduced to the Western world by Steve Nison (Nison, 1991) in his book, Japanese Candlestick Charting Techniques. A candlestick is a graphical representation of price movements for a given period of time. It is commonly formed by the opening, high, low, and closing prices of a financial instrument. A typical candlestick chart is usually composed of the real body and shadows: the area between the open and the close is called the real body, price excursions above and below the real body are called shadows. The real body illustrates the opening and closing trades. If the security closed higher than it opened, the body is white or unfilled, with the opening price at the bottom of the body and the closing price at the top. If the security closed lower than it opened, the body is black, with the opening price at the top and the closing price at the bottom. The shadow illustrates the highest and lowest traded prices of a security during the time interval represented. The shadow above the real body is called upper shadow and the shadow below the real body is known as lower shadow. Accordingly, the peak of the upper shadow is the highest price and the bottom of the lower shadow is the lowest price. Figure 3.1 presents the plot of a typical candlestick. Figure 3.2 presents what the black and white candlesticks look like. In this book we define real body, upper and lower shadows as follows: Real body: real body in candlestick is defined as the absolute difference between opening price and closing price. In this book the real body is defined as the natural logarithmic difference. RBt ¼ lnðOt Þ – lnðCt Þ;

Ot ≥ Ct ;

ð3:1Þ

RBt ¼ lnðCt Þ – lnðOt Þ;

Ct ≥ O t ;

ð3:2Þ

or

where RBt is the real body and ln(x) is the natural logarithm of x.

16

Candlestick

upper shadow

real body

lower shadow

Figure 3.1 A typical candlestick

Upper shadow: upper shadow is the price excursions above the real body. In this book, the upper shadow is defined as USt ¼ lnðHt Þ – lnðOt Þ;

O t ≥ Ct ;

ð3:3Þ

USt ¼ lnðHt Þ – lnðCt Þ;

Ct ≥ Ot ;

ð3:4Þ

or

where USt is the upper shadow, Ot and Ct are, respectively opening and closing prices. Lower shadow: lower shadow is the price excursions below the real body. In a similar way, the lower shadow is defined as LSt ¼ lnðCt Þ – lnðLt Þ;

Ot ≥ Ct ;

ð3:5Þ

LSt ¼ lnðOt Þ – lnðLt Þ;

Ct ≥ Ot ;

ð3:6Þ

or

where LSt is the lower shadow. From the concepts of real body, upper and lower shadows, we can construct the Parkinson range (Parkinson, 1980).

Basic concepts

17

high price

high price

upper shadow

upper shadow

close price

open price

real body

real body

open price

close price

lower shadow lower shadow low price

low price

Figure 3.2 Black (left) and white (right) candlesticks

Parkinson range: Parkinson range is defined as the difference between the log highest price and the log lowest price PRt ¼ lnðHt Þ – lnðLt Þ;

ð3:7Þ

where PRt is the Parkinson range. Assuming that the asset price follows a simple diffusion model without a drift term, Parkinson (1980) proved that this range is a very efficient volatility estimator s^2t ¼

PRt2 ; 4lnð2Þ

where s^t is the volatility estimator.

ð3:8Þ

18

Candlestick

It is clear that the Parkinson range is the sum of upper shadow, real body and lower shadow PRt ¼ lnðHt Þ – lnðOt Þ þ lnðOt Þ – lnðCt Þ þ lnðCt Þ – lnðLt Þ ¼ USt þ RBt þ LSt ;

ð3:9Þ

if Ot > Ct, and PRt ¼ lnðHt Þ – lnðCt Þ þ lnðCt Þ – lnðOt Þ þ lnðOt Þ – lnðLt Þ ¼ USt þ RBt þ LSt

ð3:10Þ

if Ct>Ot. Similar to Parkinson range, the difference between the highest price and the lowest price is known as the technical range. Technical range: technical range gauges the variability of price movement, which also is a direct indicator of price uncertainty, or risk: the larger is the tech­ nical range the more risk the investors are facing. TRt ¼ Ht – Lt ;

ð3:11Þ

where Ht and Lt are, respectively, the highest and lowest prices over a specified time period. TRt is the technical range. Technical range is equivalent to the length of the candlestick chart. Candlestick charts are a group of candlesticks, serving as a cornerstone of tech­ nical analysis. As can be observed from the drawing of the candlestick that can­ dlestick charts usually convey more information than other forms of charts, such as the moving average charts which use only the closing price information. The candlestick charts not only display the absolute values of the open, high, low, and closing price for a given period but also show how those prices are relative to the prior periods’ prices. For example, when the candlestick is white and high relative to other time periods, it means buyers are very bullish. The opposite is true for a black candlestick. In practice, the candlestick charts are always used with the combination of the other technical indicators, such as the moving average. There are many candle­ stick chart patterns. This section is only a brief introduction to the construction of candlestick. Readers who are interested in more details about candlestick charts and candlestick trading techniques are advised to refer to Nison (1991) and Morris (2006).

4

Statistical properties

Closing price records the price level of a speculative asset, while price range mea­ sures the price variability. Despite the difference, these two indicators are highly related. It is straightforward that Ht – Lt Ht – Ct Ct – Lt ¼ þ ; Ct Ct Ct

ð4:1Þ

where Ht, Lt, Ct are, respectively, high, low and closing prices. Note that (Ht−Ct)/Ct can be approximated by ln(Ht/Ct), and (Ct−Lt)/Ct can be approximated by ln(Ct/Lt). Thus Eq. (4.1) can be approximated by the fol­ lowing equation ( ) ( ) ( ) Ht – Lt Ht Ct Ht ≈ ln þ ln ¼ ln : Ct Lt Lt Ct

ð4:2Þ

Taking natural logarithmic operation on both sides of Equation (4.2) and rear­ ranging, we obtain lnðCt Þ ≈ lnðHt – Lt Þ þ ln½lnðHt Þ – lnðLt Þ]:

ð4:3Þ

Substituting respectively TRt and PRt for Ht−Lt and ln(Ht)−ln(Lt), we get lnðCt Þ ≈ lnðTRt Þ þ lnðPRt Þ;

ð4:4Þ

where TRt and PRt are, respectively, the technical range and the Parkinson range. Eq. (4.4) shows that the closing price can be approximated by a linear combination of technical range and Parkinson range. Under some mild assumptions, this chapter presents some interesting statisti­ cal properties of the candlestick. The remainder of this chapter is organized as follows. Section 1 presents the statistical properties of technical range. Sections 2–3 consolidate these properties with simulations and empirical evidence. Section 4 summarizes the contents of this chapter.

20

Candlestick

4.1 Propositions To obtain the statistical properties of the technical range, the following two assumptions are needed: Assumption 1: the natural logarithm of the closing price is a unit root process. Assumption 2: the natural logarithm of the Parkinson range is a weakly sta­ tionary process. The above mentioned two assumptions are not difficult to accept. For Assumption 1, it is accepted that the financial market is quite efficient and the asset price behaves almost like a random walk, which is definitely a unit root process. For Assumption 2, it is also accepted that the volatility is a stationary process. Since the Parkinson range is a volatility estimator, it thus could be accepted as a stationary process. With these 2 assumptions, the properties of the technical range are presented as follows: Proposition 1: log technical range is a unit root process. Proposition 2: closing price and technical range are co-integrated of order (1,1) with the co-integration vector of (1, −1). Proposition 3: two technical range series are co-integrated if their corre­ sponding closing prices are co-integrated. The proofs to Propositions 1–3 are presented as follows: Proof to Proposition 1: Given the facts that the log closing price is a unit root process and the Parkinson range is a stationary process, it is selfevident that the technical range must be a unit root process. Proof to Proposition 2: Since ln(Ct) − ln(TRt) can be approximated by ln (PRt) (see Eq. (4.4)) which, by Assumption 2, is a stationary process, it is easy to obtain, by the definition of co-integration, that closing price and technical range must be co-integrated of order (1, 1). Proof to Proposition 3: Suppose two closing price series C1t and C2t are co-integrated of order (1, 1) with a co-integration vector of (α1, α2). By the definition of co-integration, there exists a linear combination ot ¼ α1 lnðC1t Þ þ α2 lnðC2t Þ

ð4:5Þ

such that ωt is stationary. Substituting Eq. (4.4) for Cit (i = 1, 2) in Eq. (4.5), one obtains ot ¼ α1 lnðC1t Þ þ α2 lnðC2t Þ ≈ α1 ½lnðTR1t Þ – lnðPR1t Þ] þ α2 ½lnðTR2t Þ – lnðPR2t Þ] ¼ ½α1 lnðTR1t Þ þ α2 lnðTR2t Þ] – ½α1 lnðPR2t Þ þ α2 lnðPR2t Þ]

ð4:6Þ

Statistical properties

21

Rearranging Eq. (4.6), we get ½α1 lnðTR1t Þ þ α2 lnðTR2t Þ] ≈ ot þ ½α1 lnðPR2t Þ þ α2 lnðPR2t Þ]

ð4:7Þ

Given that ωt and ln(PRt) are stationary, it is clear from Eq. (4.7) that ln(TR1t) and ln(TR2t) are also co-integrated of order (1, 1).

4.2 Simulations This section is designed to evaluate these properties with simulations. Following the usual assumption, we assume stock price follows a geometric Brownian motion (GBM) with a drift: dBt ¼ mBt þ sBt dWt

ð4:8Þ

where Bt is the stock price, μ is the drift, Wt is a Winner process, dWt is normally distributed with mean 0 and standard deviation dt, dWt * N(0, dt). To generate discrete stock price, the following differential equation is used: Btþ1 – Bt ¼ Bt ðm þ sztþ1 Þ

ð4:9Þ

where zt+1*N(0, 1). When simulating, we assume μ = 0.000001, σ = 0.1. Suppose the initial price is 1, the open, high, low and closing prices are generated as follows: 1 2 3

Generate a path of stock price with the length of 1000,000. Divide the 1000,000 stock prices into 1000 groups. Within each group, the first and the last stock prices are used as opening and closing prices. The maximum and minimum prices within each group are used as high and low prices.

With the simulated four pieces of price information, the technical range can be calculated. To scrutinize the unit root property of the technical range, we employ the aug­ mented Dickey-Fuller (ADF) test. The results are presented in Table 4.1. Con­ sistent with Proposition 1, the ADF test shows that the null hypothesis that there is a unit root can not be rejected. Table 4.2 reports the co-integration test results. Table 4.1 Unit root test results on simulated technical range t-Statistic

Prob.

Augmented Dickey-Fuller test statistic

−2.174

0.216

Test critical values:

−3.437 −2.864 −2.568

1% level 5% level 10% level

Note: when performing unit root test on simulated technical range, the parameter β is specified to be 0. The lag length determined by SIC is p = 2.

22

Candlestick

Table 4.2 Co-integration rank test on simulation observations Unrestricted Co-integration Rank Test (Trace) Hypothesized No. of CE(2) None* At most 1

Trace Eigenvalue 0.054 0.002

Statistic 57.037 1.708

0.05 Critical Value 12.321 4.130

Prob**. 0.000 0.225

Note: Trace test indicates 1 co-integrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnon-Haug-Michelis (1999) p-value; ‘No. Of CE(s)’ means number of co-integration equation.

Consistent with Proposition 2, the result shows that the null hypothesis that there exists co-integration between closing price and technical range can not be rejected.

4.3 Empirical evidence There is huge evidence that stock price is not a purely random walk. For example, the density of stock return is found to be of high kurtosis and large skewness, and the volatility of stock return is clustering, etc. Therefore, the results obtained on the simulated random walk process are not very convincing. Empirical studies performed on real asset prices are necessary to further consolidate the above sta­ tistical properties of the technical range. In this section, empirical studies are performed on different stock indices to see if the above listed theoretical properties are correct on real data. We collect the monthly data of such stock indices as DAX of Germany, S&P500 of the United States, and SSEC index of China for the sample period over 1991.01–2013.12 with 276 observations. The collected data sets are typical, covering both developed financial markets (DAX, S&P500) and the largest developing financial market (SSEC). The data sets are downloaded from www.finance.yahoo.com. For each data set, four pieces of price information are reported, the opening, high, low and closing prices. The technical range is calculated on these prices. Table 4.3 presents the augmented Dickey-Fuller (ADF) unit root test results for the closing price of each stock index. The null hypothesis of the ADF test is that there is a unit root. The results show that the null hypothesis for S&P500 and DAX can not be rejected at the significance level of 5%. However, the ADF test result shows that the null hypothesis of the unit root process for the Chinese stock market index, SSEC, is rejected at a significant level of 5%.1 Table 4.4 reports the ADF test on the technical range. Consistent with the results in Table 4.3, ADF tests performed on the technical range also show that the hypothesis of unit root in S&P500 and DAX can not be rejected at a significance level of 5%. The unit root process in SSEC is rejected at a significance level of 1%.

Statistical properties

23

Table 4.3 Unit root test results on closing price of S&P500, DAX and SSEC Augmented Dickey-Fuller test statistic S&P500 DAX SSEC Test critical values:

1% level 5% level 10% level

t-Statistic

Prob.

−1.700 −1.422 −3.340

0.430 0.572 0.014

−3.437 −2.864 −2.568

Note: when performing the unit root test on the closing price, we specify there is a constant but no trend.

Table 4.4 Unit root test results on technical range of S&P500, DAX and SSEC Augmented Dickey-Fuller test statistic Technical range of S&P500 Technical range of DAX Technical range of SSEC

−2.174 −2.108 −2.570 −4.479

Test critical values:

−3.437 −2.864 −2.568

1% level 5% level 10% level

0.216 0.242 0.101 0.000

Note: when performing the unit root test on the closing price, we specify there is a constant but no trend.

Table 4.5 Co-integration rank test on S&P500 stock index Unrestricted Co-integration Rank Test (Trace) Hypothesized No. of CE(2) None* At most 1

Eigenvalue 0.083 0.007

Trace

0.05

Statistic 25.335 1.950

Critical Value 15.495 3.841

Prob**. 0.001 0.163

Note: Trace test indicates 1 co-integrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnon-Haug-Michelis (1999) p-value; ‘No. of CE (s)’ means the number of the co-integration equation.

Co-integration tests are only performed on the S&P500 and DAX indices for the reason that the ADF test reports that SSEC is not a unit root process. Tables 4.5– 4.6 report the testing results. Consistent with Proposition 2, the null hypothesis of no co-integration is rejected at a significance level of 5%. The trace test indicates 1 co-integration equation between the closing price and the technical range.

4.4 Summary Technical range is an important component of the candlestick, and it has been widely used in technical analysis. However, its properties have never been, to our knowledge, scrutinized by academic research.

24

Candlestick

Table 4.6 Co-integration rank test on DAX stock index Unrestricted Co-integration Rank Test (Trace) Hypothesized No. of CE(2) None* At most 1

Eigenvalue 0.086 0.006

Trace

0.05

Statistic 26.042 1.610

Critical Value 15.495 3.841

Prob**. 0.001 0.205

Note: Trace test indicates 1 co-integrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnon-Haug-Michelis (1999) p-value; ‘No. of CE (s)’ means the number of the co-integration equation.

Under some mild assumptions, this chapter for the first time obtains the sta­ tistical properties of the technical range. Both simulations and empirical studies are performed to scrutinize these properties, and the results confirm the theoret­ ical propositions. The properties of the technical range obtained in this chapter are of great interest and importance to both academic researchers and investment practition­ ers. In the following chapters, we will show through empirical studies that these properties can be used in risk spillover investigation, technical range forecasting, and return forecasting. Ignorance of these properties will result in inefficient forecasts.

Note 1 The reason may be that the Chinese stock market is not informationally efficient. Jiang, et al. (2011) find significant predictability of the Chinese stock market, and the predictability can not be explained by an asset pricing model. Han et al. (2013) find there is a significant momentum effect in the Chinese stock market.

Part III

Statistical models

5

DVAR model

The variability of stock prices is fascinating to both academic researchers and investment practitioners. The problem of how to describe the variability of stock price is of great interest, and many time series modeling techniques have been proposed. The first and the most important modeling technique for univariate time series is the ARMA model. An ARMA model of order (p, q), ARMA(p, q) is given by: rt ¼ c þ

p X

αi rt–i þ εt þ

i¼1

q X

bj εt–j ;

ð5:1Þ

j¼1

where c is a constant term, rt is the stock return, εt is a white noise. αi and βj are parameters. The ARMA model is totally data driven and capable of capturing the linear dependence among rt. However, this model can not explain the volatility cluster­ ing effect, a well documented fact in empirical finance. To simultaneously capture the linear dependence and the volatility clustering effect in returns, the following ARMA-GARCH (p, q) model can be used: P P rt ¼ c þ i αi rt–i þ j bj εt–j þ εt ; εt ¼

pffiffiffiffi ht zt ;

ht ¼ o þ

i:id:

Pp i¼1

ð5:2Þ

zt ~ N ð0; 1Þ

αi ht–i þ

Pq j¼1

bj ε2t –j ;

where zt is the noise term of standard normal distribution, ht is the stock return variance. To ensure positivity and stationarity of return variance, it is required that o > 0;

αi ≥ 0;

bj ≥ 0;

p X i¼1

αi þ

p X

bj < 1:

j¼1

The intertemporal capital asset pricing model (ICAPM) of Merton (1973) sug­ gests that the conditional expected excess return on the stock market should vary positively with the market conditional variance. To capture the risk–return

28

Statistical models

tradeoff, the ARMA-GARCH model can be further generalized to the following GARCH-in-Mean model P P rt ¼ c þ i αi rt–i þ j bj εt –j þ ght þ εt ; pffiffiffiffi ht zt ; i:id: zt ~ N ð0; 1Þ Pp Pq ht ¼ o þ i¼1 αi ht –i þ j¼1 bj ε2t–j ; εt ¼

ð5:3Þ

where γ is the coefficient of relative risk aversion, reflecting the risk–return tradeoff. The above mentioned modeling techniques are all based on closing price, giving no attention to other available price information such as the high and the low price extremes.1 Recent academic literature shows that high and low prices have significant effect on stock returns due to limited attention (Hirshleifer and Teoh, 2003; Hirshleifer et al., 2011). George and Hwang (2004) found that nearness to the 52-week high dominates and improves upon the forecasting power of past returns (both individual and industry returns) for future returns. Huddart et al. (2009) found that past price extremes (around a stock’s 52­ week highs and lows) influence investors’ trading decisions. Li and Yu (2012) found that nearness to the 52-week high positively predicts future aggregatemarket returns, while nearness to the historical high negatively predicts future market returns. Xie and Wang (2018) found high and low prices are informative for return forecasting. This chapter shows how price extremes can be used for stock return modeling using the range decomposition technique presented in Chapter 4. This chapter is organized as follows. Section 1 proposes a decomposition-based vector autore­ gressive (DVAR) model for return modeling. Section 2 presents the statistical foundations of the DVAR model. Sections 3–4 scrutinize the results through simulations and empirical studies. We summarize in Section 5.

