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Essentials of Econometrics Fifth Edition
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I dedicate this book to Joan Gujarati, Diane Gujarati-Chesnut, Charles Chesnut, and my grandchildren, "Tommy" and Laura Chesnut, and to my dear friend Karen Low.
Sara Miller McCune founded SAGE Publishing in 1965 to support the dissemination of usable knowledge and educate a global community. SAGE publishes more than 1000 journals and over 600 new books each year, spanning a wide range of subject areas. Our growing selection of library products includes archives, data, case studies and video. SAGE remains majority owned by our founder and after her lifetime will become owned by a charitable trust that secures the company’s continued independence. Los Angeles | London | New Delhi | Singapore | Washington DC | Melbourne
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Essentials of Econometrics Fifth Edition
Damodar N. Gujarati Professor Emeritus of Economics The United States Military Academy at West Point
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FOR INFORMATION: SAGE Publications, Inc. 2455 Teller Road Thousand Oaks, California 91320 E-mail: [email protected] SAGE Publications Ltd. 1 Oliver’s Yard 55 City Road London EC1Y 1SP United Kingdom SAGE Publications India Pvt. Ltd. B 1/I 1 Mohan Cooperative Industrial Area Mathura Road, New Delhi 110 044
Copyright © 2022 by Damodar N. Gujarati All rights reserved. Except as permitted by U.S. copyright law, no part of this work may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without permission in writing from the publisher. All third-party trademarks referenced or depicted herein are included solely for the purpose of illustration and are the property of their respective owners. Reference to these trademarks in no way indicates any relationship with, or endorsement by, the trademark owner. Printed in the United States of America Library of Congress Cataloging-in-Publication Data Names: Gujarati, Damodar N., author. Title: Essentials of econometrics / Damodar N. Gujarati, The United States Military Academy at West Point.
India
Description: Fifth edition. | Thousand Oaks, California : SAGE, [2022] | Includes bibliographical references and index.
SAGE Publications Asia-Pacific Pte. Ltd.
Identifiers: LCCN 2021012627 | ISBN 978-1-0718-5039-8 (paperback ; alk. paper) | ISBN 978-1-0718-5040-4 (epub) | ISBN 978-1-0718-5041-1 (epub) | ISBN 978-1-0718-5042-8 (pdf)
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Subjects: LCSH: Econometrics. | Economics—Statistical methods. Classification: LCC HB139 .G85 2022 | DDC 330.01/5195—dc23 LC record available at https://lccn.loc.gov/2021012627 This book is printed on acid-free paper.
Acquisitions Editor: Helen Salmon Product Associate: Kenzie Offley Production Editor: Rebecca Lee Copy Editor: Gillian Dickens Typesetter: C&M Digitals (P) Ltd. Indexer: Integra Cover Designer: Scott Van Atta Marketing Manager: Victoria Velasquez
21 22 23 24 25 10 9 8 7 6 5 4 3 2 1
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BRIEF CONTENTS Acknowledgmentsxvii Prefacexix About the Author
Chapter 1 •
PART I
The Nature and Scope of Econometrics
• THE LINEAR REGRESSION MODEL
xxiii
1
23
Chapter 2 •
Basic Ideas of Linear Regression: The Two-Variable Model
25
Chapter 3 •
The Two-Variable Model: Hypothesis Testing
61
Chapter 4 •
Multiple Regression: Estimation and Hypothesis Testing
105
Chapter 5 •
Functional Forms of Regression Models
147
Chapter 6 •
Qualitative or Dummy Variable Regression Models
193
PART II • REGRESSION ANALYSIS IN PRACTICE
239
Chapter 7 •
Model Selection: Criteria and Tests
241
Chapter 8 •
Multicollinearity: What Happens If Explanatory Variables Are Correlated?
279
Chapter 9 •
Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?313
Chapter 10 •
Autocorrelation: What Happens If Error Terms Are Correlated?
PART III • ADVANCED TOPICS IN ECONOMETRICS
359
393
Chapter 11 •
Elements of Time-Series Econometrics
395
Chapter 12 •
Panel Data Regression Models
419
Appendix A: Review of Statistics: Probability and Probability Distributions
441
Appendix B: Characteristics of Probability Distributions
475
Appendix C: Some Important Probability Distributions
505
Appendix D: Statistical Inference: Estimation and Hypothesis Testing
533
Appendix E: Statistical Tables
565
Index 593
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DETAILED CONTENTS Acknowledgmentsxvii Prefacexix About the Author
Chapter 1 • The Nature and Scope of Econometrics
xxiii
1
1.1 What Is Econometrics?
1
1.2 Why Study Econometrics?
2
1.3 The Methodology Of Econometrics
4
1. The Object of Research 2. Collecting Data 3. Specifying the Mathematical Model of Labor Force Participation 4. Specifying the Statistical, or Econometric, Model of Labor Force Participation 5. Estimating the Parameters of the Chosen Econometric Model 6. Checking for Model Adequacy: Model Specification Testing 7. Testing Hypothesis Derived From the Model 8. Using the Model for Prediction or Forecasting
4 5 8 9 11 11 13 14
1.4 The Road Ahead
15
Key Terms and Concepts
16
Questions16 Problems17 Appendix 1A: Economic Data on the World Wide Web
19
PART I • THE LINEAR REGRESSION MODEL
23
Chapter 2 • Basic Ideas of Linear Regression: The Two-Variable Model
25
2.1 The Meaning of Regression
25
2.2 The Population Regression Function (PRF): A Hypothetical Example
26
2.3 Statistical or Stochastic Specification of The Population Regression Function30 2.4 The Nature of the Stochastic Error Term
31
2.5 The Sample Regression Function (SRF)32 2.6 The Special Meaning of the Term Linear Regression Linearity in the Variables Linearity in the Parameters
37 37 38
2.7 Two-Variable Versus Multiple Linear Regression
39
2.8 Estimation of Parameters: The Method of Ordinary Least Squares
39
The Method of Ordinary Least Squares
40
Get Complete eBook Download by email at [email protected] 2.9 Putting It All Together Interpretation of the Estimated Math SAT Score Function
2.10 Some Illustrative Examples
44 44
45
2.11 Summary
52
Key Terms and Concepts
53
Questions53 Problems55 Optional Questions
60
Appendix 2A: Derivation of Least Squares Estimators
60
Chapter 3 • The Two-Variable Model: Hypothesis Testing
61
3.1 The Classical Linear Regression Model
62
3.2 Variances and Standard Errors of Ordinary Least Squares Estimators
66
Variances and Standard Errors of the Math SAT Example Summary of the Math SAT Score Function
3.3 Why OLS? Properties of OLS Estimators Gauss–Markov Theorem
3.4 The Sampling, or Probability, Distributions of OLS Estimators Central Limit Theorem
3.5 Hypothesis Testing Testing H0:B 2 = 0 Versus H1:B 2 ≠ 0: The Confidence Interval Approach The Test of Significance Approach to Hypothesis Testing Math SAT Example Continued
3.6 Hypothesis Testing: Some Practical Aspects
67 68
69 70
70 71
72 74 76 77
80
3.7 How Good Is The Fitted Regression Line: The Coefficient of Determination, r280 Formulas to Compute r283 r2 for the Math SAT Example 84 The Coefficient of Correlation, r84
3.8 Reporting the Results of Regression Analysis
85
3.9 Illustrative Examples
87
1. Relationship Between Wages and Productivity in the Business Sector, USA, 1959–2006 2. Expenditure on Education and Income in the 50 U.S. States for 2000 3. CEO Salaries of 447 Fortune 500 Companies for 1999 4. Impact of Advertising Expenditure on Viewers
87 88 90 91
3.10 Comments on the Illustrative Examples
92
3.11 Forecasting
92
3.12 Normality Tests
96
Histograms of Residuals Jarque–Bera Test
96 96
3.13 Summary
98
Key Terms and Concepts
98
Questions98 Problems100
Get Complete eBook Download by email at [email protected] Chapter 4 • Multiple Regression: Estimation and Hypothesis Testing 4.1 The Three-Variable Linear Regression Model The Meaning of Partial Regression Coefficient
105 106 107
4.2 Assumptions of the Multiple Linear Regression Model
108
4.3 Estimation of the Parameters of Multiple Regression
111