5.1 The model In Chapter 4, we demonstrated that a closing price can be approximated by the following equation, lnðCt Þ ≈ lnðTRt Þ – lnðPRt Þ

ð5:4Þ

where Ct, TRt and PRt are, respectively, the closing price, the technical range and the Parkinson range. Denoting the error resulted from this approximation as Wt, we can rewrite the log closing price as follows lnðCt Þ ¼ lnðTRt Þ – lnðPRt Þ þ Wt ; where Wt ¼ –lnð1 þ Et =PRt Þ

ð5:5Þ

DVAR model

29

and Et ¼ ½ðHt – Ct Þ=Ct – lnðHt =Ct Þ] þ ½ðCt – Lt Þ=Ct – lnðCt =Lt Þ]: Denoting ln(TRt)−ln(PRt) as Rt, the log closing price can be rewritten as lnðCt Þ ¼ Rt þ Wt :

ð5:6Þ

Taking differential operation on both sides of Eq. (5.6), we obtain rt ¼ DRt þ DWt

ð5:7Þ

where Δ is the differential operator. Eq. (5.7) indicates that stock return can be decomposed into a couple of components. Therefore modeling stock return is equivalent to a modeling of ΔRt and ΔWt. Instead of modeling ΔRt and ΔWt one by one, we suggest ΔRt and ΔWt are treated as a system, and propose to use a vector autoregressive (VAR) model for a simultaneous modeling of ΔRt and ΔWt. The VAR model is popularized by Sims (1980) as an atheoretical forecasting technique. This technique is underpinned by statistical methodology and not subject to contemporary macroeconomic theory. A VAR model of order p, VAR(p) is given by yt ¼ C þ

p X Ai yt–i þ εt ;

ð5:8Þ

i¼1

where yt is a k × 1 vector and εt is a k × 1 vector of error terms. Here in this book yt = (ΔRt, ΔWt)T. As this VAR model is based on price decomposition, we thus call this model the decomposition-based vector autoregressive model, or DVAR for short.

5.2 Statistical foundation In the absence of prior information, the VAR assumes that every series interacts linearly with both its own past values as well as those of every other included series. In this section, we will show, both theoretically and empirically, that ΔRt and ΔWt interact linearly with each other. The tool we use to detect the linear interaction is the Granger causality test. Before presenting the proof of bidirectional Granger causality between ΔRt and ΔWt, some symbols need to be clarified first. The symbols used in this section are Oh;l t : Information set released from Ht to Lt; Oc;l tþ1 : Information set released from Ct to Lt+1; h;l Otþ1 : Information set released from Ht+ 1 to Ct+1. According to neoclassical finance, asset price changes are due to new information being reflected. Therefore, price changes are functions of information sets. To be

30

Statistical models

specific, lnðTRt Þ ¼ f ðOht;l Þ; lnðPRt Þ ¼ gðOht;l Þ;

ð5:9Þ

l;h h;c rtþ1 ¼ hðOc;l tþ1 ; Otþ1 ; Otþ1 Þ;

where ln(TRt) and ln(PRt) are specified with different functions for the consid­ eration that these two indicators are assumed to manufacture information sets in different ways although TRt and PRt have the same information set. To make the above notations easy to understand, a random walk simulation is presented. The simulation is produced as follows: Step 1: An i.i.d sample of size 2000 from normal distribution N(0, 0.1) is generated; Step 2: The first 1000 sample is used as the trading process on day t, and the remaining sample is used as the trading process on day t+1; Step 3: Each day the largest data observation is used as the high price and the lowest data observation is used as the low price. Figure 5.1 presents the simulation results. It is the argument of neoclassical financial economics that the price movement is totally due to new information being reflected. According to Eq. (5.9), ΔRt and ΔWt can be represented by the following functions. DRtþ1 ¼ DlnðTRtþ1 Þ – DlnðPRtþ1 Þ ¼ ½f ðO

l ;h t þ1

h;l t

h;l t

Þ – f ðO Þ] – ½gðO

¼ pðO ; O

l ;h t þ1

(5.10) l ;h t þ1

h;l t

Þ – gðO Þ]

(5.11)

Þ;

(5.12)

ΔRt+1

Δω

t+1

μt+1 Δω

Δω t

t

β O

α

ΔR

t

Figure 5.1 Information sets and price changes

O

ΔRt

DVAR model 3

31

A random walk Ht

2.5

Ht+1 2

Ωh,l t

1.5

h,l

Ωt+1 l,h

Ωt+1

1

O

t

0.5

C

t+1

0

C

t

L 0

t

200

400

600

800

c,l Ωt+1

1000

L

t+1

1200

1400

1600

1800

2000

Figure 5.2 Granger causality: an illustration

and DWtþ1

¼

rtþ1 – DRtþ1 c;l t þ1

;O

l ;h t þ1

;O

(5.13) h;c tþ1

h;l t

Þ – pðO ; O

¼

hðO

¼

c;l h;c l;h qðOh;l t ; Otþ1 ; Otþ1 ; Otþ1 Þ:

l ;h t þ1

Þ

(5.14) (5.15)

Before proceeding to the rigorous mathematical statement, we present an intuitive explanation to the bi-directional Granger causality between ΔRt+1 and ΔWt+1. The argument that ΔWt Granger causes ΔRt+1 is manifested through Equations (5.12) and (5.15) since ΔWt and ΔRt+1 contain the same information set Oht;l .2 The same reasoning is also applicable to explaining why ΔRt Granger causes ΔWt+1. The intuitive explanation presented above shows some hints on how to prove the bidirectional Granger causality between ΔRt and ΔWt in mathematical sense. The proof consists of three stages: Stage 1: ΔRt and ΔWt contribute to forecasting ΔRt+1; Stage 2: ΔRt and ΔWt are not independent; Stage 3: ΔRt and ΔWt are not linear correlated. The proof of Stages 1–2 is obvious given that ΔRt+1, ΔRt and ΔWt have the same information set Oh;l t . An indirect proof is given to Stage 3. Suppose that ΔRt is perfectly linearly correlated with ΔWt, then the following equity

32

Statistical models

establishes DRt ¼ a þ b * DWt :

ð5:16Þ

l;h c;l l;h h;c h;l l;h Substituting pðOh;l t–1 ; Ot Þ and hðOt ; Ot ; Ot Þ – pðOt–1 ; Ot Þ for ΔRt and ΔWt respectively yields

ð5:17Þ

l;h c;l l;h h;c h;l l;h pðOh;l t–1 ; Ot Þ ¼ a þ b * ½hðOt ; Ot ; Ot Þ – pðOt–1 ; Ot Þ]:

Equation (5.17) indicates perfectly linear correlation between pðOht–;l 1 ; Olt;h Þ and l;h h;c hðOc;l t ; Ot ; Ot Þ, which seems impossible given that these two random variables are determined by different information sets. In a similar way, it can be proved that ΔRt Granger causes ΔWt+1. The relationships among ΔRt+1, ΔWt+1, ΔRt and ΔWt can be seen clearly in Figure 5.2. μt+1 denotes the residual term which can not be explained by ΔRt and ΔWt.

5.3 Simulations Section 2 has demonstrated that the bi-directional Granger causality test is due to information overlapping, regardless of the data generating process of the stock price. To confirm the demonstration, this section resorts to simulations. The sim­ ulating process proceeds as follows: Step 1: An i.i.d sample of size 100000 is generated from normal distribu­ tion, N(0, 0.01); Step 2: With these 100000 sample data, a geometric Brownian motion without drift term is obtained; Step 3: Divide these geometric Brownian motion points into 1000 groups, thus each group includes 100 points. Within each group the last data is used as the close price, the maximum price as the highest price and the minimum as the lowest price; Step 4: From Step 3, the ΔRt and ΔWt are calculated. Table 5.1 reports Granger causality test results between ΔRt and ΔWt. The null hypothesis is there is no Granger causality from ΔRt (ΔWt) to ΔWt (ΔRt). Since the Granger causality test is very sensitive to the lags, different lags are selected to consolidate the results. Feige and Pearce (1979), Christiano and Ljungqvist (1988) and Stock and Watson (1989) study the sensitivity of Granger causality Table 5.1 Granger causality tests between ΔRt and ΔWt: simulation results Lags Granger causality ΔWt doesn’t cause ΔRt ΔRt doesn’t cause ΔWt

2

4

6

F-Statistic

F-Statistic

F-Statistic

332.841*** 125.390***

187.013*** 44.387***

134.598*** 23.134***

Note: We use *** to mean significance at the level of 1%.

DVAR model

33

Table 5.2 Granger causality tests between ΔRt and ΔWt: empirical results Lags Granger Causality S&P500: ΔWt doesn’t cause ΔRt S&P500: ΔRt doesn’t cause ΔWt F100: ΔWt doesn’t cause ΔRt F100: ΔRt doesn’t cause ΔWt NK225: ΔWt doesn’t cause ΔRt NK225: ΔRt doesn’t cause ΔWt

2

4

6

F-Statistic

F-Statistic

F-Statistic

188.607*** 88.234*** 69.399*** 35.266*** 85.549*** 32.994***

99.070*** 26.650*** 36.840*** 12.139*** 43.435*** 9.093***

70.318*** 13.805*** 25.854*** 6.757*** 31.767*** 4.044***

Note: We use *** to mean significance at the level of 1%.

to lags selection. The F-statistics indicate significant evidence of bi-directional Granger causality between ΔRt and ΔWt.

5.4 Empirical results The real data generating process is more complicated than the simulations. Hence, empirical studies are needed to further scrutinize the results obtained in Section 3. To perform empirical studies, the monthly index data of different stock markets are collected. We collected the Standard and Poors 500 (S&P500) in the U.S. for the sample period from January, 1950 to December, 2008 with 708 observations, the FTSE100 (F100) in Great Britain for the sample period from April, 1982 to December, 2008 with 297 observations, and the Japanese NIKKEI225 stock index (NK225) for the sample period from January, 1984 to December, 2008 with 300 observations. For each month, four pieces of price information, opening, high, low and closing, are reported. The data set is downloaded from the finance subdirectory of the website www.finance. yahoo.com. Table 5.2 reports the Granger causality tests results. Consistent with both simulation results and the theoretical ones, the F-statistics show that the null hypothesis of no Granger causality between ΔRt and ΔWt is rejected. Also we find the results are quite robust to the lags.

5.5 Summary Traditionally, stock return modeling is closing price-based. This modeling tech­ nique, though simple in application, fails to incorporate the other price informa­ tion, such as the high and low price extremes. Based on the range decomposition-based technique, this chapter proposes the DVAR model for return modeling. The DVAR model makes full use of the high, low and close prices, and thus is more efficient in information employment

34

Statistical models

compared with the classic return modeling technique. The DVAR model pro­ vides a new framework for return modeling and forecasting. The statistical foundations of the DVAR model are also presented in this chapter. We use both theoretical explanations, simulations, and empirical evi­ dence to confirm the statistical foundations.

Notes 1 High–low price extremes have been widely used in financial econometrics. Parkin­ son (1980) proposed the high–low price range as a volatility estimator. Instead of using two points data, Garman and Klass (1980) further extended the range esti­ mator by incorporating the high, low, opening and closing prices into a volatility estimator. Rogers and Satchell (1991) and Rogers et al. (1994) proposed an alter­ native estimator which is drift-independent. Other references on range include Beckers (1983), Wiggins (1991), Kunitomo (1992), and more recently Yang and Zhang (2000). Corwin and Schultz (2012) used high–low range to estimate bid–ask spreads. Another noticeable effort in using high-low price information to model financial markets goes to Han et al. (2008), who suggested “interval” time series, instead of “point”, to describe financial markets. 2 Information sets Oht;l and Olt;h contain the same information since they both denote the information released over the time spanning Ht and Lt.

6

Shadows in DVAR

It has been demonstrated that stock return can be decomposed into a couple of components, ΔRt and ΔWt. A decomposition-based vector autoregressive (DVAR) model is proposed for a simultaneous modeling of ΔRt and ΔWt to obtain a return forecast. In this chapter, we will further show that shadows in the candlestick charts are informative for forecasting ΔRt and ΔWt. In Chapter 5, we show theoretically that stock return can be decomposed into a couple of components and propose a DVAR model for return modeling. The DVAR model establishes a new framework for analyzing the dynamics of stock return. In this chapter, we step further by showing that shadows in candlestick charts inform the forecasts of ΔRt and ΔWt, and thus the forecasting of stock return. The main finding of this chapter is that both lower and upper shadows Granger cause ΔRt and ΔWt. This finding is important as it means that the infor­ mation contained in the lower and upper shadows should be used if we model stock return using the DVAR model. This chapter is organized as follows: Section 1 presents simulation results. Section 2 presents the theoretical explanations. Empirical results are presented in Section 3. Summaries are given in Section 4.

6.1 Simulations In this section, we are going to argue through simulation that both upper and lower shadows Granger cause ΔRt and ΔWt. The simulating process proceeds as follows: Step 1: An i.i.d. sample of size 1,000,000 is generated from N(0, σ2); Step 2: A geometric Brownian motion of size 1,000,000 is generated from the above data sample; Step 3: Divide the geometric Brownian motion into 1,000 periods; within each period there are 1,000 data observations which simulate how the price evolves within each period; Step 4: Within each period, the first data observation is used as the opening price, the last as the closing price, the maximum as the high price, and the minimum as the low price;

36

Statistical models Step 5: From 4, ΔRt, ΔWt, lst and ust are calculated.

To be robust, we use different σs (σ=0.01, 0.05, 0.1) to represent low, medium, and high volatility respectively. We repeat the above simulations for 1000 times for each σ. Since the Granger causality test is very sensitive to the lag selection, different lags are used to consolidate the results. Feige and Pearce (1979), Christiano and Ljungqvist (1988), and Stock and Watson (1989) studied the sensitivity of Granger causality to lag selection. Tables 6.1–6.3 report the summary statistics of F-statistic and the p-value for the Granger causality test. Symbol ‘A ↛ B’ means that ‘A does not Granger causes B’. The statistics as a whole confirm that upper and lower shadows Granger cause ΔRt and ΔWt, however there is still something interesting. The results reported in Tables 6.1–6.3 reject the hypothesis that lst and ust do not Granger cause ΔRt+1 as the p values are reported to be 0, and the results are very robust, free from lags and volatility. The results on the hypothesis that lst and ust do not Granger cause ΔWt+1 are a little bit more complex: first, the hypothesis can not be consistently rejected that lst and ust do not Granger cause ΔWt+1 since the maximum p values are above 10% for all the simulation results; second, the mean p values increase with the lags. Table 6.1 Granger causality tests: σ = 0.01 Lags

2

4

6

lst ↛ ΔRt

F-Statistic

p-value

F-Statistic

p-value

F-Statistic

p-value

Mean Max Min Std

90.540 114.345 49.983 14.743

0.000 0.000 0.000 0.000

50.869 75.551 30.566 7.282

0.000 0.000 0.000 0.000

34.374 52.528 22.844 4.932

0.000 0.000 0.000 0.000

ust ↛ ΔRt

F-Statistic

p-value

F-Statistic

p-value

F-Statistic

p-value

Mean Max Min Std

88.379 132.616 52.065 14.159

0.000 0.000 0.000 0.000

49.953 79.739 29.255 7.521

0.000 0.000 0.000 0.000

33.503 52.283 21.359 4.889

0.000 0.000 0.000 0.000

lst ↛ ΔWt

F-Statistic

p-value

F-Statistic

p-value

F-Statistic

p-value

5.129 14.191 1.061 1.930

0.0103 0.374 0.000 0.0346

3.076 8.646 0.363 1.188

0.0421 0.903 0.000 0.0907

F-Statistic

p-value

F-Statistic

p-value

5.033 12.398 0.283 1.894

0.0139 0.889 0.000 0.0547

3.084 9.223 0.614 1.220

0.0456 0.719 0.000 0.100

Mean Max Min Std ust ↛ ΔWt Mean Max Min Std

11.183 28.933 1.657 4.365 F-Statistic 11.158 26.848 1.224 4.481

0.00237 0.191 0.000 0.0115 p-value 0.00256 0.295 0.000 0.0145

Shadows in DVAR 37 Table 6.2 Granger causality tests: σ = 0.05 Lags

2

4

6

lst ↛ ΔRt

F-Statistic

p-value

F-Statistic

p-value

F-Statistic

p-value

Mean Max Min Std

80.165 127.320 40.421 13.177

0.000 0.000 0.000 0.000

43.238 66.692 25.506 6.763

0.000 0.000 0.000 0.000

28.783 44.802 16.763 4.354

0.000 0.000 0.000 0.000

ust ↛ ΔRt

F-Statistic

p-value

F-Statistic

p-value

F-Statistic

p-value

Mean Max Min Std

75.855 117.905 46.218 12.844

0.000 0.000 0.000 0.000

40.558 64.317 23.442 6.802

0.000 0.000 0.000 0.000

27.593 42.804 13.735 4.399

0.000 0.000 0.000 0.000

lst ↛ ΔWt

F-Statistic

p-value

F-Statistic

p-value

F-Statistic

p-value

10.991 30.756 1.992 4.168

0.00206 0.137 0.000 0.00937

5.006 13.679 0.821 1.789

0.0114 0.512 0.000 0.0398

3.106 8.716 0.586 1.228

0.0481 0.742 0.000 0.104

F-Statistic

p-value

F-Statistic

p-value

F-Statistic

p-value

10.630 28.306 1.038 4.397

0.00464 0.354 0.000 0.0251

4.819 12.778 1.047 1.876

0.0151 0.382 0.000 0.0425

2.967 7.744 0.407 1.091

0.0455 0.875 0.000 0.0925

p-value

F-Statistic

p-value

23.533 40.131 13.256 4.002

0.000 0.000 0.000 0.000

F-Statistic

p-value

22.341 37.375 12.084 4.198

0.000 0.000 0.000 0.000

Mean Max Min Std ust ↛ ΔWt Mean Max Min Std

Table 6.3 Granger causality tests: σ = 0.1 Lags lst ↛ ΔRt Mean Max Min Std ust ↛ ΔRt Mean Max Min Std lst ↛ ΔWt Mean Max Min Std ust ↛ ΔWt Mean Max Min Std

2 F-Statistic 66.480 108.804 37.795 11.413 F-Statistic 63.468 106.682 33.023 11.753 F-Statistic 10.687 35.249 1.369 4.286

4 p-value 0.000 0.000 0.000 0.000 p-value 0.000 0.000 0.000 0.000 p-value 0.0250 0.255 0.000 0.0141

F-Statistic 34.854 54.921 19.122 5.967 F-Statistic 43.238 66.692 25.506 6.763

6 0.000 0.000 0.000 0.000 p-value 0.000 0.000 0.000 0.000

F-Statistic

p-value

F-Statistic

p-value

4.700 16.346 0.783 1.880

0.0157 0.536 0.000 0.0448

3.001 8.750 0.515 1.158

0.0473 0.797 0.000 0.0971

F-Statistic

p-value

F-Statistic

p-value

F-Statistic

p-value

10.283 27.438 1.043 4.393

0.00416 0.353 0.000 0.0191

4.639 12.554 0.598 1.852

0.0189 0.664 0.000 0.0575

2.939 8.309 0.253 1.177

0.0512 0.958 0.000 0.105

38

Statistical models

Figures 6.1–6.3 present the histograms of p values when σ takes different values. The histograms demonstrate the distributions of p values center on the value of less than 5% and diverge with the increase of lags, which are consistent with the summary statistics.