Ordinary Least Squares Estimators Variance and Standard Errors of OLS Estimators Properties of OLS Estimators of Multiple Regression
111 113 114
4.4 Goodness of Fit of Estimated Multiple Regression: Multiple Coefficient of Determination, R2114 4.5 Antique Clock Auction Prices Revisited Interpretation of the Regression Results
116 116
4.6 Hypothesis Testing In A Multiple Regression: General Comments
117
4.7 Testing Hypotheses About Individual Partial Regression Coefficients
118
The Test of Significance Approach The Confidence Interval Approach to Hypothesis Testing
118 120
4.8 Testing the Joint Hypothesis That B2 = B3 = 0 Or R2 = 0
121
4.9 Two-Variable Regression In the Context of Multiple Regression: Introduction to Specification Bias
125
An Important Relationship Between F and R2124
4.10 Comparing Two R2 Values: The Adjusted R2127 4.11 When to Add an Additional Explanatory Variable to a Model
128
4.12 Restricted Least Squares
130
4.13 Illustrative Examples
131
Discussion of Regression Results
132
4.14 Summary
138
Key Terms and Concepts
139
Questions140 Problems000 Appendix 4A.1: Derivations of OLS Estimators144 Appendix 4A.2: Derivation of Equation (4.31)
145
Appendix 4A.3: Derivation of Equation (4.49)
145
Chapter 5 • Functional Forms of Regression Models
147
5.1 How to Measure Elasticity: The Log-Linear Model
148
Hypothesis Testing in Log-Linear Models
153
5.2 Multiple Log-Linear Regression Models
154
5.3 How to Measure the Growth Rate: The Semilog Model
157
Instantaneous Versus Compound Rate of Growth The Linear Trend Model
160 161
5.4 The Lin-Log Model: When the Explanatory Variable Is Logarithmic
162
5.5 Reciprocal Models
164
5.6 Polynomial Regression Models
169
5.7 Regression Through the Origin: The Zero Intercept Model
173
Get Complete eBook Download by email at [email protected] 5.8 A Note on Scaling and Units of Measurement
175
5.9 Regression on Standardized Variables
177
5.10 Summary of Functional Forms
180
5.11 SUMMARY
180
Key Terms and Concepts
181
Questions182 Problems183 Appendix 5A: Logarithms
Chapter 6 • Qualitative or Dummy Variable Regression Models
190
193
6.1 The Nature of Dummy Variables
193
6.2 ANCOVA Models: Regression on One Quantitative Variable and One Qualitative Variable With Two Categories
197
6.3 Regression on One Quantitative Variable and One Qualitative Variable With More Than Two Classes or Categories
200
6.4 Regression on One Quantiative Explanatory Variable and More Than One Qualitative Variable
203
Interaction Effects A Generalization
204 205
6.5 Comparing Two Regessions
207
6.6 The Use of Dummy Variables In Seasonal Analysis
212
6.7 What Happens if the Dependent Variable Is Also a Dummy Variable? The Linear Probability Model (LPM)214 6.8 The Logit Model Estimation of the Logit Model
219 221
6.9 Summary
228
Key Terms and Concepts
229
Questions229 Problems230
PART II • REGRESSION ANALYSIS IN PRACTICE Chapter 7 • Model Selection: Criteria and Tests
239 241
7.1 The Attributes of a Good Model
242
7.2 Types of Specification Errors
243
7.3 Omisson of Relevant Variable Bias: “Underfitting” a Model
243
7.4 Inclusion of Irrelevant Variables: “Overfitting” a Model
248
7.5 Incorrect Functional Form
251
7.6 Errors of Measurement
253
Errors of Measurement in the Dependent Variable Errors of Measurement in the Explanatory Variable(s)
254 254
7.7 Detecting Specification Errors: Tests of Specification Errors
255
Detecting the Presence of Unnecessary Variables Tests for Omitted Variables and Incorrect Functional Forms
255 258
Get Complete eBook Download by email at [email protected] Choosing Between Linear and Log-Linear Regression Models: The MWD Test Regression Error Specification Test: RESET
260 262
7.8 Outliers, Leverage, and Influence Data
265
7.9 Probabity Distribution of the Error Term
268
7.10 Model Evaluation Criteria
270
7.11 Nonnormal Distribution of the Error Term
272
7.12 Fixed Versus Random (or Stochastic) Explanatory Variables
272
7.13 Summary
273
Key Terms and Concepts
274
Questions275 Problems275
Chapter 8 • Multicollinearity: What Happens if Explanatory Variables Are Correlated?
279
8.1 The Nature of Multicollinearity: The Case of Perfect Multicollinearity
280
8.2 The Case of Near, or Imperfect, Multicollinearity
283
8.3 Theoretical Consequences of Multicollinearity
285
8.4 Practical Consequences of Multicollinearity
287
8.5 Detection of Multicollinearity
289
8.6 Is Multicollinearity Necessarily Bad?
294
8.7 An Extended Example: The Demand for Chickens In The United States, 1960 To 1982
295
Collinearity Diagnostics for the Demand Function for Chickens
8.8 What to Do With Multicollinearity: Remedial Measures Dropping a Variable(s) From the Model Acquiring Additional Data or a New Sample Rethinking the Model Prior Information About Some Parameters Transformation of Variables Other Remedies
297
299 300 301 302 303 304 305
8.9 Summary
305
Key Terms and Concepts
306
Questions306 Problems307
Chapter 9 • Heteroscedasticity: What Happens if the Error Variance Is Nonconstant? 9.1 The Nature of Heteroscedasticity Reasons for Heteroscedasticity
313 313 315
9.2 Consequences of Heteroscedasticity
316
9.3 Detection of Heteroscedasticity: How Do We Know When There Is a Heteroscedasticity Problem?
319
1. Nature of the Problem 2. Graphical Examination of Residuals
320 320
Get Complete eBook Download by email at [email protected] 3. Park Test 4. Glejser Test 5. White’s General Heteroscedasticity Test 6. Breusch-Pagan (BP) Test of Heteroscedasticity Other Tests of Heteroscedasticity
9.4 What to Do if Heteroscedasticity Is Observed: Remedial Measures
323 327 328 330 332
332
When s2i Is Known: The Method of Weighted Least Squares (WLS) When True s2i Is Unknown Respecification of the Model
333 334 339
9.5 White’s Heteroscedasticity-Corrected Standard Errors and t Statistics
340
9.6 Some Concrete Examples of Heteroscedasticity
342
9.7 Summary
349
Key Terms and Concepts
350
Questions350 Problems351
Chapter 10 • Autocorrelation: What Happens If Error Terms Are Correlated? 10.1 The Nature of Autocorrelation
359 360
Inertia361 Model Specification Error(s) 362 The Cobweb Phenomenon 362 Data Manipulation 362
10.2 Consequences of Autocorrelation
364
10.3 Detecting Autocorrelation
364
The Graphical Method The Durbin–Watson d Test
365 367
10.4 Remedial Measures
372
10.5 How to Estimate r374 ρ = 1: The First Difference Method 375 ρ Estimated From Durbin–Watson d Statistic 375 ρ Estimated From OLS Residuals, et376 Other Methods of Estimating ρ377
10.6 A Large Sample Method of Correcting OLS Standard Errors: The Newey–West (NW) Method
378
10.7 A General Test of Autocorrelation: The Breusch–Godfrey (BG) Test
383
10.8 Summary
386
Key Terms and Concepts
386
Questions387 Problems388
PART III • ADVANCED TOPICS IN ECONOMETRICS Chapter 11 • Elements of Time-Series Econometrics
393 395
11.1 The Phenomenon of Spurious Regression: Nonstationary Time Series
395
11.2 Tests of Stationarity
398
1. Graphical Analysis
398
Get Complete eBook Download by email at [email protected] 2. Autocorrelation Function (ACF) and Correlogram 3. The Unit Root Test of Stationarity
399 402
11.3 Cointegrated Time Series
406
11.4 The Random Walk Model
408
11.5 Causality In Economics: The Granger Causality Test
411
The Granger Causality Test
411
11.6 Summary
415
Key Terms and Concepts
416
Problems416
Chapter 12 • Panel Data Regression Models
419
12.1 The Importance of Panel Data
420
12.2 An Illustrative Example: Charitable Giving
421
12.3 Pooled OLS Regression of the Charity Function
423
12.4 The Fixed-Effects Least Squares Dummy Variable (LSDV) Model
424
12.5 Limitations of the Fixed-Effects LSDV Model
427
12.6 The Fixed-Effects Within-Group (WG) Estimator
428
12.7 The Random-Effects Model (REM) or Error Components Model (ECM)430 Some Guidelines About REM and FEM
434
12.8 Properties of Various Estimators
435
12.9 Panel Data Regressions: Some Concluding Comments
436
12.10 Summary and Conclusions
436
Key Terms and Concepts
437
Problems438
INTRODUCTION TO APPENDIXES A, B, C, AND D: BASICS OF PROBABILITY AND STATISTICS Appendix A: Review of Statistics: Probability and Probability Distributions A.1 Some Notation The Summation Notation Properties of the Summation Operator
A.2 Experiment, Sample Space, Sample Point, and Events
441 442 442 442 443
444
Experiment444 Sample Space or Population 444 Sample Point 445 Events445 Venn Diagrams 445
A.3 Random Variables
446
A.4 Probability
447