6.2 Theoretical explanation We have demonstrated with simulations that upper and lower shadows Granger cause ΔRt. In this section, we are going to present a theoretical explanation. Before presenting the results, some preliminary knowledge on Taylor expansion is needed. Suppose function f(x) has continuous derivative of order n + 1. Thus f(x) can be expanded at point x0 as follows: f ðxÞ ¼ f ðx0 Þ þ

n X 1 k¼1

k!

k

f k ðx0 Þðx – x0 Þ þ

1 nþ1 f nþ1 ðyÞðx – x0 Þ ; ðn þ 1Þ!

ð6:1Þ

where f k is the kth derivative of f (x), and θ 2 (x0, x) if x > x0 or θ 2 (x, x0) if x0 > x. With the fact that the kth derivative of ex is still ex, we can expand the high price Ht+1 at the low price Lt+1 as follows: Htþ1

¼

e lnðHtþ1 Þ ¼ e htþ1

¼

e ltþ1 þ e ltþ1 ðhtþ1 – ltþ1 Þ þ

¼

Ltþ1 þ Ltþ1 ðhtþ1 – ltþ1 Þ þ

’

Ltþ1 þ Ltþ1 ½ðhtþ1 – ltþ1 Þ];

1 x 2 e ðhtþ1 – ltþ1 Þ 2! 1 x 2 e ðhtþ1 – ltþ1 Þ 2!

where ht+1 = ln(Ht+1), lt+1 = ln(Lt+1), ξ 2 (lt+1, ht+1). Rearranging and taking logarithm on both sides of the above equality, we obtain lnðHtþ1 – Ltþ1 Þ ’ ltþ1 þ lnðhtþ1 – ltþ1 Þ

ð6:2Þ

Rtþ1 ¼ lnðTRtþ1 Þ – lnðPRtþ1 Þ ’ ltþ1 :

ð6:3Þ

or

Through Eq. (6.3), we get DRtþ1 ’ ðltþ1 – lt Þ:

ð6:4Þ

0

0

0.05

0.1

0.05

0.15

0.1

0.2

0.25

0.15

0.3

lag=2

lag=2

0.2

0

100

200

300

400

500

600

700

800

900

0

100

200

300

400

500

600

700

800

900

0

0

0.1

0.2

0.3

lag=4

t

0.2

0.4

0.6

0.8

lag=4

ust doesn’t Granger cause ΔWt

t

ls doesn’t Granger cause ΔW

Figure 6.1 Granger causality test: the histogram of p values when σ = 0.01

0

200

400

600

800

1000

0

200

400

600

800

1000

0.4

0

100

200

300

400

500

600

700

0

100

200

300

400

500

600

700

800

0

0

0.2

0.2

0.4

0.4

0.6

0.6

lag=6

0.8

lag=6

0.8

1

Shadows in DVAR 39

0

0

0.1

0.05

0.2

0.3

0.1

lag=2

lag=2

0.4

0

100

200

300

400

500

600

700

800

900

0

100

200

300

400

500

600

700

800

900

0

0

0.2

0.3

0.4

0.5

0.6

lag=4

t

0.1

0.2

0.3

lag=4

ust doesn’t Granger cause ΔWt

0.1

t

ls doesn’t Granger cause ΔW

Figure 6.2 Granger causality test: the histogram of p values when σ = 0.05

0

200

400

600

800

1000

0

200

400

600

800

1000

0.4

0

100

200

300

400

500

600

700

0

100

200

300

400

500

600

700

0

0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

lag=6

lag=6

0.8

40 Statistical models

0

0

0.05

0.1

0.1

0.2

0.15

0.2

0.3

0.25

lag=2

0.3

lag=2

0.4

0

100

200

300

400

500

600

700

800

900

0

100

0

0

200

300

400

500

600

700

800

900

0.2

0.3

0.4

0.5

0.6

t

lag=4

t

0.1

0.2

0.3

0.4

0.5

0.6

lag=4

ust doesn’t Granger cause ΔW

0.1

t

ls dosen’t Granger cause ΔW

Figure 6.3 Granger causality test: the histogram of p values when σ = 0.1

0

200

400

600

800

1000

0

200

400

600

800

1000

0

100

200

300

400

500

600

700

0

0

0

100

200

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700

0.2

0.2

0.4

0.4

0.6

lag=6

0.8

0.6

lag=6

1

0.8

Shadows in DVAR 41

42

Statistical models

In the same way, we can expand the low price Lt+1 at the high price Ht+1 as follows, Ltþ1 ¼ e lnðLtþ1 Þ ¼ e ltþ1

1 Z 2 e ðltþ1 – htþ1 Þ 2! 1 2 ¼ Htþ1 þ Htþ1 ðltþ1 – htþ1 Þ þ e Z ðltþ1 – htþ1 Þ 2! ’ Htþ1 þ Htþ1 ðltþ1 – htþ1 Þ; ¼ e htþ1 þ e htþ1 ðltþ1 – htþ1 Þ þ

where η 2 (lt+1, ht+1). Rearranging and taking logarithm on both sides yields lnðHtþ1 – Ltþ1 Þ ’ htþ1 þ lnðhtþ1 – ltþ1 Þ

ð6:5Þ

Rtþ1 ¼ lnðTRtþ1 Þ – lnðPRtþ1 Þ ’ htþ1 :

ð6:6Þ

or

By Eq. (6.6), we get DRtþ1 ’ ðhtþ1 – ht Þ:

ð6:7Þ

With the above preparations, we present the explanations to the Granger causal­ ity as follows: (1) In case of Ot > Ct, lt+1 − lt = lt+ 1−ct+ ct−lt = lt+ 1−ct+ lst. Since lt+ 1−lt includes the lower shadow lst, which thus makes lower shadow Granger cause ΔRt+ 1. (2) In case of Ot < Ct, ht+1 − ht = ht+1 − lt − (ht − lt), lst = ot − lst = ot − ht + (ht − lt). In this case, both lst and ht+1 − ht include ht − lt, which indicates lst gets the predictive information for forecasting ΔRt+1. In other words, lst contributes to forecasting ΔRt+ 1. In both cases, lst Granger causes ΔRt+1. (3) In case of Ot < Ct, ht+1 − ht = ht+ 1 − ct−(ht − ct) = ht+1 − ct − ust. Since ht+1 − ht includes the lower shadow ust, which thus makes upper shadow Granger cause ΔRt+1. Although simulations are performed with the assumption that stock prices follow the random walk model, the theoretical explanations presented above actu­ ally require no specific assumptions for the data generating process (DGP) on the stock price. The reason why upper and lower shadows are Granger causality to the DVAR model is actually due to the information overlapping: ΔRt+1 overlaps with lower shadow (upper shadow). The information overlapping is well illustrated in Figure 6.4. To further consolidate these findings, it is of great necessity to perform empirical studies on real stock prices.

Shadows in DVAR 43 O >C t

Ht+1

t

H

t

O

t+1

C

t+1

Ot

Ct L

Lt

t+1

Ct>Ot H

t+1

Ht

Ct

Ct+1

Ot+1 Ot Lt

Lt+1

Figure 6.4 Shadows in DVAR: Granger causality

6.3 Empirical evidence The following are the well documented facts: (1) there is no significant linear autocorrelation in stock returns; (2) the volatility of stock returns are clustering and highly persistent; (3) the distribution of stock returns are far from being normal. They are of high kurtosis, negative skewness, and so on. All these facts indicate that the real DGP of the stock prices is unknown. Thus, simulations based on the random walk hypothesis might produce biased results. To lower down the risk of potential bias, empirical studies performed on real stock prices are needed. The empirical studies performed in this section fulfil two

44

Statistical models

Table 6.4 Empirical studies on S&P500 Daily Index Data: S&P500 Lags 2 4 6

lst ↛ ΔRt

lst ↛ ΔWt

ust ↛ ΔARt

ust ↛ ΔWt

0.000 0.000 0.000

0.0585 0.0192 0.0217

0.000 0.000 0.000

0.000 0.000 0.000

lst ↛ ΔRt

lst ↛ ΔWt

ust ↛ ΔRt

ust ↛ ΔWt

0.000 0.000 0.000

0.0727 0.0845 0.0629

0.000 0.000 0.000

0.000 0.000 0.000

lst ↛ ΔRt

lst ↛ ΔWt

ust ↛ ΔRt

ust ↛ ΔWt

0.000 0.000 0.000

0.551 0.480 0.446

0.000 0.000 0.000

0.0017 0.0039 0.0297

Weekly Index Data: S&P500 Lags 2 4 6

Monthly Index Data: S&P500 Lags 2 4 6

purposes: first, they are used to consolidate the simulations and the theoretical explanations; second, they are used to confirm that the Granger causality is due to information overlapping, free from the real DGP of the stock prices. We collected the daily, weekly, and monthly index data of the U.S. Standard and Poors 500 (S&P500) index data for the sample period from January, 1990 to December, 2011. For each frequency data, four pieces of price information, opening, high, low and closing, are reported. The data set is downloaded from the finance subdirectory of the website http://finance.yahoo.com. The observa­ tions for daily, weekly and monthly data are respectively 5547, 1147 and 264. The empirical results performed on the S&P500 index are reported in Table 6.4. For each frequency data observation, the p values of the Granger cau­ sality test are presented with different lags. For the sake of consistency, the lags are selected to be 2, 4, or 6. Highly consistent with the simulations, the hypoth­ esis of no Granger causality from upper and lower shadows to ΔRt is rejected at a significance level of 5%.

6.4 Summary We have demonstrated in Chapter 5 that stock return can be modeled using the DVAR model. In this chapter, we show with both theoretical explanations and empirical evidence that the upper and lower shadows in the candlestick are infor­ mative for DVAR forecasting. The findings obtained in this chapter are of great importance, as they indicate that shadows in the candlestick should be considered when using the DVAR model to forecast asset returns.

Part IV

Applications

7

Market volatility timing

It has been proved in Chapter 4 that the closing price (Ct) of an asset and its technical range (TRt) are cointegrated and that lnðCt Þ – lnðTRt Þ ≈ –lnðPRt Þ;

ð7:1Þ

where PRt is the Parkinson range. Let the unconditional mean of ln(PRt) be C, E[ln(PRt)] = C. Adding C to both sides of Eq. (7.1) yields lnðCt Þ – lnðTRt Þ þ C ≈ –lnðPRt Þ þ C:

ð7:2Þ

From Eq. (7.2) we can see that ln(Ct) = ln(TRt) − C is the equilibrium relation between ln(Ct) and ln(TRt), and that the absolute value of −ln(PRt) + C mea­ sures the deviation from the equilibrium. Larger deviation from equilibrium implies higher risk in this system. Since PRt is a volatility estimator, we will show in this chapter that PRt can be used in market volatility timing strategy.

7.1 Introduction Volatility has always been an interesting and important topic in financial econo­ metrics. Ever since the seminal paper of Engle (1982), a large number of ARCHlike models have been proposed. Among them, the GARCH (Bollerslev, 1986), the EGARCH (Nelson, 1991) and the GJR-GARCH (Glosten, et al. 1993) have been emphasized most in volatility estimating and forecasting. For a critical review with a thorough survey of the ARCH literature, see Bollerslev, et al. (1992). Traditionally, ARCH-like models are return-based. Recent literature shows a rising interest in using price range to estimate volatility. The price range is far more informative about the current level of volatility than is the squared return.1 Chou (2005) proposed the conditional autoregressive range model (henceforth CARR) to describe the dynamics of range-based volatility. They find the CARR model does provide a sharper volatility estimate compared with the standard GARCH model. Brandt and Jones (2006) formulated a model that is analogous to Nelson’s (1991) EGARCH model, but used the square root of the intra-day price range in place of the absolute return. They find

48

Applications

much better predicting power with the range-based volatility model over the return-based model for out-of-sampling forecasts. For a comprehensive review on range-based volatility, see Chou et al. (2009). A critical issue with both the CARR model and the range-based EGARCH model of Brandt and Jones (2006) is that they are “incomplete”. The CARR model only explains the varia­ tion in the price range that is indirectly related to the return volatility, while the range-based EGARCH model does not explain the variation in the price range, so both the CARR model and the range-based EGARCH model are partial (incomplete) models. Recently, a new range-based volatility model, GARCH@CARR is proposed by Xie, et al. (2019) to study the dynamics of asset volatility. The GARCH@CARR model is a joint modeling of the asset return and the price range. In this model, the dynamics of asset volatility are specified to be subordinated to the CARR model. This chapter examines the economic value of GARCH@CARR model in vol­ atility timing. Following Marquering and Verbeek (2004), we consider an inves­ tor with mean-variance utility function who uses the conditional volatility forecast to allocate his wealth between two assets: the risky asset and the riskfree asset.2 Empirical studies are performed on the monthly S&P500 stock index, and the results show that the GARCH@CARR model produces a valuable volatility forecast. Using the EGARCH (Nelson, 1991) and GJR-GARCH (Glosten et al. 1993) as benchmark models, we find portfolio returns formulated using the GARCH@CARR model report larger mean and smaller standard devia­ tion, and thus the higher Sharp ratio. The utility statistics show that an investor would be willing to pay 1.41% (1.09%) portfolio management fee to have access to the GARCH@CARR volatility forecast relative to the EGARCH (GJR­ GARCH) volatility forecast. The chapter is organized as follows. Section 2 presents the econometric meth­ odologies with some discussions. Section 3 presents the volatility timing strategy. In Section 4, an empirical example is performed on the S&P500 index to inves­ tigate the volatility timing ability of the GARCH@CARR model. Both in-sample and out-of-sample forecast comparisons between the GARCH@CARR model and the EGARCH and GJR-GARCH models are presented. Section 5 concludes.

7.2 GARCH@CARR model In this chapter the GARCH@CARR model of Xie, et al. (2019) is used to esti­ mate the assets’ volatility. The GARCH@CARR model of order (p, q) is pre­ sented as follows rt ¼ m þ rlt zt ; zt jF t –1 ~ i:i:d:N ð0; 1Þ; lnðlt Þ ¼ o þ

p X i¼1

αi lnðlt–i Þ þ

q X

bj lnðPRt–j Þ þ kzt–1 ;

(7.3) (7.4)

j¼1

PRt ¼ lt εt ; εt jF t –1 ~ i:i:d:LN ð–s2 =2; sÞ;

(7.5)

Market volatility timing

49

where rt and PRt are respectively the return and the Parkinson range. The distri­ bution of the disturbance term zt is assumed to be distributed with a standard normal. The distribution of the disturbance term εt is assumed to be distributed with a log-normal density with unit mean. Further it is assumed that zt and εt are mutually independent. Parameter κ is used to capture the leverage effect which has been widely documented in empirical literature. The first two equations are referred to as the return equation and the GARCH equation, and the last equation is referred to as the measurement equation as it relates the observed real­ ized measure (price range) to the latent volatility. The measurement equation is an important component because it “completes” the model. In the return equa­ tion, λt = E(PRt jF t –1 ) is scaled by a factor ρ to make sure that ρλt is an unbiased conditional volatility estimator of rt. The persistence parameter of the Pp Pq GARCH@CARR model is calculated as i¼1 αi + j¼1 bj . The GARCH@CARR can be estimated by maximum likelihood estimation (MLE) method. The log-likelihood function of the GARCH@CARR model (conditionally on F t–1 ) is given by n

logLðfrt ; PRt gt¼1 ; YÞ ¼

n X

logf ðrt ; PRt jF t–1 Þ;

ð7:6Þ

t¼1

where Θ is the parameter set to be estimated. With the assumption that zt is inde­ pendent of εt, we can factorize the joint conditional density for (rt, PRt) by f ðrt ; PRt jF t –1 ; lt –1 Þ ¼ f ðrt jF t–1 ; lt –1 Þf ðPRt jF t –1 ; lt –1 Þ:

ð7:7Þ

Thus the log-likelihood function can be rewritten as n

logLðfrt ; PRt gt¼1 ; YÞ ¼ lðrÞ þ lðRÞ;

ð7:8Þ

where lðrÞ :¼

n X

logf ðrt jF t –1 ; lt–1 Þ; lðRÞ :¼

t¼1

n X

logf ðPRt jF t –1 ; lt –1 Þ:

t¼1

The partial log-likelihood, l(r) can be used to compare with that of a returnbased GARCH-like model. More discussions about the GARCH@CARR model are available in Xie, et al. (2019).