Probability of an Event: The Classical or A Priori Definition Relative Frequency or Empirical Definition of Probability Probability of Random Variables
A.5 Random Variables and Their Probability Distributions Probability Distribution of a Discrete Random Variable Probability Distribution of a Continuous Random Variable Cumulative Distribution Function (CDF)
448 448 455
455 455 457 459
Get Complete eBook Download by email at [email protected] A.6 Multivariate Probability Density Functions Marginal Probability Functions Conditional Probability Functions Statistical Independence
462 464 465 467
A.7 Summary and Conclusions
468
Key Terms and Concepts
469
References469 Questions470 Problems470 Appendix B: Characteristics of Probability Distributions
475
B.1 Expected Value: A Measure of Central Tendency
475
Properties of Expected Value Expected Value of Multivariate Probability Distributions
B.2 Variance: A Measure of Dispersion Properties of Variance Chebyshev’s Inequality Coefficient of Variation
B.3 Covariance Properties of Covariance
B.4 Correlation Coefficient Properties of the Correlation Coefficient Variances of Correlated Variables
B.5 Conditional Expectation Conditional Variance
477 479
479 481 483 484
484 486
486 487 489
489 491
B.6 Skewness and Kurtosis
491
B.7 From Population to the Sample
494
Sample Mean Sample Variance Sample Covariance Sample Correlation Coefficient Sample Skewness and Kurtosis
495 496 496 498 498
B.8 Summary
499
Key Terms and Concepts
499
Questions500 Problems501 Optional Exercises
Appendix C: Some Important Probability Distributions C.1 The Normal Distribution Properties of the Normal Distribution The Standard Normal Distribution Random Sampling From a Normal Population – The Sampling or Probability Distribution of the Sample Mean X The Central Limit Theorem (CLT)
C.2 The t Distribution Properties of the t Distribution
503
505 506 506 508 512 512 518
519 519
Get Complete eBook Download by email at [email protected] C.3 The Chi-Square (χ2) Probability Distribution Properties of the Chi-Square Distribution
C.4 The F Distribution Properties of the F Distribution
523 524
526 527
C.5 Summary
529
Key Terms and Concepts
530
Questions530 Problems531 Appendix D: Statistical Inference: Estimation and Hypothesis Testing
533
D.1 The Meaning of Statistical Inference
533
D.2 Estimation and Hypothesis Testing: Twin Branches of Statistical Inference
535
D.3 Estimation of Parameters
536
D.4 Properties of Point Estimators
541
Linearity541 Unbiasedness542 Minimum Variance 543 Efficiency543 Best Linear Unbiased Estimator (BLUE) 544 Consistency545
D.5 Statistical Inference: Hypothesis Testing The Confidence Interval Approach to Hypothesis Testing Type I and Type II Errors: A Digression The Test of Significance Approach to Hypothesis Testing A Word on Choosing the Level of Significance, α, and the p Value The χ 2 and F Tests of Significance
546 547 548 551 555 556
D.6 Summary
559
Key Terms and Concepts
560
Questions560 Problems561 Appendix E: Statistical Tables
565
Index 593
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ACKNOWLEDGMENTS
I
would like to thank Inas R. Kelly, Associate Professor of Economics at Loyola Marymount University, and Michael Grossman, Distinguished Professor of Economics at the City University of New York, for reading and providing feedback on this fifth edition, and also Helen Salmon at SAGE for her behind-the-scenes help and encouragement. SAGE and the author are grateful for feedback from the following reviewers in the development of this fifth edition: Prasad V. Bidarkota, Florida International University Chinyere Emmanuel Egbe, Medgar Evers College, City University of New York Kyungkook Kang, University of Central Florida C. Burc Kayahan, Acadia University Tom Means, San Jose State University Elias Shukralla, Siena College Robert Sonora, University of Montana Patricia Kay Smith, University of Michigan–Dearborn Della Lee Sue, Marist College W. Scott Trees, Siena College
x v ii