7.3 Economic value of volatility timing Consider an investor maximizing a mean-variance utility function and composing his portfolio from a risky asset and a risk-free asset. For a given level of (initial) wealth, the investor’s optimization problem is given by g U ðot Þ ¼ max fEt–1 ðrp;t Þ – Vt–1 ðrp;t Þg; ot 2 rp;t

¼ ot rt þ ð1 – ot Þrf ;t ;

(7.9) (7.10)

50

Applications

where U(•) is the quadratic utility function, rt, rf,t and rp,t are respectively the return on the risky asset, the return on the risk-free asset and the portfolio return, ωt is the wealth proportion allocated on the risky asset, and γ is the risk aversion parameter. In this paper, we take γ = 3, which is also commonly used in empirical studies. Et−1(•) and Vt−1(•) are respectively the conditional expected portfolio return and the conditional expected portfolio variance. It is straightfor­ ward that the optimal proportion allocated on the risky asset is given by ot ¼

1 Et–1 ðrt – rf ;t Þ ; g Vt –1 ðrt – rf ;t Þ

ð7:11Þ

where Et−1(rt−rf,t) is the expected risk premium over t−1 to t, and Vt−1(rt − rf,t) is the expected variance of risk premium over t−1 to t. Following Marquering and Verbeek (2004), we assume that short selling and borrowing at the risk-free rate is not allowed, the portfolio weights must lie between 0 and 1, and the optimal portfolio weight becomes ot ¼ 0 ¼ ot ¼ 1

if

ot ≤ 0; 0 < ot ≤ 1;

if if

ot > 1:

Over the out-of-sample period, the investor realizes an average utility level of 1 2 ^n 0 ¼ m ^ 0 – gs ^ ; 2 0

ð7:12Þ

^ 0 and s ^ 02 are the sample mean and variance, respectively, over the out-of­ where m sample period for the return on the benchmark portfolio. In this chapter, the popular EGARCH (Nelson, 1991) and GJR-GARCH (Glosten, et al. 1993) models are used as our benchmark models, and the portfolios formed using these two models are used as the benchmark portfolios. For the same investor when he or she forecasts the asset volatility using the GARCH@CARR model, the realized average utility level is given by 1 2 ^n g ¼ m ^ g – gs ^ ; 2 g

ð7:13Þ

^ 2g are the sample mean and variance, respectively, over the out-of­ ^ g and s where m sample period for the return on the portfolio formed using the forecasts of the asset volatility based on the GARCH@CARR model. The economic value of volatility timing is measured by utility gain which is defined as the difference between Equation (7.13) and Equation (7.12). We multiply the utility gain by 1200 to get the average annualized percentage return. The utility gain can be interpreted as the portfolio management fee that an investor would like to pay to have access to the additional information available in a predictive model relative to the benchmark model.

Market volatility timing

51

7.4 Empirical results This section describes the data and presents both the in-sample and the out-of­ sample results for volatility timing performance of the GARCH@CARR model relative to the benchmark models.

7.4.1 The data We collected the monthly data of the S&P500 stock index for the sample period from January 1983 to December 2016 with 408 observations.3 For each month four pieces of price information, the high, low, opening and closing prices were reported. The S&P500 index data sets were downloaded from the website www. finance.yahoo.com. The risk-free interest rate was downloaded from the homepage of Amit Goyal at http://www.hec.unil.ch/agoyal/. In this analysis, the S&P500 stock index is used as the risky asset. The high–low Parkinson range, PRt is calculated by PRt ¼ lnðHt Þ – lnðLt Þ;

ð7:14Þ

where ln(•) is natural logarithm, Ht and Lt are respectively the high and low prices. The risky asset return is calculated as rt ¼ lnðCt Þ – lnðCt –1 Þ;

ð7:15Þ

where rt and Ct are respectively the risky asset return and the closing price. Excess return is obtained by the risky asset return less the risk-free interest rate rpt ¼ rt – rf ;t ;

ð7:16Þ

where rpt and rf,t are respectively the excess return and the risk-free interest rate. Figure 7.1 presents the time series plots of the Parkinson price range and the excess return. It is clear from Figure 7.1 that price range is clustering and is more volatile in negative return than in positive return (the leverage effect). We multiply both the range and the excess return by 100. Table 7.1 presents the summary statistics for the range, the excess return, the absolute excess return and the squared excess return. The range, the absolute excess return and the squared excess return can be used as volatility estimator candidates. Consistent with the widely documented facts, we find high kurtosis and large negative skew­ ness of the excess return, which indicates a strong deviation from the normal dis­ tribution. The Ljung-Box Q statistics indicate strong persistence in both range, absolute excess return and the squared excess return, which implies the predict­ ability of the volatility. It is more interesting to compare the Q statistics for the range with those for the absolute excess return and for the squared excess return. The Q statistics show that there is a significant difference between the range and the absolute excess return (squared excess return), which means the volatility information in the range is different from those in the absolute excess return and in the squared excess return.

52

Applications

50

40

30 Range Excess Return 20

10

0

−10

−20

−30

0

50

100

150

200

250

300

350

400

Figure 7.1 Time series plots of Parkinson price range and excess return Table 7.1 Summary statistics of the range, the excess return, the absolute excess return and the squared excess return Range

Excess return

Absolute excess return

Mean 6.724 0.000 3.190 Median 5.692 0.417 2.530 Std 4.190 0.012 2.915 Min 2.110 −25.417 0.011 Max 41.847 11.562 25.417 Skewness 3.371 −1.024 2.343 Kurtosis 21.526 6.708 13.570 Jarque-Bera 6607.443*** 305.026*** 2272.566*** ρ(1) ρ(2) ρ(3) ρ(6) ρ(12) Q(12)

0.476 0.381 0.279 0.210 0.123 325.53***

0.052 −0.042 0.025 −0.060 0.029 7.528

0.185 0.096 0.175 0.094 0.033 61.436***

Squared excess return 18.647 6.402 44.606 0.000 646.045 8.751 108.287 193658*** 0.158 0.042 0.076 0.028 −0.010 23.338**

Notes: We use ρ(i) as the sample autocorrelation function coefficient at lag i, Q as the LjungBox Q statistics. The symbols *, **, *** mean respectively significance at 10%, 5%, and 1%.

7.4.2 In-sample volatility timing Maximum likelihood estimation of the GARCH@CARR model is performed over the whole sample for in-sample volatility timing.4 For comparison we use the

Market volatility timing

53

Table 7.2 Estimates of GARCH@CARR, GJR-GARCH and EGARCH Sample

GARCH@CARR

GJR-GARCH

EGARCH

0.611E – 04 (0.351) 0.843 (0.047) 0.112 (0.065) 0.046 (0.054)

−2.281 (0.447) 0.643 (0.067) 0.296 (0.098) −0.344 (0.064)

Panel A: Point Estimates and Log-Likelihood −0.335 (0.053) 0.610 (0.002) 0.263 (0.019) −0.097 (0.014) 0.619 (0.025) 0.369 (0.013)

ω α1 β1 κ ρ σ

Panel B: Auxiliary Statistics ψ l(r)

0.873 742.574

0.978 724.375

0.643 730.285

Notes: The return-based volatility model is specified as

rpt

¼

m þ εt ;

εt

¼

st zt ;

zt

e

N ð0; 1Þ; i:i:d:

In this chapter, we assume constant expected risk premium, which is removed by demeaning the excess return. For the GJR-GARCH model, the dynamics of the volatility is presented as

s2t ¼ o þ α1 s2t–1 þ b1 ε2t–1 þ kIt –1 ε2t–1 ; where It−1 is an indicator function of εt−1: It−1=0 if εt−10. The persistence parameter for the GJR-GARCH model is ψ=α1+β1+κ/2. For the EGARCH model, the volatility dynamics is given by

lnðs2t Þ ¼ o þ α1 lnðs2t–1 Þ þ b1 ðzt–1 –

rffiffiffi 2 Þ þ kzt–1 p

The persistence parameter for the EGARCH model is ψ=α1. The numbers in the brackets are the standard errors. When performing estimation, we only consider the models of order (1, 1) because there is evidence that the order (1, 1) is sufficient for capturing the persistence in the range and in the volatility (Chou, 2005).

GJR-GARCH (Glosten, et al. 1993) and EGARCH (Nelson, 1991) as bench­ mark models. We favor the GJR-GARCH and EGARCH models for two reasons: (i) the flexibility in capturing the leverage effect and (ii) the ability in fitting the volatility dynamics. Table 7.2 reports the estimation results. In this section, we only consider models of order (1, 1) because it has been well noted that order (1, 1) is sufficient for capturing the persistence in volatility.

54

Applications

The estimation results show that the GJR-GARCH model has the highest per­ sistence (ψ) in volatility dynamics, then comes the GARCH@CARR model, while the EGARCH model reports the lowest persistence. The log-likelihood statistic l(r) shows that the GARCH@CARR model, although it does not maximize the log likelihood function of return, still produces a better empirical fit than both the GJR-GARCH and the EGARCH models. The estimates also report signifi­ cant leverage effect in both the GARCH@CARR model and the EGARCH model since κs are reported to be negative and statistically significant. Interest­ ingly, the GJR-GARCH model reports no significant leverage effect. Figure 7.2 presents the in-sample volatility forecasts reported by the the GJR­ GARCH, EGARCH and the GARCH@CARR models. It is clear from Figure 7.2 that the GARCH@CARR model is more adaptive to the price changes.5 For in-sample volatility timing, the optimal weight allocated on the risky asset is presented as oin;t ¼

̄ 1 m ; g s2in;t

t ¼ 1; 2; ::; T :

ð7:17Þ

where ωin,t is the in-sample optimal weight, s2in;t is the in-sample volatility fore­ ̄ is the sample mean of the excess return which is presented as cast, and m ̄¼ m

T 1X rp : T t¼1 t

ð7:18Þ

Table 7.3 presents in Panel A the summary statistics of the in-sample portfolio returns formulated using different volatility models. It is clear that portfolio returns formulated using volatility models have lower standard deviations than the market portfolio.6 Both the utility and the Sharpe ratio statistics show the outperformance of the GARCH@CARR model over the other three benchmark models, which are consistent with the estimation results. Figure 7.3 presents the cumulative portfolio returns formulated using different volatil­ ity models. Obviously, portfolio returns formulated using volatility models are less volatile.

7.4.3 Out-of-sample volatility timing In practice, the ultimate way to evaluate a model is through its performance in out-of-sample forecasting. In this section a recursive (expanding) window fore­ casting procedure is carried out on the S&P500 stock index. To be specific, the whole T data observations are divided into an in-sample portion composed of the first m observations and an out-of-sample portion composed of the last q observations. The initial out-of-sample forecast sfmþ1 is based on the first m observations. The next out-of-sample forecast sfmþ2 is based on the first m+1 observations. Proceeding in this manner through the end of the out-of-sample period, we generate a series of q out-of-sample forecasts. The recursive predicting

0

50

Volatility Proxy

GJR

EGARCH

GARCH@CARR

100

150

200

250

300

Figure 7.2 In-sample volatility forecasting: GARCH@CARR, GJR-GARCH and EGARCH

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

350

400

450

Market volatility timing 55

56

Applications

Table 7.3 Summary statistics of the portfolio returns formulated using different vol­ atility models GARCH@CARR

Market Portfolio

EGARCH

GJRGARCH

6.78E-03 0.011 0.043 −0.245 0.124 −1.027 3.733 0.397 0.774 0.084

5.87E-03 8.6E-03 0.030 −0.165 0.075 −0.892 3.252 0.455 0.072 0.092

5.59E-03 8.34E-03 0.029 −0.168 0.082 −1.041 4.672 0.437 0.298 0.086

Panel A: In-sample volatility timing Mean 5.87E-03 Median 9.79E-03 Std 0.029 Min −0.154 Max 0.085 Skewness −0.755 Kurtosis 2.770 Utility (%) 0.462 Annual utility gain (%) – Sharp ratio 0.094 Panel B: Out-of-sample volatility timing Mean 3.54E-03 Median 6.52E-03 Std 0.029 Min −0.158 Max 0.082 Skewness −0.859 Kurtosis 3.323 Utility (%) 0.226 Annual utility gain (%) – Sharp ratio 0.061

4.61E-03 2.83E-03 2.91E-03 9.625E-03 5.621E-03 5.395E-03 0.045 0.032 0.032 −0.186 −0.158 −0.158 0.102 0.085 0.092 −0.801 −0.951 −0.808 1.518 2.866 2.636 0.163 0.130 0.135 0.763 1.152 1.101 0.064 0.034 0.036

Notes: (1) For in-sample volatility timing, the market portfolio is formulated by allocating all the wealth to the risky asset (ωin,t = 1), which is consistent with the buy-and-hold trading strategy. (2) For out-of-sample volatility timing, the wealth proportion allocated to the risky asset is determined by Equation (7.11). (3) The utility gain is obtained by the GARCH@CARR utility less the benchmark model utility. For example, if the benchmark model is EGARCH, the utility gain is calculated by (0.226%−0.130%), and the annual utility gain is given by (0.226%−0.130%) × 1200 = 1.152.

procedure simulates the situation of a forecaster in real time. In this section, we choose q = 240 which expands twenty years of data observations, from January 1997 to December 2016. The wealth proportion allocated to the risky asset is determined by oout;mþk ¼

̄out;mþk 1m ; g s2out;mþk

k ¼ 1; 2; 3; :::; 240;

ð7:19Þ

where σout,m+k is the out-of-sample volatility forecasts, and ̄ out;mþk ¼ m

mþk–1 X 1 rp ; m þ k – 1 t¼1 t

ð7:20Þ

0

50

GJR EGARCH GARCH@CARR Market Portfolio

100

150

200

250

300

Figure 7.3 Cumulative return using different in-sample volatility timing strategies: 1983.01–2016.12

0

0.5

1

1.5

2

2.5

3

350

400

450

Market volatility timing 57

58

Applications

which is the historical mean of the excess return. Goyal and Welch (2003, 2008) ̄ out;mþk is a stringent benchmark: predictive regression forecasts based show that m on macroeconomic variables frequently fail to outperform the historical average forecast in out-of-sample tests. Figure 7.4 presents the time series plots of the optimal allocation weights on risky asset. Compared with the GJR-GARCH model, the optimal weights reported by the EGARCH and GARCH@CARR models are more volatile, which indicates that the GARCH@CARR model and EGARCH model are more adaptive to the volatility changes. Figure 7.5 presents the time series plots of the out-of-sample cumulative portfolio returns. We find the GARCH@­ CARR model not only reports a larger cumulative return but also lower volatility compared with other competing models. The summary statistics for the out-of-sample portfolio returns formulated using different volatility models are reported in Panel B in Table 7.3. From the results two empirical facts emerge: (1) the EGARCH and the GJR­ GARCH models have non-distinguishable out-of-sample volatility forecasting performance since the summary statistics of the portfolio returns produced by these two models are very close. This finding is consistent with Figure 7.5 as the time series plots of the cumulative portfolio returns reported by these two models are quite similar. (2) The GARCH@CARR model delivers better out­ of-sample volatility forecasts. The summary statistics show that the portfolio returns formulated using the GARCH@CARR model report larger sample mean, lower standard deviation, higher utility and sharp ratio compared with the other two competing models. For example, the annual utility gain statistic shows that an investor would be willing to pay 1.15% (1.10%) portfolio manage­ ment fee to have access to the GARCH@CARR volatility forecast relative to the EGARCH (GJR-GARCH) volatility forecast. These two empirical facts reveal that the GARCH@CARR model has sharper volatility timing ability than the EGARCH and GJR-GARCH models have.

7.5 Summary From Proposition (2) (see Chapter 4, p.20) we obtain that the Parkinson range measures the deviation from the equilibrium between closing price and technical range. Therefore, the we can use the Parkinson range as a risk measure of the system of closing price and technical range. Taking the Parkinson range as a measure of volatility risk, this chapter inves­ tigates its economic value in volatility timing strategy. An empirical study is per­ formed on the monthly S&P500 stock index, and the results show that the range-based volatility model, GARCH@CARR model is more informative than the return-based EGARCH and GJR-GARCH models for both in-sample and out-of-sample volatility timing.

1

0

0

0

GJR

EGARCH

GARCH@CARR

50

50

50

100

100

100

150

150

150

Figure 7.4 Out-of-sample optimal allocation weight on risky Asset: 1997.01–2016.12

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

200

200

200

250

250

250

Market volatility timing 59

0

Market Portfolio GARCH@CARR GJR EGARCH

50

100

150

200

Figure 7.5 Cumulative portfolio return formulated using different out-of-sample volatility timing strategies: 1997.01–2016.12

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

250

60 Applications

Market volatility timing

61

Notes 1 Parkinson (1980) forcefully argued and demonstrated the superiority of using range as a volatility estimator as compared with the standard methods. Beckers (1983), among others, further extended the range estimator to incorporate infor­ mation about the opening and closing prices and the treatment of a time-varying drift, as well as other considerations. Other references concerning price range vol­ atility include Garman and Klass (1980), Wiggins (1991), Rogers and Satchell (1991), Kunitomo (1992) and Yang and Zhang (2000). Alizadeh, et al. (2002) presented theoretically and empirically that a range-based volatility estimator is not only highly efficient but also approximately Gaussian and robust to microstruc­ ture noise. Degiannakis and Livada (2013) even found the range-based volatility estimator is more accurate than the realized volatility estimator. 2 Other research papers concerning the economic value of volatility timing include Busse (1999), Fleming et al. (2001), Thorp and Milunovich (2007) and Chou and Liu (2010). 3 Only the data from 1983 to 2016 is used for the reason that a change in the data compilation occurred around the end of April, 1982 (Chou, 2005). 4 In this paper, we assume the expected excess return (risk premium) to be constant, thus we demean the excess return when performing maximum likelihood estimation. pffiffiffiffiffiffiffiffi pffiffi 5 Since E(|xt|) = p2s if xt * N(0, σ), so we use jrt j p=2 as the volatility proxy. 6 For in-sample volatility timing, the market portfolio is formulated by allocating all the wealth to the risky asset, which is consistent with the buy-and-hold trading strategy.

8

Technical range forecasting

In Chapter 4 we demonstrated that closing price and technical range are co­ integrated. In this chapter, we will show how this property can be used to improve technical range forecasting.

8.1 Introduction Technical range, defined as the difference between the high and low prices, is an important component of the candlestick, and gauges the variability of the price movement during a specific time period. The larger the technical range, the higher risk investors are confronted with. Therefore, the question of how to describe the dynamics of technical range is of great interest to investors. Voluminous literature is available investigating high–low price range. Early application of range in the field of finance can be traced to Mandelbrot (1971), and academic work on the range-based volatility estimator started from the early 1980s. Many authors, such as Parkinson (1980), Garman and Klass (1980), developed from range several volatility estimators which are found to be far more efficient than the return-based volatility estimators. Also, Alizadeh et al. (2002) found that the conditional distribution of the log range is approximately Gaussian, which facilitates the maximum likelihood estimation of stochastic volatility models. Moreover, as pointed by Alizadeh et al. (2002), and Brandt and Diebold (2006), the range-based volatility estimator is robust to microstructure noise such as bid–ask bounce. Through Monte Carlo simula­ tion, Shu and Zhang (2006) found that range estimators are fairly robust toward microstructure effects, which is consistent with the finding of Alizadeh et al. (2002). Using a proper dynamic structure for the conditional expectation of range, Chou (2005) proposed the conditional autoregressive range (CARR) model to describe the dynamics of range. Empirical studies performed on the S&P500 index, both in-sample and out-of-sample, show that the CARR model does provide a more accurate volatility estimator compared with the GARCH model. Brandt and Jones (2006) formulated a model that is analogous to Nelson’s (1991) EGARCH model, but uses the square root of the intra-day price range in place of the absolute return and find that the range-based volatility estimators

Technical range forecasting 63 offer a significant improvement over their return-based counterparts. Using the dynamic conditional correlation (DCC) model proposed by Engle (2002), Chou et al. (2007) extended CARR to a multivariate context and find that this rangebased DCC model performs better than other return-based volatility models in forecasting covariances. Other models concerning range-based volatility include the ACARR (asymmetric CARR) of Chou (2006), the FACARR (feed­ back ACARR) of Xie (2019), the extended CARR of Xie and Wu (2019). However, all the available academic literature concerning range is on the Par­ kinson range. No academic research, to the best of our knowledge, is available investigating the technical range. This chapter, based on the theoretical proper­ ties of technical range, proposes a vector error correction model (VECM) to describe and forecast the dynamics of technical range. The chapter is organized as follows: Section 2 presents the econometric meth­ odologies with some discussions. We present the empirical results in Section 3. The summary is presented in Section 4.

8.2 Econometric methods 8.2.1 The model The first benchmark model is a simple Moving Average (MA) model which is widely used in technical analysis. A MA model of order q is given by 1X x ; q i¼1 tþ1–i q

xtþ1 ¼

ð8:1Þ

where xt is ln(TRt). Another benchmark model is the ARMA model. The ARMA model is used for the reason that it is the most commonly used technique in univariate time series modeling and forecasting. An ARMA model of order (p, q) is presented as follows: xt ¼ c þ

p X i¼1

αi xt–i þ εt þ

q X

bj εt –j

ð8:2Þ

j¼1

where xt is Δln(TRt). We don’t model ln(TRt) because it is a unit root process. Both MA and ARMA models are univariate time series models, they don’t take into consideration the co-integration between the closing price and the technical range. To see if this co-integration can be used to improve technical range fore­ casting, we propose to use a vector error correction model (VECM). The VECM model is presented as follows DYt ¼

k X

Gj DYt–j þ αbT Yt –1 þ m þ εt ;

ð8:3Þ

j¼1

where k is a lag structure, Yt is a p × 1 vector of variables and is integrated of order one, I(1), μ is a p × 1 vector of constants, and εt is a p × 1 vector of

64

Applications

white noise error terms. Γj is a p × p matrix that represents short-term adjust­ Pk ments among variables across p equations at the jth lag. j¼1 Gj DYt–j and αβTYt−1 are the vector autoregressive (VAR) components in first differences and error-correction components respectively. β is a p × r matrix of co-integrating vectors, and α is a p × r matrix of speed of adjustment parameters. The co­ integrating vector β shows the long-term equilibrium relationship between the concerned variables while the adjustment factor α shows the speed of adjustment towards equilibrium in case there is any deviation. A larger α suggests a faster con­ vergence toward long-run equilibrium in cases of short-run deviations from the long-run equilibrium. In this chapter, Yt = (ln(Ct), ln(TRt))T, where Ct and TRt are respectively closing price and technical range.

8.2.2 Out-of-sample forecast evaluation To evaluate the out-of-sample forecasting accuracy, both mean absolute error (MAE) and root mean squared error (RMSE) are used MAE

¼

RMSE

¼

n 1X jX – XFt ðMi Þj; n t¼1 t sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1X 2 ðX – XFt ðMi ÞÞ ; n t¼1 t

(8.4)

(8.5)

where Xt and XFt(Mi) are, respectively, the observations and forecasts reported by model i. Models that report smaller MAE (RMSE) are said to have better forecasts. To see if there is significant difference between two competing models for out­ of-sample forecasting, the DM statistic (Diebold and Mariano, 1995) is used. Let the forecasting error of model i be εi;t ¼ Xt – XFt ðMi Þ:

ð8:6Þ

We test the superiority of model j over model i with a t-test of μi,j coefficient in 2 ε2i;t – εj;t ¼ mi;j þ Zt ;

ð8:7Þ

where a positive estimate of μi, j indicates support for model j. To further gain insight into the difference between two competing models, we follow the approach of Mincer and Zarnowitz (1969) in running the following regression: Xt ¼ a þ bXFt ðMi Þ þ nt ;

ð8:8Þ

A test of the unbiasedness of the predicted volatility can be performed by a joint test of a = 0 and b = 1.

Technical range forecasting 65 To determine the relative information content of two competing volatility models, we also run a forecast encompassing regression: Xt ¼ a þ bXFt ðMi Þ þ cXFt ðMj Þ þ nt :

ð8:9Þ

If model i dominates model j, then it is expected that b is statistically significant while c is not.

8.3 An empirical study 8.3.1 The data We collected the monthly data of Standard and Poors 500 (S&P500) stock index for the sample period from 1950.01 to 2014.12 with 780 observations. For each month, four pieces of price information, opening, high, low and closing prices were collected. The data set was downloaded from website http://finance.yahoo.com. Table 8.1 presents the summary statistics of log closing price, stock return, log technical range, and differenced log technical range. The Jarque-Bera statistics reject the null hypothesis of normal distribution for both closing price and techni­ cal range. However, the null hypothesis of normal distribution for differenced log technical range can not be rejected. The ADF statistics show that the unit root hypothesis for closing price and technical range can not be rejected. The unit root hypothesis is rejected for both return and differenced log technical range.

8.3.2 In-sample estimation For the ARMA model, the SIC criteria prefers the ARMA (0, 1) model. The esti­ mation result is presented as follows DlnðTRt Þ ¼ 6:537** – 0:763*** εt –1 þ εt ;

R2 ¼ 0:373:

ð8:10Þ

The R-square shows that the ARMA(0, 1) model can explain respectively 37.3% variation of the total variance of differenced log technical range.

Table 8.1 Summary statistics of closing price and technical range

Mean St.D. Max. Min. Kurt. SKew Jarque-Bera ADF

ln(Ct)

ln(TRt)

rt

Δln(TRt)

5.345 1.328 7.634 2.836 1.756 0.140 52.859*** −0.788

2.473 1.494 5.791 −1.109 2.144 0.001 23.790*** −1.473

0.006 0.042 0.151 −0.245 5.468 −0.670 256.134*** 26.501***

0.007 0.469 1.561 −1.515 3.113 0.035 0.569 −19.178***

Note: Ct is the closing price, TRt is the technical range, and rt is the stock return. We use ***, **, and * to mean significance at the level of 1%, 5%, 10%.

66

Applications

Table 8.2 Vector error correction estimates on S&P500 stock index Vector Error Correction Estimates t-statistics in [ ] Co-integrating Eq:

CointEq1

ln(Ct−1):

1 −0.933 [−28.564]

ln(TRt−1):

−3.033

C: Error Correction: CointEq1

Δln(Ct) −0.007 [−1.497]

Δln(TRt) 0.267 [ 6.392]

Δln(Ct−1) Δln(Ct−2) Δln(Ct−3) Δln(Ct−4) Δln(Ct−5) Δln(TRt−1) Δln(TRt−2) Δln(TRt−3) Δln(TRt−4) Δln(TRt−5) C

0.045 [ 1.214] −0.050 [−1.317] 0.067 [ 1.769] 0.055 [ 1.448] 0.095 [ 2.517] −0.016 [−2.950] −0.009 [−1.588] −0.007 [−1.140] −0.003 [−0.669] −0.003 [−0.619] 0.005 [ 3.204]

−2.4187 [−7.776] −0.127 [−0.393] 0.370 [ 1.151] 0.385 [ 1.195] 0.054 [ 0.169] −0.607 [−13.093] −0.388 [−7.830] −0.217 [−4.485] −0.119 [−2.714] −0.077 [−2.238] 0.025 [ 1.877]

0.026

0.447

R-square

The VECM model estimates are reported in Table 8.2. We select the lag k = 5 by the SIC criteria. The co-integration vector is given by β = (1.000, −0.933, −3.033)T. The co-integration relationship between closing price and technical range is presented as: lnðCt Þ ¼ 3:033 þ 0:933*** lnðTRt Þ:

ð8:11Þ

The speed of adjustment factor is given by α = (−0.007, 0.267). The fact that the adjustment factor α = −0.007 is small and insignificant in the closing price sug­ gests that closing price is exogenous to the changes in technical range. The large and significant adjustment factor α = 0.267 in the technical range means that technical range responds quickly to the changes in closing price. For technical range the R-square statistic is reported to be 0.447, which is much larger than 0.373 of the ARMA model. This result indicates that VECM reports better insample technical range forecasts than ARMA does.

8.3.3 Out-of-sample forecast For practical purposes, the more important thing is the out-of-sample forecasting performance. For out-of-sample predicting, the whole T data observations are divided into an in-sample portion composed of the first m observations and an

Technical range forecasting 67 Table 8.3 Out-of-sample MAE and RMSE for VECM, ARMA and MA

MAE RMSE

VECM

ARMA

MA(12)

0.282 0.351

0.304 0.381

0.323 0.401

Table 8.4 Out-of-sample forecasting evaluation: DM test

ARMA MA(12)

ARMA

VECM

– 2.252**

2.903*** 3.207***

We use ***, **, and * to mean significance at the level of 1%, 5%, 10%.

Table 8.5 Out-of-sample forecasting evaluation: regression and encompassing regression Cons.

MA(12)

1.112*** 0.745* −0.634 0.858** −0.610 −0.738* −0.983**

0.741***

ARMA

VECM

0.819*** −0.371 −0.086 0.336

1.126*** 1.162*** −0.289 −0.706*

1.205*** 1.436*** 1.574***

R2 0.258 0.327 0.433 0.334 0.444 0.450 0.455

We use ***, **, and * to mean significance at the level of 1%, 5%, 10%.

out-of-sample portion composed of the last q observations. A static forecasting procedure is used. To be specific, we use the first m observations to obtain the estimates of the parameters. Then these estimates are kept fixed for out-of­ sample forecasting period. For the simple MA model, we set q=12. We used the data over 2001.01–2014.12 as the out-of-sample evaluation period. Table 8.3 reports the out-of-sample forecasting results. It is clear both MAE and RMSE show that VECM dominates ARMA, and ARMA dominates the simple MA model. Table 8.4 reports the results of the DM test. The DM statis­ tics show that VECM significantly outperforms the ARMA, and ARMA signifi­ cantly dominates the MA model. Table 8.5 reports the regression [Eq. (8.8)] and encompassing regression [Eq. (8.9)] results. The regression results show that VECM reports the least biased forecasts and has the largest R-square. The encompassing regression results show that VECM has the most informative forecasts. Once the VECM forecasts are included, the coefficient on the forecasts reported by ARMA (MA) becomes negative or insignificant. Figure 8.1 presents the time series plots of technical range forecasts reported by different models. It is clear from Figure 8.1 that

68

Applications 6.0

5.5

ln(TR )

V EC M

ARMA

MA(12)

5.0

4.5

4.0

3.5

3.0 2002

2004

2006

2008

2010

2012

2014

Figure 8.1 Out-of-sample technical range forecasting

the forecasts given by the VECM model are more flexible and adaptive to evo­ lution of real technical range.

8.4 Summary Using the co-integration relation between the closing price and the technical range, a VECM model is proposed for technical range forecasting. An empirical study is performed on the monthly S&P500 stock index. The results show that the VECM dominates both ARMA and MA for both in-sample and out-of­ sample forecasting. The results obtained in this chapter are interesting and important as they indi­ cate that the statistical properties of the technical range can be used for improv­ ing technical range forecasting.

9

Technical range spillover

The question of how changes on asset price and volatility are propagated across markets is of great interest in academic research. This question is highly related to the efficiency of financial markets. The efficient market hypothesis (EMH) inter­ prets the interdependence of stock returns and volatility as an informational link across markets: news revealed in one country is decoded as informative to funda­ mentals of stock price in another country. This view can be attributed to real and financial linkage of economies.1 While the behavioral finance (BF) claims the spillover as market contagion: stock prices in one country are affected by changes in another country beyond what is conceivable by connections through economic fundamentals.2 This chapter will demonstrate how the statis­ tical properties of the technical range can be used to investigate the spillover effect across markets.

9.1 Introduction Financial markets have witnessed in both developed and developing countries liberalized capital movements, financial reforms, advances in computer technol­ ogy and information processing, which greatly reduce the isolation of domestic markets and increase their ability to react promptly to news and shocks originat­ ing from the rest of the world. This indicates that the linkages across stock markets around the world may have grown stronger. Modern portfolio theory says that gains from international portfolio diversifi­ cation are inversely related to the correlation between equity returns. Investors hold various securities in the expectation of achieving a reduction in risk via diversification. In the mean variance framework, correlation is the measure of co-movement in returns. Errunza (1977) demonstrated the advantage of inter­ national diversification based on low correlation between the equity markets. Therefore, understanding the interdependence in returns and volatility across different markets not only adds to insights related to diversification and hedging strategies but also contributes to devising policies related to capital inflow in the market. There is voluminous academic literature concerning information transmission across markets. Some of the literature focuses on the long-term interdependence

70

Applications

and causality among stock markets, in order to find long-term or short-term cor­ relation among these markets (Eun and Shim, 1989; Nath and Verma, 2003; and Constantinou et al., 2005). Since information transmission across markets can take place through not only price changes but also price volatility, there is also growing literature concerning volatility spillover (Bekaert and Harvey, 1997; Ng, 2000; Baele, 2003; Christiansen, 2003; and Worthington and Higgs, 2004). This chapter is neither to investigate the mean spillover nor the volatility spil­ lover, instead it is designed with an empirical study to show how the statistical properties of technical range can be used to scrutinize the information transmis­ sion across financial markets. An empirical study is performed on the German stock index (DAX) and the French stock index (CAC40). The results show persuasive evidence that there is information spillover from the German stock market to the French market, while not vice versa. This chapter is organized as follows: Section 2 presents the econometric method with some discussions. Section 3 presents the empirical results. In Section 4, we summarize.

9.2 Econometric method We have demonstrated in Chapter 4 that technical ranges are co-integrated if their corresponding closing prices are co-integrated. With this property in mind, we will present in the following that technical range spillover can be scru­ tinized through a vector error correction (VECM) model. Suppose two speculative assets, closing prices c1t and c2t (in log form) are co­ integrated of order (1, 1) with a co-integration vector of β = (β1, β2)T. By the definition of co-integration, there exists a linear combination ot ¼ b1 c1;t þ b2 c2;t

ð9:1Þ

such that ωt is I(0). Given the facts that ci,t ≈ ln(TRi,t) − ln(PRi,t) and that ln(PRit) (i = 1, 2) is I(0), we can get b1 lnðTR1;t Þ þ b2 lnðTR2;t Þ ≈ ot þ b1 lnðPR1;t Þ þ b2 lnðPR2;t Þ;

ð9:2Þ

where TRi,t and PRi,t are the technical range (see Eq.(4.1)) and Parkinson range (see Eq.(4.2)). Since ωt and β1ln(PR1,t) + β2ln(PR2,t) are stationary processes, we can obtain that β1ln(TR1,t) + β2ln(TR2,t) is stationary. Thus we get the result that log technical ranges, ln(TRi,t) (i = 1, 2) are co-integrated so long as their closing prices are co-integrated. It is claimed that price changes are due to new information being reflected. Technical range as a measurement of price variation can thus be treated as an information proxy. In this chapter, the following vector error correction model

Technical range spillover

71

(VECM) is proposed to investigate the information spillover effect across finan­ cial markets DYt ¼

k X

Gj DYt–j þ αbT Yt–1 þ m þ εt ;

ð9:3Þ

j¼1

where k is a lag structure, Yt is a p × 1 vector of variables and is integrated of order one, I(1), μ is a p × 1 vector of constants, and εt is a p × 1 vector of white noise error terms. Γj is a p × p matrix that represents short-term adjustments among var­ Pk iables across p equations at the jth lag. j¼1 Gj DYt –j and αβTYt−1 are the vector autoregressive (VAR) component in first differences and error-correction compo­ nents respectively. β is a p × r matrix of co-integrating vectors, and α is a p × r matrix of speed of adjustment parameters. The co-integrating vector β shows the long-term equilibrium relationship between the concerned variables while the adjustment factor α shows the speed of adjustment towards equilibrium in case there is any deviation. A larger α suggests a faster convergence toward long-run equilibrium in cases of short-run deviations from the long-run equilib­ rium. In this chapter, Yt = (ln(TR1,t), ln(TR2,t))T.