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PREFACE PURPOSE OF THE FIFTH EDITION OF ESSENTIALS OF ECONOMETRICS As in the first four editions of this book, my main purpose of the fifth edition is to provide a user-friendly introduction to econometric theory and practice to a wide variety of students. The intended audience is undergraduate economics majors, undergraduate business administration majors, MBA students, and others in social and behavioral sciences where econometrics techniques, especially the techniques of linear regression analysis, are used. It is no exaggeration to say that regression analysis has become an integral part of study in any discipline where one is interested in studying the relationship between a variable of interest, called the dependent or response variable, and a set of explanatory or predictor variables. Sir Francis Galton (1822–1911) used it in the study of heredity, particularly the height of grownup children in relationship to the height of their parents. He used the method of least squares, the workhorse of linear regression analysis, for this purpose. Since then, the methodology of regression analysis has been improved and developed in many ways. Regression analysis is the most widely used social science research tool. The book is designed to help beginning students understand econometric techniques through extensive examples, careful explanations, and a wide variety of problem material. In each of the editions of Essentials of Econometrics (EE), I have tried to incorporate major developments in the field in an intuitive and informative way without resorting to matrix algebra, calculus, or statistics beyond the introductory level. The fifth edition of EE continues this tradition. Students wishing to pursue this subject at a higher mathematical level can consult my book, Linear Regression: A Mathematical Introduction (Sage, 2018).
DESCRIPTION OF THE SPECIFIC MARKET (COURSES) FOR THIS BOOK As noted, the bread-and-butter tool of econometrics is linear regression analysis. A perusal of the books published in this field shows a variety of titles: Introduction to Econometrics, Introductory Econometrics, A First Course in Linear Regression, Regression
xix
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Analysis for the Social Sciences, Statistical Methods in the Social Sciences, Understanding Econometrics, Econometrics Models and Economic Forecasting, Applied Regression Models, Running Regressions, Economic Analysis of Financial Data, Linear Models in Statistics, Understanding Econometrics, and Principles of Econometrics. Despite the variety of names, they all deal with linear regression analysis at various levels of mathematical sophistication. The fifth edition of EE basically provides the foundation of linear analysis for the beginning student. A search on the Internet will reveal that various editions of my book have been used or are still being used in universities and colleges all over the world. A growing trend now is that the subject is now offered on several online courses offered by various colleges and universities. EE has also been used in private business, government and nongovernment entities, and research organizations as a reference book.
THE MAJOR FEATURES OF THE BOOK AND THE BENEFITS OF THESE FEATURES The foundation of linear regression is the classical linear regression model (CLRM). The CLRM is based on several simplifying assumptions, as is true of many other disciplines. I discus these assumptions one by one, pointing out the reasons for the assumption. After discussing CLRM carefully, I look at each assumption critically—how realistic is the assumption? What are the consequences if the assumption is not met in any concrete situation, and what remedies are available? I provide several real data-based examples to shed light on the robustness of the CLRM. Regression outputs of several examples, using statistical packages, such as Eviews and Stata, are included in each chapter. The regression outputs at a glance tell the reader the main feature of the data underlying the various examples. The long longevity of the book probably shows why students and teachers find my book so appealing. With the knowledge of linear regression, the reader will be able to undertake projects involving regression modeling. The reader will also be able to follow empirical research in their field of interest as well as read professional journals in their field. The appendix to Chapter 1 lists various sources of governmental and nongovernmental data that can be easily accessed. The Federal Reserve Bank of St. Louis has a very extensive set of data on a variety of subjects that can be easily downloaded in Excel or other formats and used with several statistical packages. And this is all free!