9.3 An empirical study: DAX and CAC40 9.3.1 The data We collected the monthly data of the German stock index DAX and the French stock index CAC40 for the sample period 1994.01–2014.12 with 252 observa­ tions. For each stock index, four pieces of price information, opening, high, low and closing prices were reported. The data sets were downloaded from http:// finance.yahoo.com. We do not use the data after 2007 for the reason that there seems to be a structure change in time series data. Figure 9.1 presents the time series plot of log closing prices of DAX and CAC40 over 1994.01–2014.12. From the plot, it can be observed that the deviation between DAX and CAC40 sharply increases after 2007. Table 9.1 presents the summary statistics of the log technical range and the differenced log technical range. The Ljung-Box Q statistics show high persis­ tence of log technical range. The Ljung-Box Q statistics of differenced log tech­ nical range, though not so much larger as the technical range, are still significant at a level of 1%. The ADF testing results show: (1) at the significance level of 5% the null hypothesis of unit root process can not be rejected for the log technical range of DAX. The null hypothesis of unit root process can be rejected for the the differenced log technical range of DAX; (2) at the significance level of 5%, the null hypothesis of unit root process is rejected for the technical range of CAC40. However, the ADF testing results performed on the log closing prices of both DAX and CAC40 show that the null hypothesis of unit root process

72

Applications 9.6 C AC 40

9.2

DAX

8.8

8.4

8.0

7.6

7.2 94

96

98

00

02

04

06

08

10

12

14

Figure 9.1 Time series plot of log closing prices, DAX and CAC40: 1994.01– 2014.12 Table 9.1 Summary statistics of technical range

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability Ljung-Box Q(12) ADF test

ln(TRdax)

Δln(TRdax)

ln(TRcac)

Δln(TRcac)

5.830 5.920 7.427 3.995 0.712 −0.440 2.614 6.463 0.039 1013.8*** −2.621*

0.003 0.004 1.250 −1.075 0.438 0.330 2.993 3.047 0.218 63.034*** −13.436***

5.620 5.590 7.130 4.236 0.589 0.022 2.447 2.157 0.340 738.31*** −3.235**

0.004 −0.027 1.101 −1.036 0.419 0.412 3.148 4.903 0.086 39.558*** −13.230***

We use ***, **, and * to mean significance at the level of 1%, 5%, 10%.

can not be rejected. By Proposition 1 in Chapter 4 (p.20), technical range is a unit root conditional on the closing price being a unit root. Therefore in this chapter, we still take the technical range of CAC40 as a unit root process. The Johansen Trace test performed on the technical ranges of DAX and CAC40 shows that there is a co-integration equation at the level of 5%. Figure 9.2 pre­ sents the time series plot of technical ranges of DAX and CAC40 over 1994.01– 2007.12. The co-integration relation is clear in Figure 9.2.

Technical range spillover

73

7.6 T R_CAC

T R_DAX

7.2 6.8 6.4 6.0 5.6 5.2 4.8 4.4 4.0 3.6 94

96

98

00

02

04

06

08

10

12

14

Figure 9.2 Time series plot of log technical range, DAX and CAC40: 1994.01– 2014.12

9.3.2 Estimation The VECM model estimates are reported in Table 9.2. The lags of the VECM model are determined by SIC criteria. In this VECM model, the lagged returns on the closing prices of DAX and CAC40 are used as exogenous vari­ ables. The reason is that closing price is exogenous to technical range and is informative for predicting technical range (see the empirical results in Chapter 8). The co-integration vector in the technical range for DAX and CAC40, is given by β = (1.000, -0.933, 1.605)T. The co-integration relationship is presented as follows: lnðTRdax;t Þ ¼ 1:324lnðTRcac;t Þ – 1:605;

ð9:4Þ

Since all the variables are in logarithmic function form, the coefficients in βT can be interpreted as long-term elasticities: technical range on DAX increases 1.324 percent with respect to 1 percent increase of technical range on CAC40. The matrix of speed of adjustment parameters α for DAX and CAC40, is given by α = (−0.042, 0.520). The fact that α (αdax = −0.042) in the technical range on DAX is small and insignificant suggests that technical range on DAX is exoge­ nous to the technical range on CAC40. The large and significant α (αcac = 0.520) in the technical range on CAC40 means that technical range on CAC40 responds quickly to the changes in the technical range on DAX.

74

Applications

Table 9.2 Vector error correction estimates on DAX and CAC40 Vector Error Correction Estimates t-statistics in [ ] Co-integrating Eq:

CointEq1

ln(TRdax,t−1):

1

ln{TRcac,t−1):

−1.324 [−21.119]

C:

1.605

Error Correction: CointEq1

Δln(TRdax,t) −0.042 [−0.296]

Δln(TRcac,t) 0.521 [3.947]

Δln(TRdax,t−1) Δln(TRdax,t−2) Δln(TRcac,t−1) Δln(TRcac,t−2) C

0.045 [ 1.214] −0.050 [−1.317] 0.067 [ 1.769] 0.055 [ 1.448] 0.013 [0.438]

−2.4187 [−7.776] −0.127 [−0.393] 0.370 [ 1.151] 0.385 [ 1.195] 0.017 [0.598]

rdax,t−1 rcac,t−1

−0.335[−0.359] −0.128[−0.119]

0.070[ 0.080] −1.364[−1.350]

In econometrics and other applications of multivariate time series analysis, a var­ iance decomposition or forecast error variance decomposition is used to aid in the interpretation of a vector autoregressive (VAR) model once it has been fitted. The variance decomposition indicates the amount of information each variable con­ tributes to the other variables in the autoregressive. It determines how much of the forecast error variance of each of the variables can be explained by exogenous shocks to the other variables. Figure 9.3 presents the plots of variance decomposition. The results show that the forecast error variance of technical range on DAX due to exogenous shocks to CAC40 is almost zero, most of its forecast error var­ iance is accounted by exogenous shocks to DAX itself. The forecast error variance of technical range on CAC40 due to exogenous shocks to CAC40 itself decreases very fast with the lags: after about 20 lags the exogenous shocks to CAC40 itself contributes no more than 20% of the forecast error variance. While the forecast error variance of technical range on CAC40 due to exogenous shocks to DAX increases sharply with the lags: after about 20 lags contributes more than 80%. All these findings indicate that, in terms of technical range, the DAX index is exogenous to the CAC40 index, and there is significant technical range spillover from DAX to CAC40, not vice versa.

9.4 Summary This chapter is designed to show how the statistical properties of the technical range can be used to investigate the information spillover effect. We scrutinize the range spillover effect between the German stock market and the French stock market with the vector error correction model.

Technical range spillover

75

Variance Decomposition 100

100

80

80

60

Percent DAX variance due to DAX

60

40

40

20

20

0

Percent DAX variance due to CAC

0 5

10

15

20

25

30

5

100

100

80

80

60

60

Percent CAC variance due to DAX 40

40

20

20

0

10

15

20

25

30

Percent CAC variance due to CAC

0 5

10

15

20

25

30

5

10

15

20

25

30

Figure 9.3 Plots of variance decomposition of technical range, DAX and CAC40

An empirical study performed on the DAX (Germany) and CAC40 (France) stock index shows that there is range spillover effect from the German stock market to the French market while not vice versa. The German stock market is exogenous to the French stock market. Of course, this chapter only serves as an illustration that the statistical proper­ ties of the technical range can be used to investigate the information spillover effect.

Notes 1 An international asset pricing model can incorporate correlations between stock returns in different countries, see, for example, Adler and Dumas (1983) and Solnik (1974). 2 According to behavioral finance (BF), overreaction, speculation, and/or noise trading are transmissible across borders. For example, DeLong et al. (1990).

10 Stock return forecasting U.S. S&P500

For hundreds of years, investors have been fascinated by the variability of spec­ ulative prices. Investment practitioners have developed many a tool to forecast the future path of the prices in the hope that they can make great fortunes with good forecasts. One of the most commonly used forecasting tools is tech­ nical analysis. Despite its popularity among technicians, the value of technical analysis is still controversial due to its subjective nature. In contrast to funda­ mental analysis, which was quick to be adopted by the scholars of modern quan­ titative finance, academic scrutiny of technical analysis is still in its infancy. It has been argued that the difference between fundamental analysis and technical anal­ ysis is not unlike the difference between astronomy and astrology. Among some circles, technical analysis is even known as “voodoo finance”. However, several academic studies suggest that technical analysis may well be an effective means for extracting useful information from market prices. For example, Lo and MacKinlay (1988, 1999) have shown that past prices may be used to forecast future returns to some degree, a fact that all technical analysts take for granted. Studies by Tabell and Tabell (1964), Treynor and Ferguson (1985), Brown and Jennings (1989), Jegadeesh and Titman (1993), Blume et al. (1994), Chan et al. (1996), Lo and MacKinlay (1997), Grundy and Martin (1998), and Rouwenhorst (1998) have also provided indirect support for technical analysis, and more direct support has been given by Pruitt and White (1988), Neftci (1991), Brock et al. (1992), Neely et al. (1997), Neely and Weller (1998), Chang and Osler (1994), Osler and Chang (1995), and Allen and Karjalainen (1999). Lo et al. (2000) found that over the 31-year sample period, several technical indicators do provide incremental information and may have some practical value. Recent academic results show that high– low price extremes are valuable for forecasting speculative prices (Xie et al., 2012; Xie et al., 2012; Xie and Wang, 2013; Xie et al., 2013; Xie et al., 2014; Xie et al., 2015; Xie and Wang, 2018). Despite the growing evidence supporting the practical value of technical anal­ ysis, it is still frequently criticized due to its highly subjective nature and its lack of theoretical underpinnings. In this chapter, we are going to scrutinize the forecasting power of the candle­ stick using the DVAR model proposed in Chapter 5. The main purpose of this

Stock return forecasting

77

chapter is to see if the statistical properties of the candlestick can be used to improve stock return forecasting.

10.1 Introduction The predictability of stock market returns is of great interest to both academic researchers and investment practitioners, and numerous economic and financial variables have been identified as predictors of stock returns in academic literature. Examples include valuation ratios, such as the dividend-price (Dow, 1920; Fama and French, 1988, 1989), earnings-price (Campbell and Shiller, 1988, 1989), and book-to-market (Kothari and Shanken, 1997; Pontiff and Schall, 1998), as well as nominal interest rates (Fama and Schwert, 1977; Campbell, 1987; Breen et al., 1989; Ang and Bekaert, 2007), the inflation rate (Nelson, 1976; Fama and Schwert, 1977; Campbell and Vuolteenaho, 2004), term and default spreads (Campbell, 1987; Fama and French, 1989), corporate issuing activity (Baker and Wurgler, 2000; Boudoukh et al., 2007), consumption– wealth ratio (Lettau and Ludvigson, 2001), stock market volatility (Guo, 2006).1 Almost all of these existing studies focus on the in-sample tests and con­ clude significant evidence of in-sample return predictability. Despite the consistent agreement on the in-sample predictability of stock returns, evidence of out-of-sample predictability remains controversial. Bossaerts and Hillion (1999), Ang and Bekaert (2007), and Goyal and Welch (2003) casted doubt on the in-sample evidence documented by the early authors by showing that these variables have negligible out-of-sample predictive power. Among these studies, Goyal and Welch (2008) take a comprehensive look at the empirical performance of stock returns, and show that a long list of predictors from the literature is unable to deliver consistently superior out-of-sample fore­ casts of the U.S. stock returns relative to a simple historical mean forecast. In contrast, recent empirical studies confirm new predictor variables and economet­ ric methods that can improve the out-of-sample predictability of stock returns. New predictor variables include technical indicators (Neely et al., 2014; Han et al., 2013; Goh et al., 2013; Huang and Zhou, 2013), sentiment index (Baker and Wurgler, 2006; Stambaugh et al., 2012; Huang et al., 2013). New econometric methods include support vector machine method (Huang et al., 2005), economically motivated model restrictions (Campbell and Thompson, 2008; Ferreira and Santa-Clara, 2011), combination forecast (Rapach et al., 2010), diffusion index (Ludvigson and Ng, 2007; Kelly and Pruitt, 2012; Neely et al., 2014), regime shifts (Guidolin and Timmermann, 2007; Henkel et al., 2011; Dangl and Halling, 2012), sequential learning (Johannes et al., 2014). Rapach and Zhou (2013) made a more extensive survey of the vast liter­ ature on predicators and methodologies concerning stock return predictability. Different from the existing the methods, we use the DVAR model proposed in Chapter 5 to scrutinize the the out-of-sample predictability of the U.S. stock market and the economic value of candlestick forecasting. Out-of-sample pre­ dictability of the U.S. stock market is performed on the monthly stock returns

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Applications

of the U.S. S&P500 index over 1995.01–2015.12. To mitigate the concern of data-mining, we use the out-of-sample R-square of Campbell and Thompson (2008) as the statistical performance measure and the certainty equivalent return (CER) gain as the economic performance measure. We find the DVAR model reports significant out-of-sample predictability for the U.S. stock market. The rest of the chapter is organized as follows. Section 2 describes the econo­ metric methodologies. Section 3 presents the empirical results on the predictabil­ ity of the U.S. stock market. Section 4 presents some details of U.S. stock market predictability. A summary is presented in Section 5.

10.2 Econometric methods 10.2.1 The model For univariate time series modeling, one of the most commonly used benchmark models is the ARMA-GARCH-in-Mean model (see Eq. (5.3), p.28). This model can simultaneously capture the linear autocorrelation in return series and the risk– return tradeoff. In this chapter, only the following ARMA-GARCH-in-Mean of order (1, 1) is used as it has been well documented that the order (1, 1) is suf­ ficient to capture the return dynamics. rt ¼ c þ α1 rt–1 þ b1 εt–1 þ ght þ εt ; pffiffiffiffi εt ¼ ht zt ; i:id: zt ~ N ð0; 1Þ

ð10:1Þ

2 ; ht ¼ o þ α1 ht–1 þ b1 εt–1

where γ is the coefficient of relative risk aversion, reflecting the risk–return tradeoff. Another model used in this chapter is the DVAR model proposed in Chapter 6. A DVAR model of order p is given by Yt ¼ C þ

p X Ai Yt–i þ CXt–1 þ εt ;

ð10:2Þ

i¼1

where Yt = (ΔRt, ΔWt)T, Xt−1 is a vector of exogenous variables. The exogenous variables used in this chapter include the upper shadow, the lower shadow, and other variables. Upper shadow and lower shadow are used because we have shown in Chapter 6 that these two variables are informative for predicting both ΔRt and ΔWt. For more about DVAR, readers can refer to Chapter 5. Note that the DVAR model does not directly predict the stock return. To obtain the return forecasts, we proceed in an indirect way. To be specific, the return forecasts are constructed through the following equation rtf ¼ DRtf þ DWtf ;

ð10:3Þ

Stock return forecasting

79

where rtf is the return forecast, DRft and DWtf are respectively the forecasts of DRft and DWtf reported by the DVAR model.

10.2.2 Out-of-sample evaluation A potential problem with in-sample predictability is over-fitting. In a comprehen­ sive study, Goyal and Welch (2008) found that many macroeconomic variables, though they deliver significant in-sample forecasts, perform poorly out of sample. 2 Following Goyal and Welch (2008), we also use the out-of-sample R2, Roos sta­ tistic which is defined as PT 2 f t¼M þ1 ðrt – rt ðmÞÞ 2 Roos ¼ 1 – P ; ð10:4Þ T 2 rt Þ t¼M þ1 ðrt – ̄ where rtf ðmÞ is the return forecast given by model m, ̄r t is the historical mean forecast given by Pt–1 r ð10:5Þ ̄r t ¼ i¼1 i : t –1 The historical mean forecast is equivalent to an efficient market model, and thus 2 indicates the better performance of serves as a natural benchmark. A positive Roos 2 model m forecast over the simple historical mean forecast, while a negative Roos indicates the opposite. 2 The Roos statistic measures the reduction in mean squared forecast error (MSFE) for model m relative to the historical mean forecast. To see if the reduc­ 2 tion is significant, we test the null hypothesis that Roos ≤ 0 against the alternative 2 hypothesis that Roos > 0. Following Rapach et al. (2010), we also test this hypothesis by using the Clark and West (2007) MSFE-adjusted statistic. Define 2

2

2

ft ¼ ðrt – ̄r t Þ – ½ðrt – rtf ðmÞÞ – ðrtf ðmÞ – ̄r t Þ ]; t ¼ M þ 1; :::; T

ð10:6Þ

the Clark and West (2007) MSFE-adjusted statistic is the t-statistic from the regression of ft on a constant.

10.3 Statistical evidence This section describes the data and presents the in-sample and out-of-sample forecasting results.

10.3.1 The data We scrutinized the monthly return predictability of the U.S. stock market using the S&P500 index. The data spans from January 1950 through December 2015 with 792 observations and was downloaded from the website, www.finance.yahoo.com.

80

Applications

For each month, the high, low, close and open prices were reported. The risk-free bill rates came from www.bus.emory.edu/AGoyal/Research.html. From these prices, we constructed the lower shadow (LSt) by Eq. (3.5)–(3.6), the upper shadow (USt) by Eq. (3.3)–(3.4), ΔRt and ΔWt by Eq. (5.6)–(5.7), and the asset returns. Candlestick chart forecasting is often used with market states. We define the market is in up-trend state if the market index is above its 200-day moving average and in down-trend state otherwise. The 200-day moving average has been widely used by practitioners and is available in investment letters, trading software, and newspapers, which can thus mitigate the concerns of data mining and data snooping. The 200-day moving average has also been used in Huang et al. (2013) to define market states. To be specific, the market state var­ iable, mst is constructed from the daily stock price, given by 8 200 1 X > > < 1; if Pt ≥ P 200 i¼1 tþ1–i ð10:7Þ mstþ1 ¼ > > : 0; otherwise where Pt is the daily price level of the market index.2 Table 10.1 reports the summary statistics of stock returns together with some other variables. We find no significant linear autocorrelation in the return series, which is consistent with our intuition that the U.S. stock market is quite efficient. For the other time series variables, the Ljung-Box Q statistics report significant linear autocorrelations. The ADF statistics show that all these time series are stationary.

10.3.2 In-sample estimation Table 10.2 reports the estimates of the ARMA-GARCH-in-Mean model. Panel A reports the estimates of the mean equation in the ARMA-GARCH-in-Mean model and Panel B presents the estimates of the GARCH equation. The results indicate Table 10.1 Summary statistics of stock return and other variables rt

ΔRt

ΔWt

lst

ust

Mean 0.006 0.006 0.000 0.018 0.014 St.D. 0.042 0.035 0.32 0.021 0.012 Max. 0.151 0.125 0.154 0.190 0.072 Min. −0.245 −0.190 −0.110 0.000 0.000 Kurt. 5.435 6.324 4.320 16.857 4.731 SKew −0.655 −1.017 0.406 2.907 1.313 Jarque-Bera 251.91*** 500.579*** 79.117*** 7452.278*** 326.507*** Ljung-Box 16.412 78.184*** 227.59*** 102.82*** 330.29*** Q(12) ADF −26.802*** −18.591*** −15.199*** −15.733*** −7.948*** We use ***, **, and * to mean significance at the level of 1%, 5%, 10%.

Stock return forecasting

81

Table 10.2 Estimates of the ARMA-GARCH-in-Mean model Panel A: The Mean Equation ARMA-GARCH-in-Mean c α β mst−1 γ

Coef. −0.001 −0.970 0.997

t-Statistic −0.106 −137.651 533.805

0.200

0.999

ARMAX-GARCH-in-Mean t-Statistic −0.579 −120.160 367.531 1.813 1.504

Coef. −0.003 −0.970 0.997 0.005 3.456

Panel B: The GARCH Equation ω α β

9.61E-05 0.834 0.115

2.568 26.398 4.583

9.89E-05 0.834 0.113

2.550 25.494 4.535

R-square

R-square

0.015

0.024

Table 10.3 Estimates of the DVAR model: S&P500 ΔRt Coef. c α β lst−1 ust−1 mst−1 rt−1 * lust−1

ΔWt t-Statistic

−0.003 0.475 0.524 0.473 −0.484 0.006 2.528

−1.045 12.764 16.439 7.815 −6.173 2.467 4.416

Coef.

t-Statistic

0.004 −0.599 −0.505 −0.607 0.597 0.004 0.335

1.982 −19.183 −18.849 −11.949 9.070 1.848 0.696

R-square

R-square

0.465

0.541

there is a significant volatility clustering effect in stock return. Also we can see from the estimates of the mean equation that there is a positive but insignificant risk–return tradeoff. The low R-square statistics (R2 = 1.5%) demonstrate that the forecasts reported by the ARMA-GARCH-in-Mean model contain almost no information about the true return observations. We also present in Table 10.2 the estimates of the ARMA-GARCH-in-Mean model with the state variable, mst−1. The coefficient on state variable is reported to be 0.005 and sta­ tistically significant at the level of 10%. The R-square reported by ARMAXGARCH-in-Mean model sharply increases to 2.4%. This result means that the state variable is very informative for forecasting stock return.

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Applications

Table 10.3 presents the estimates of the DVAR model. The lag k=1 in the DVAR model is chosen by the SIC criteria. Consistent with the theoretical results given in Chapter 6, we find both upper shadow and lower shadow are informative for predicting ΔRt and ΔWt. Also we find the state variable is very important for forecasting ΔRt and ΔWt. The high R-square statistics indicate the high predictability of ΔRt and ΔWt. We also calculate the in-sample 2 R-square, Rin using P ðr – rtf Þ2 2 Rin ¼ 1 – Pt t ð10:8Þ 2 ; rÞ t ðrt – ̄ where ̄r is the mean of rt over the whole sample, and rtf is the forecast given by DVAR model (see Eq. (10.3)). The in-sample R-square reported by DVAR model is 2.98%.

10.3.3 Out-of-sample forecast For out-of-sample forecasting, the total T observations are divided into two por­ tions. The first portion from 1 to M observations is used to estimate the coeffi­ cients, and the remaining portion from M + 1 to T is used for forecasting evaluation. A static forecasting procedure is used. To be specific, we first use the M obser­ vations to obtain the estimates of the parameters, and then make out-of-sample­ forecasts with these estimates being fixed. In other words, the estimates of the parameters in the DVAR model are not updated with new information Ytþ1

^ Y þC ^ X ; t ¼ M ; M þ 1; :::; T ^ þA ¼ C 1 t t

ð10:9Þ

^ and C, ^, A ^ are estimates of the parameters using the only the first M where C 1 observations. We use the data over 1995.01–2015.12 as the out-of-sample time period. We compute the cumulative squared forecast error of each competing model Sq ¼

q X

2

ðrt – rtf ðmÞÞ ; q ¼ M þ 1; M þ 2; :::; T

ð10:10Þ

t¼M þ1

where Sq is the cumulative squared forecast error, rt and rtf ðmÞ are respectively the return observation and the return forecast given by model m. Figure 10.1 presents the time series plots of the cumulative squared forecast error. We use CUM_M to mean the cumulative squared forecast error reported by model M. It is clear that the DVAR model reports the lowest cumulative squared forecast error and then comes the ARMA-GARCH-in-Mean model, and the historical mean (HM) model has the largest cumulative squared forecast error. Following Goyal and Welch (2008), we also compute the difference between the cumulative squared forecast error for the historical mean model and the

Stock return forecasting .5

C UM_ARMA

C UM_DVAR

83

C UM_HM

.4

.3

.2

.1

.0 96

98

00

02

04

06

08

10

12

14

Figure 10.1 Time series of cumulative squared forecast error: 1995.01–2015.12

cumulative squared forecast error for the competing model. DSq ¼

q X

2

2

½ðrt – rtf ðHM ÞÞ – ðrt – rtf ðmÞ ]; q ¼ M þ 1; M þ 2; :::; T

ð10:11Þ

t¼M þ1

where DSq is a series of difference. Figure 10.2 presents the time series plots of the differences. We use HM_M to mean the difference between the cumulative squared forecast error for the histor­ ical mean model and the cumulative squared forecast error for the competing model M. This is an informative graphical device that provides a visual impression of the consistency of a competing model’s out-of-sample forecasting perfor­ mance relative to the historical mean model over time. When the curve increases, the competing model outperforms the historical mean model, while the opposite holds when the curve decreases. The plots conveniently illustrate whether a com­ peting model has a lower mean squared forecast error (MSFE) than the historical mean model for any particular out-of-sample period by redrawing the horizontal zero line to the start of the out-of-sample period. A competing model that always dominates the historical mean model for any out-of-sample period will have a curve with a slope that is always positive; the closer a competing mode is to this ideal, the greater its ability to consistently beat the historical mean model in terms of MSFE.

84

Applications .024 .020

HM_DV AR

HM_AR MA

.016 .012 .008 .004 .000 –.004 –.008 96

98

00

02

04

06

08

10

12

14

Figure 10.2 Cumulative squared forecast error for the historical mean benchmark forecasting model minus the cumulative squared forecast error for the competing model: 1995.01–2015.12

Several findings emerge from Figure 10.2. First, both DVAR and ARMAGARCH-in-Mean models outperform the historical mean benchmark model as the curves have ending points higher than starting points. Second, DVAR out­ performs ARMA-GARCH-in-Mean since the ending point of the DVAR curve is higher than the ending point of the ARMA-GARCH-in-Mean curve. Third, the DVAR model is more robust than the ARMA-GARCH-in-Mean model for out-of-sample forecasting as the DVAR curve is less volatile than the ARMA-GARCH-in-Mean curve. The out-of-sample R-squares reported by the DVAR model and the ARMAGARCH-in-Mean are respectively 4.82% and 1.70%, indicating outperformance of DVAR and ARMA-GARCH-in-Mean over the historical mean. To see if the outperformance is statistically significant, we calculate the MSFE-adjusted t-statistic by Eq. (10.6). The MSFE-adjusted t-statistic for the DVAR is 2.633, which is sig­ nificant at the level of 1%. The MSFE-adjusted t-statistic for the ARMA-GARCH­ in-Mean model is 1.612, which is not significant at the level of 10%.

10.4 Economic evidence 2 A limitation to the Roos measure is that it does not explicitly account for the risk borne by an investor over the out-of-sample period. To address this, following

Stock return forecasting

85

Campbell and Thompson (2008), we also calculate realized utility gains for a mean-variance investor on a real-time basis. More specifically, we first compute the average utility for a mean-variance investor with relative risk aversion param­ eter γ who allocates his or her portfolio monthly between stocks and risk-free bills using forecasts of the equity premium based on the historical sample mean. This exercise requires the investor to forecast the variance of stock returns, and similar to Campbell and Thompson (2008), we assume that the investor estimates the variance using a ten-year rolling window. A mean-variance investor who forecasts the equity premium using the historical average will decide at the end of period t to allocate the following share of his or her portfolio to equities in period t+1: ( )( ) 1 ̄r tþ1 – rftþ1 o0;t ¼ ; ð10:12Þ ^2tþ1 g s 2 ^ tþ1 where rft+ 1 and s are respectively risk-free rate and the rolling-window esti­ mate of the variance of stock returns.3 Over the out-of-sample period, the inves­ tor realizes an average utility level of

1 2 ^n 0 ¼ m ^ 0 – g^ s ; 2 0

ð10:13Þ

^ 0 and s ^ 20 are the sample mean and variance, respectively, over the out-of­ where m sample period for the return on the benchmark portfolio formed using forecasts of the equity premium based on the historical sample mean. We then compute the average utility for the same investor when he or she fore­ casts the equity premium using the DVAR or ARMA-GARCH-in-Mean model. He or she will choose an equity share of ) ( )( f 1 rt þ1 ðmÞ – rftþ1 ð10:14Þ op;t ¼ ; ^2tþ1 g s and realizes an average utility level of 1 2 ^np ¼ m ^ p – g^ s ; 2 p

ð10:15Þ

^ p and s ^ 2p are the sample mean and variance, respectively, over the out-of­ where m sample period for the return on the portfolio formed using forecasts of the equity premium based on DVAR or ARMA-GARCH-in-Mean model. We measure the utility gain as the difference between Eq. (10.15) and Eq. (10.13), and we multiply this difference by 1200 for monthly observations to express it in an average annualized percentage return. The utility gain (or cer­ tainty equivalent return, CER) can be interpreted as the portfolio management fee that an investor would be willing to pay to have access to the additional infor­ mation available in a predictive regression model relative to the information in the historical sample mean. We report results for γ = 3; the results are qualita­ tively similar for other reasonable γ = values.

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Applications

Table 10.4 presents the realized utilities of the historical mean model, DVAR model and ARMA-GARCH-in-Mean model and the CER gains relative to the historical mean (HM). The realized utility of DVAR model is 0.618 which is almost twice as much as that (0.313) of the ARMA-GARCH-in-Mean model. Of course, both earn a higher realized utility than the HM model has. The annual CER gain of the DVAR (ARMA-GARCH-in-Mean) model in 5.098 (1.443). We also calculate the sharp ratio of each trading strategy, and the results are presented in Table 10.4. We find clear dominance of the DVAR model over the other models. For comparison, we also calculate the realized utility of CER gain of the simple buy-and-hold (BH) trading strategy. We find a trading strategy based on the DVAR model even outperforms the BH method. Figure 10.3 presents the dynamic weights allocated on equities by different models. Figure 10.4 presents the out-of-sample cumulative returns given by Table 10.4 Realized utilities and CER gains

Sharp ratio Realized utility (%) CER gain

1.6

HM

ARMA-GARCH-in-Mean

DVAR

BH

0.051 0.193 –

0.095 0.313 1.443

0.172 0.618 5.098

0.088 0.303 1.322

W _ARMA

W _DVAR

W _HM

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 96

98

00

02

04

06

08

10

12

14

Figure 10.3 Dynamic weights allocated on equities over time: 1995.01–2015.12

Stock return forecasting

87

2.4 ARMA HM

2.0

DVAR BH

1.6

1.2

0.8

0.4

0.0 96

98

00

02

04

06

08

10

12

14

Figure 10.4 Dynamic cumulative portfolio returns formed by different trading strategies over time: 1995.01–2015.12

different trading strategies. Figure 10.4 shows clear dominance of the trading strategy based on the DVAR forecast over the other trading strategies.

10.5 More details Rapach et al. (2010) found the predictability of the U.S. stock market depends highly on business cycle: stock returns are more predictable in recession than in expansion. To see how the predicting power of the DVAR model is related to the business cycle. We also divide the out-of-sample forecasts by the NBER-dated business cycle phases.4 For economic expansion there are 226 months, and for economic recession there are 26 months. Figure 10.5 presents the cumulative squared forecast error of the DVAR model over expansion in left panel and over recession in right panel. It is clear that the DVAR model has lower forecast error than the historical mean model in both economic expansion and recession. Figure 10.6 presents the difference between the cumulative squared forecast error for the historical mean benchmark forecasting model and the cumulative squared forecast error for the DVAR model over expansion in the left panel and over recession in the right panel. It seems that the outperformance of the DVAR model over the historical mean

88

Applications .16

.4 C UM_DVAR

C UM_HM C UM_DVAR

.3

.12

.2

.08

.1

.04

C UM_HM

.00

.0 100

5

200

10

15

20

25

Figure 10.5 Time series of cumulative squared forecast error over business cycle .012

.016

.010

HM_DV AR

HM_DV AR

.012

.008 .006

.008

.004 .004

.002 .000

.000 –.002 –.004

–.004 50

100

150

200

5

10

15

20

25

Figure 10.6 Cumulative squared forecast error for the historical mean benchmark forecasting model minus the cumulative squared forecast error for the DVAR model over business cycle

model is quite robust, regardless of the economic cycle since the curve increases in a quite steady way. We also calculate the MSFE-adjusted t-statistics by Eq. (10.6) over business cycle. The MSFE-adjusted t-statistic over expansion is 2.293, which is significant

Stock return forecasting 2.4

.1 DVAR

HM

BH

2.0

DVAR

HM

89

BH

.0 –.1

1.6

–.2 –.3

1.2

–.4 0.8

–.5 –.6

0.4 –.7 0.0

–.8 50

100

150

200

5

10

15

20

25

Figure 10.7 Dynamic cumulative portfolio return formed by different trading strategies over business cycle

at the level of 5%. The MSFE-adjusted t-statistic over recession is 1.781, which is significant at the level of 10%. This result confirms that the forecasts given by the DVAR model significantly dominate those reported by the historical mean model over both economic expansion and recession. To see if there is any difference in economic value of the DVAR model relative to the historical mean model over the business cycle we compute the CER gains. The CER gain over expansion is 2.654 and 25.638 over recession. This result indicates that the forecasts reported by the DVAR model are more valuable in economic recession than in economic expansion. Figure 10.7 presents the out­ of-sample cumulative returns over expansion in the left panel and over recession in the right panel.

10.6 Summary This chapter scrutinizes the performance of the candlestick in return forecasting using the DVAR model. The empirical study is performed on the monthly S&P500 index. The results show that the DVAR model outperforms both the historical mean model and the ARMA-GARCH-in-Mean model. Moreover, we find the outperformance is not only statistically significant but also econom­ ically valuable. Further evidence indicates that the dominance of the DVAR model is robust to the business cycle. The results obtained in this chapter provide statistical evidence that the candlestick chart is valuable for predicting stock returns. We believe that the

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evidence presented in this chapter confirms, more or less, that candlestick chart forecasting is not a “voodoo” finance.

Notes 1 The list of studies is not meant to be exhaustive; see Goyal and Welch (2008) for more extensive surveys of the vast literature on return predictability. 2 The market state of next month is determined by the last trading day’s 200-day moving average indicator of current month. 3 Following Campbell and Thompson (2008), we constrain the portfolio weight on stocks to lie between 0% and 150% (inclusive) each month, so that ω0,t = 0 (ω0,t = 1.5) if ω0,t < 0 (ω0,t > 1.5) in Equation (10.12). 4 According to the NBER-dated business cycle phases, the period 2001.04–2001.11 and the period 2008.01–2009.06 are used as economic recession periods.

11 Oil price forecasting WTI crude oil

In this chapter, the forecasting power of the candlestick is further scrutinized on the crude oil price. The econometric method used in this chapter is also the DVAR model.