A DISCUSSION OF SPECIAL PEDAGOGICAL AIDS AND HIGH-INTEREST FEATURES Each chapter has a summary of the main points discussed in the chapter as well as a glossary of the key terms. The problem set in each chapter includes some analytical
Get Complete eBook Download by email at [email protected] Preface
questions as well as real-world data sets that will let the reader solve problems using a variety of techniques discussed in the text. The variety of examples discussed in the text as well as those included in the exercises will show the reader how regression analysis has been used in a variety of disciplines. There are approximately 54 fully worked illustrative examples, about 82 analytical questions, and about 56 data sets in the exercises in the book. The teacher can assign one or more of the data sets as a class project. I firmly believe in learning by doing! A salient feature of EE is that in four statistical appendices, it provides the basics of probability and statistics for the benefit of students whose knowledge of these subjects has become a little rusty or who have studied these subjects a long time ago. Some teachers might want to cover the material in these appendices before covering the rest of the book.
CHANGES IN THE FIFTH EDITION New to this edition, and in response to useful feedback, are Chapter 11 on time-series econometrics and Chapter 12 on panel data econometrics. Some of the material in these chapters is rather advanced, but I have tried to explain it in a way that is understandable to beginning students. Since time-series data are generally collected sequentially, there is every likelihood that adjacent observations are correlated. This leads to the problem of autocorrelation. This topic is discussed at length in Chapter 10. However, a more fundamental problem with time series is that the series may not be stationary. The concept of stationary time series is discussed in this chapter more intuitively, with the warning that if we regress a nonstationary time series on another nonstationary time series, such regressions may lead to the so-called spurious or nonsense regressions. In this chapter, I discuss methods to find out if a time series is stationary or not. Another topic discussed in this chapter is the topic of causality, a kind of chicken and egg problem: Which comes first, the chicken or the egg? Thus, does money supply cause gross domestic product, or it is the other way round? The so-called Granger causality test is often used to answer this question, although there is controversy about this test. In panel data modeling, we have data with both time and cross-sectional dimensions. Thus, we may have data on profits and sales for 50 firms over, say, 10 years, for a total of 500 observations. In finding out the relationship between sales and profits, how do we handle such data? We can regress profits on sales for each of the 50 firms using 10 years of data for each firm, obtaining 50 time-series regressions. We can also regress profits on sales for each year for the 50 firms, obtaining 10 cross-sectional regressions. How do we reconcile these regressions? The topics discussed in Chapter 12 show the various alternatives.
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The chapter on dummy, or qualitative explanatory, variables now also includes the discussion of dummy dependent variables. For example, how do we model the decision to smoke or not? Smoking is a binary variable—you either smoke or you do not smoke. Whereas, in the traditional regression model the dependent variable is generally quantifiable. In this chapter, I discuss how the dummy explanatory variables enhances the linear regression model. I also discuss the problems entailed in estimating a regression model involving dummy dependent variables by the method of least squares and suggest alternatives, such as the logit model.
DIGITAL RESOURCES TO ACCOMPANY THE TEXT An instructor website at edge.sagepub.com/gujarati5e contains the figures from the book and answers to all the problems in the text, together with PowerPoint slides. Students can access the data sets for many of the larger tables for the book on the student section of this website.