11.1 Introduction Crude oil, known as the blood of industries, plays an important role in world economies. As one of the main focal points in the world, oil price has become an increasingly essential topic of concern to governments, enterprises and inves­ tors. For example, Hamilton noted in a paper published in Macroeconomic Dynamics in 2011 that at that time, 10 out of the 11 postwar U.S. recessions had been preceded by a sharp increase in the price of crude petroleum (Hamilton, 2011). Improving the forecasting accuracy of the oil price is of great interest to both researchers and practitioners. For example, central banks and private sector forecasters take the oil price as one of the key variables in assessing macro­ economic risks. Thus, a more accurate forecast of the oil price is of great impor­ tance to both policy-makers and investors. Influenced mainly by the fundamental supply and demand shocks, the crude oil price is also shocked by many other factors such as environmental disasters, political events, speculations, and so on. All these factors contribute to the huge volatility of the oil price and make oil price forecasting a difficult and chal­ lenging job. Despite the difficulty, voluminous approaches have been devoted to concern­ ing crude oil price forecasting and analyzing. Generally, these approaches can be classified into two categories: structural models and data-driven methods. Struc­ tural models attempt to analyze and forecast oil price in terms of supply–demand equilibrium schedule (e.g. Bacon, 1991; Huntington, 1994; Ye et al., 2004; Yang et al., 2002; He et al., 2010). In a different way, the data-driven models try to capture the real data generating process (DGP) from the historical price information using either statistical analysis or artificial intelligence. Data-driven approaches include linear models such as Autoregressive Moving Average (ARMA), Autoregressive Conditional Heteroscedasticity (ARCH) type models (e.g. Sadorsky, 2002; Morana, 2001) etc., and nonlinear models such as Artificial

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Neural Network (e.g. Mirmirani and Li, 2004; Moshiri, 2004; Nelson et al., 1994), pattern matching approach (Fan et al., 2008), soft computing approach (Ghaffari and Zare, 2009), wavelet decomposition and neural network modeling (Jammazi and Aloui, 2012). Recent academic research shows a rising interest in using interval data to forecast oil price, including Sun et al. (2018), Yang et al. (2012), Qiao et al. (2019). Other references on oil price forecasting and analyz­ ing include Abosedra and Baghestani (2004), Stevens (1995), Pindyck (1999), Chaudhuri (2001), Zhang et al. (2008), Yu et al. (2008), Huang et al. (2009), Kang and Yoon (2013), and Yang et al. (2013, 2016). Different from the existing method, this chapter scrutinizes the predictability of crude oil price with the DVAR model. We compare the DVAR modeling tech­ nique with the efficient market model and the classic ARMA model. Previewing the results, we obtain the following interesting findings. First, both in-sample and out-of-sample forecasts demonstrate that the DVAR model does report statistically significant and informative forecasts. Different evaluation methods are used to check the robustness of the forecasts, and the results confirm that the crude oil price is predictable. Second, empirical results indicate that the DVAR model performs better when the oil price is in recession, which is consistent with the findings of Li et al. (2013) in future oil price and with the information friction theory of Hong et al. (2000, 2007). This finding is important as it indicates that the oil market is not so efficient as it is commonly believed to be, especially when in recession, and that the DVAR model is more applicable in recession. Finally, out-of-sample forecasting results demonstrate the dominance of the decomposition-based VAR model over the ARMA model, indicating that the candlestick chart is informative in forecasting crude oil price. This chapter is organized as follows. Section 2 presents the DVAR model along with some discussions. Section 3 empirically investigates the performance of the DVAR model using the monthly WTI spot oil price data series and com­ pares the results with the efficient market model and with the ARMA model. Section 4 summarizes.

11.2 Econometric method In this section, we present a brief introduction to the DVAR model and the fore­ cast evaluation criteria.

11.2.1 DVAR model The decomposition-based VAR (DVAR) model will be used to investigate the predictability of the crude oil price. A DVAR model of p order can be presented as follows: Yt ¼ C þ

p X Ai Yt–i þ CXt –1 þ εt ; i¼1

ð11:1Þ

Oil price forecasting

93

where Yt=(ΔRt, ΔWt)T, Xt−1 is a vector of exogenous variables. The exogenous variables used in this chapter include the upper shadow and the lower shadow. The forecasts of the return on crude oil are constructed through the following equation rtf ¼ DRtf þ DWtf ;

ð11:2Þ

where rtf is the return forecast, DRtf and DWtf are respectively the forecasts of DRtf and DWtf reported by the DVAR model.

11.2.2 Forecast evaluation The commodity market is believed to be efficient and follow a random walk. To see if the DVAR model beats the simple random walk model in terms of out-of­ sample forecasts, the first forecast evaluation criterion used is the out-of-sample 2 R-square, Roos (Campbell and Thompson, 2008): PT 2 ðri – rif Þ 2 Roos ¼ 1 – Pi¼mþ1 ð11:3Þ T 2 ; ̄ i¼mþ1 ðri – r i Þ where rif is the return forecast, and ̄r i is the historical mean forecast Pi–1 ̄r i ¼ t¼1 ; i ¼ m þ 1; m þ 2; :::; T : i–1

ð11:4Þ

2 will be positive, which implies a If the DVAR model reports better forecasts, Roos lower mean-squared forecast error (hereafter MSFE) relative to the forecast based on the historical average return. 2 The null hypothesis of interest is therefore Roos < 0 against the alternative 2 hypothesis that Roos ≥0. We test this hypothesis by using the Clark and West (2007) MSFE-adjusted statistic. Define 2

2

2

fi ¼ ðri – ̄r i Þ – ½ðri – rif Þ – ðrif – ̄r i Þ ];

ð11:5Þ

then the Clark and West (2007) MSFE-adjusted statistic is the t-statistic from the regression of fi on a constant. Following Sadorsky (2002), we also perform the market timing ability of the DVAR model using different ways: 1:

BGJ

test : I ðri > 0Þ ¼ α0 þ α1 I ðrif > 0Þ þ n1;i ;

ð11:6Þ

where I is an indicator function, which is equal to 1 if its argument is true and 0 otherwise. The BGJ test (Breen et al., 1989) is asymptotically equivalent to a one-tailed test on the significance of the slope coefficient, α1. 2:

CM

test : ri ¼ α0 þ α1 I ðrif > 0Þ þ n2;i :

ð11:7Þ

Applications

94

The CM test (Cumby and Modest, 1987) extends the BGJ test to include not just market timing, but also the magnitude. 3:

BH

test : ri ¼ α0 þ α1 rif þ n3;i :

ð11:8Þ

The BH test (Bossaerts and Hillion, 1999) investigates if the forecasts capture any valuable information contained in the real values. In case of statistically sig­ nificant nonzero of α1, the forecasts are said to be informative. In case of α0 = 0 and α1 = 1, rif is said to be an unbiased forecast of ri. In both the CM and BH tests, the null hypothesis is that the slope coefficient is equal to zero, and the alternative hypothesis is a one-sided alternative that the slope coefficient is positive. In time series forecasting, the ARMA model is the most commonly used econometric tool. To see if high–low price information adds additional informa­ tion for improving the forecasting accuracy, we compare the performance of the DVAR model with the ARMA model. To see if the DVAR model outperforms the ARMA model, we report the root mean square error (RMSE) and the mean absolute error (MAE) i.e., rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XT 2 RMSE ¼ (11.9) ðri – rif ðmÞÞ ; i¼mþ1 T –m T X 1 MAE ¼ jr – rif ðmÞj; (11.10) T – m i¼mþ1 i where rif ðmÞ is the forecast reported by model m.

11.3 Empirical results This section investigates the empirical performance of the DVAR model forecast­ ing and compares the forecasting accuracy between the DVAR model and the random walk model, and the ARMA model.

11.3.1 The data The monthly spot price data used in this analysis is the U.S. Cushing, OK WTI Spot Price (dollars per barrel). The data sample covers 1986.01–2013.01, with a number of 325 observations. Figure 11.1 presents the time series for the WTI spot oil price. The data was downloaded from the EIA website. Since the original data set provides no high and low prices, we constructed the high and low prices from the closing price as follows: Ht ¼ maxðCt–11 ; Ct–10 ; :::; Ct Þ;

Lt ¼ minðCt –11 ; Ct–10 ; :::; Ct Þ:

ð11:11Þ

Oil price forecasting

95

140 WTI

120 100 80 60 40 20 0 88

90

92

94

96

98

00

02

04

06

08

10

12

Figure 11.1 Time series of monthly WTI crude oil price over 1986.01–2013.01 Table 11.1 Summary statistics of ΔRt, ΔWt and crude oil return: 1986.01−2013.01

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera

ΔRt

ΔWt

rt

0.006 0.000 0.148 −0.174 0.037 −0.480 7.890 323.848***

0.000 0.004 0.297 −0.255 0.073 0.037 4.716 38.463***

0.006 0.012 0.392 −0.332 0.082 −0.249 5.593 90.893***

We use ***, **, and * to mean significance at the level of 1%, 5%, 10%.

To be specific, the high price is taken to be the maximum price of the 12 con­ secutive monthly observations, and the low price is take to be the minimum price of the 12 consecutive observations. From the high and low prices, ΔRt and ΔWt can be constructed. Table 11.1 reports the summary statistics ΔRt, ΔWt, and rt. The summary statistics indicate great price volatility risk of the crude oil market: within one month, the oil price changes can go up as large as 39% and down as huge as 33%, both of which are three times larger than the standard deviation. Consistent with the well documented

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facts, the oil price changes also exhibit skewness and high kurtosis. The JarqueBera statistic reveals the abnormal distribution of ΔRt, ΔWt, and rt at a signifi­ cance level of 1%.

11.3.2 In-sample model estimation The in-sample forecast covers the whole data observations. Lag order selection is of great importance when performing a VAR model estimation. As is usual, complex models underperform the simple ones when delivering an out-of­ sample forecast. Therefore, we use the SIC (Schwarz Information Criterion) as the order selection criterion. Table 11.2 reports the estimates of the DVAR model. To quantitatively see how well the DVAR model explains the real observa­ tions, we perform a linear regression test. The result is presented as follows rt ¼ 3:46E – 10 þ 1:000*** rtf ðdvarÞ;

R2 ¼ 0:086

ð11:12Þ

where rtf are the forecasts. The result indicates good in-sample forecasts. Regres­ sion analysis shows unbiased in-sample forecasts of the DVAR model: the slope coefficient is equal to 1 and is statistically significant, while the constant is almost 0 and is insignificantly different from 0. The R-square indicates that about 8% variance can be explained away by the DVAR model.

11.3.3 Out-of-sample performance For the out-of-sample forecast, the total sample of T observations are divided into two portions: the in-sample portion composed of the first M observations and the out-of-sample portion composed of the last T-M observations. We use the static forecasting procedure to produce the out-of-sample forecast. To be specific, the first M observations are used to estimate the parameters in the DVAR model. Keeping fixed these estimated parameters, the out-of-sample fore­ casts are reported. The static forecast is employed to check the robustness of the DVAR model. It is commonly accepted that the out-of-sample forecast would be Table 11.2 Estimates of the DVAR model: WTI oil price ΔWt

ΔRt c α β lst−1 ust−1

Coef.

t-Statistic

Coef.

0.007 0.466 0.119 −0.060 0.009

2.677 8.039.657 4.265 −3.341 0.468

−0.005 −0.273 0.223 0.045 0.034

t-Statistic −0.803 −2.536 3.578 1.059 0.792

R-square

R-square

0.300

0.084

Oil price forecasting

97

poor if there is any instability in the model structure. Thus, the static forecast offers a nice tool for checking the robustness of the model structure. For this example, the time period 1986.12–2000.12 is used to estimate the parameters, and the time period 2001.01–2013.01 is used for the out-of-sample forecast. The division is typical. For one thing, the cutoff point is almost the middle point of the whole sample; for another, the portion used for the out-of­ sample forecast covers both expansion and recession in the crude oil market. The BGJ, CM, and BH tests are reported as follows: BGJ test: I(ri > 0) = 0.557***+ 0.145*I ðrif > 0Þ, CM test: ri = −0.017 + 0.043***I ðrif > 0Þ, BH test: ri = 0.006 + 0.900***rif . Contrary to Sadorsky (2002), the BGJ test result indicates there is direction forecasting ability for monthly returns. Different from Sadorsky (2002), the CM and BH tests give positive reports, which confirms that the DVAR model does report significant out-of-sample forecasts. To see in what conditions DVAR performs better, recession or expansion, we present the following regression analysis ri ¼ 0:010 þ 0:327rif ðdvarÞI ðri >¼ 0Þ þ 1:907*** rif ðdvarÞI ðri < 0Þ:

ð11:13Þ

The test result shows that when the oil price is in expansion (ri ≥ 0), the slope coefficient is positive but not significant; while the slope is significant at a level of 1% when the oil price is in recession (ri < 0). This pattern is consistent with the information friction of Hong et al. (2000, 2007) who claim that “bad news travels more slowly”, and with the slow information diffusion effect detected by Li et al. (2013) in the oil futures price. This pattern has also been widely observed in the stock markets, such as Rapach et al. (2010), Henkel et al. (2011), and Dangl and Halling (2012). To see if the DVAR model significantly outperforms the simple historical mean, we calculate the t-value of the MSFE-adjusted statistic. The t-value is reported to be 2.18, statistically significant at a level of 5%, which confirms that the DVAR model beats the simple historical average. The crude oil market is thus not informationally efficient. Another interesting question is whether or not the DVAR model outperforms the classic ARMA(p, q) model for out-of-sample forecasting. Following the same rule, we select the order (p, q) based on SIC. The ARMA model selected by the SIC is given by rt ¼ 3:06E – 03 þ εt þ 0:327*** εt –1 – 0:277*** εt –5 :

ð11:14Þ

To compare the relative performance of DVAR and ARIMA, the MAE and RMSE are computed. The results are presented in Table 11.3. The DVAR model outper­ forms the ARIMA in terms of both MAE and RMSE.

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Table 11.3 Out-of-sample prediction error comparison: DVAR v.s. ARMA Model

MAE

RMSE

ARMA DVAR

0.072 0.066

0.091 0.084

1.4

.20 DVAR

1.2

ARMA

ARMA_DV AR

.16

1.0 .12 0.8 .08 0.6 .04

0.4

.00

0.2 0.0

–.04 02

04

06

08

10

12

02

04

06

08

10

12

Figure 11.2 Out-of-sample forecasting comparison, ARMA v.s. DVAR over 2001.01–2013.01

We also run encompassing regression, and the results are presented as follows ri ¼ 6:521E – 03 þ 0:718* rif ðdvarÞ þ 0:151rif ðarmaÞ:

ð11:15Þ

The result shows clear dominance of DVAR over ARMA: once the forecasts of DVAR are included, the forecasts given by ARMA become insignificant. Figure 11.2 presents the cumulative squared forecast error in the left panel and the difference between the cumulative squared forecast error for the ARMA model and the cumulative squared forecast error for the DVAR model in the right panel. We use legend “ARMA_DVAR” to mean the difference.

11.4 Summary This chapter further scrutinizes the forecasting power of candlestick forecasting on crude oil price. An empirical study was performed on the WIT crude oil

Oil price forecasting

99

price, and the result confirms that the DVAR model out-performs not only the historical mean model but also the classic ARMA model. This finding indicates that candlestick charts have additional valuable information for oil price forecast­ ing. In summary, candlestick forecasting is informative.

Part V

Conclusions and future studies

12 Main conclusions

Technical analysis has been widely used in financial practice, however this disci­ pline has not received the same level of academic scrutiny and acceptance as the more traditional approaches such as fundamental analysis. One of the main obstacles is its lack of theoretical underpinnings. This book focuses on one of the most popular technical indicators, the candle­ stick chart. We investigate the statistical properties of the candlestick chart and empirically investigate its forecasting performance. Based on a decomposition technique and under some mild assumptions, we obtain the following statistical properties of the candlestick charts: (1) Technical range is a unit root process; (2) Technical range is co-integrated with closing price; (3) Technical ranges are co-integrated given that their corresponding closing prices are co-integrated. Empirical studies are performed on different assets, and the results show that econometric models taking into consideration of these properties report more informative in-sample and out-of-sample forecasts compared to the classic mod­ eling techniques. We believe the statistical properties obtained in this book and the empirical evidence presented in book have established, to some extent, the statistical foun­ dation of candlestick forecasting. We conclude that the statistical properties of the candlestick are valuable to financial modeling and forecasting.

13 Future studies

This book focuses on the statistical properties of candlestick charts and how these properties can be used in financial forecasting through some empirical applica­ tions. In this section, we present how the findings in this book can be further extended. 1) More empirical studies are needed to further scrutinize the forecasting power of the candlestick and its economic value in investment practice. The can­ dlestick chart has been widely used in technical analysis, however its forecasting power has not been carefully scrutinized by academic research. Comprehensive empirical studies concerning its forecasting power would be of great significance and value to both academic researchers and investment practitioners. 2) In Chapter 10, we demonstrate that candlestick forecasting is more valuable in economic recession than in economic expansion. This finding is interesting as it hints that the forecasting power of the candlestick is state-dependent. More academic research is thus required to relate the forecasting power of the candle­ stick to economic conditions. 3) Empirical studies performed in Chapters 10–11 show that candlestick reports are informative for in-sample and out-of-sample forecasts. However, the reasons behind these findings remain unknown, thus needing further inves­ tigation. We conjecture the outperformance of the DVAR model over the classic ARMA model, to some extent, is due to the clustering effect in shadows. It can be proved under pure random walk process that shadows, both upper and lower, are independent. However, in reality we often detect autocorrelation in shadows. The autocorrelation in shadows hints that information in historical shadow is valuable to forecast future shadows, and thus the future returns. 4) In this book, we only present one example to show how the statistical prop­ erties of the candlestick chart can be used to study the information spillover effect across financial markets. More applications are needed to scrutinize its ability of capturing the information spillover effect. 5) Recent academic research has provided huge evidence that the structure of speculative prices is nonlinear. Therefore, nonlinear modeling techniques should be of more value for investors in practice. In this book, all the modeling tech­ niques are linear; nonlinear models and their forecasting power have not been investigated, all of which requires more attention and application.

Future studies 105 6) The candlestick chart is always used in company with trading volume. In this book, the role of trading volume in candlestick chart forecasting has not been investigated, which also requires attention. In summary, a lot of work has not been done in this book and requires more attention and more efforts in future studies.

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Index

black and white candlesticks 17

Parkinson range 17, 19, 49

CAC40 71 candlestick 62 candlestick chart 15, 18 CARR model 48 co-integration 21, 73

real body 15, 17 return predictability 77

DAX 22, 71 DVAR 28, 29, 35, 77, 78, 91 FTSE100 33 historical mean forecast 79 information spillover 69 information transmission 69 interval data 92

S&P500 22, 33, 51, 65, 79 shadow 15, 35 spillover effect 69 SSEC 22 Taylor expansion 38 technical range 18, 20, 22, 62 technical range forecasting 62 unit root 20 upper shadow 16, 35, 44

MSFE-adjusted statistic 79, 93

VECM 63, 70 volatility spillover 70 volatility timing 47, 49

NIKKEI225 33

WTI crude oil 91, 94

lower shadow 16, 35, 44