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ABOUT THE AUTHOR Damodar N. Gujarati, MCom, University of Bombay (Mumbai), MBA and PhD, both from the University of Chicago, is Professor Emeritus of Economics at the United States Military Academy at West Point, New York. Prior to that, he taught for 25 years at the Baruch College of the City University of New York (CUNY) and at the Graduate Center of CUNY. He is the author of Government and Business (McGraw-Hill, 1984), the bestselling textbook Basic Econometrics (fifth edition, 2009, with coauthor Dawn Porter), Econometrics by Example (second edition, 2014, PalgraveMacmillan), and Essentials of Econometrics (fifth edition, 2021) and Linear Regression: A Mathematical Introduction (2018), both with SAGE. He is also the author of Pensions and New York City Fiscal Crisis (American Enterprise Institute, 1978). His books on econometrics have been translated into several languages. He has published extensively in recognized national and international journals, such as the Review of Economics and Statistics, Economic Journal, Journal of Financial and Quantitative Analysis, and Journal of Business, published by the University of Chicago. He has held visiting professorships at the University of Sheffield, United Kingdom; National University of Singapore; and the University of New South Wales, Australia. He was a Visiting Fulbright Scholar to India. He has lectured extensively on microand macroeconomic topics in Australia, China, Bangladesh, Germany, India, Israel, Mauritius, and the Republic of South Korea.
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1 THE NATURE AND SCOPE OF ECONOMETRICS Econometrics may be defined as the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observations, related by appropriate methods of inference. Paul Samuelson Econometrics may be defined as the social science in which tools of economic theory, mathematics, and statistical inference are applied to the analysis of economic phenomena. Arthur S. Goldberger
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esearch in economics, finance, management, marketing, and related disciplines is becoming increasingly quantitative. Beginning students in these fields are encouraged, if not required, to take a course or two in econometrics—a field of study that has become quite popular. This chapter gives the beginner an overview of what econometrics is all about.
1.1 WHAT IS ECONOMETRICS? Simply stated, econometrics means economic measurement. Although quantitative measurement of economic concepts such as the gross domestic product (GDP), unemployment, inflation, imports, and exports is very important, the scope of econometrics is much broader, as can be seen from the following definitions: Econometrics may be defined as the social science in which the tools of economic theory, mathematics, and statistical inference are applied to the analysis of economic phenomena.1 1Arthur S. Goldberger, Econometric Theory, Wiley, New York, 1964, p. 1.
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Econometrics, the result of a certain outlook on the role of economics, consists of the application of mathematical statistics to economic data to lend empirical support to the models constructed by mathematical economics and to obtain numerical results.2
1.2 WHY STUDY ECONOMETRICS? As the preceding definitions suggest, econometrics makes use of economic theory, mathematical economics, economic statistics (i.e., economic data), and mathematical statistics. Yet, it is a subject that deserves to be studied in its own right for the following reasons. Economic theory makes statements or hypotheses that are mostly qualitative in nature. For example, microeconomic theory states that, other things remaining the same (the famous ceteris paribus clause of economics), an increase in the price of a commodity is expected to decrease the quantity demanded of that commodity. Thus, economic theory postulates a negative or inverse relationship between the price and quantity demanded of a commodity—this is the widely known law of downwardsloping demand or simply the law of demand. But the theory itself does not provide any numerical measure of the strength of the relationship between the two; that is, it does not tell by how much the quantity demanded will go up or down as a result of a certain change in the price of the commodity. It is the econometrician’s job to provide such numerical estimates. Econometrics gives empirical (i.e., based on observation or experiment) content to most economic theory. If we find in a study or experiment that when the price of a unit increases by a dollar the quantity demanded goes down by, say, 100 units, we have not only confirmed the law of demand, but in the process, we have also provided a numerical estimate of the relationship between the two variables—price and quantity. The main concern of mathematical economics is to express economic theory in mathematical form or equations (or models) without regard to measurability or empirical verification of the theory. Econometrics, as noted earlier, is primarily interested in the empirical verification of economic theory. As we will show shortly, the econometrician often uses mathematical models proposed by the mathematical economist but puts these models in forms that lend themselves to empirical testing.
2P. A. Samuelson, T. C. Koopmans, and J. R. N. Stone, “Report of the Evaluative Committee for Economet-
rica,” Econometrica, vol. 22, no. 2, April 1954, pp. 141–146.
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The Nature and Scope of Econometrics
Economic statistics is mainly concerned with collecting, processing, and presenting economic data in the form of charts, diagrams, and tables. This is the economic statistician’s job. He or she collects data on the GDP, employment, unemployment, prices, and so on. These data constitute the raw data for econometric work. But the economic statistician does not go any further because he or she is not primarily concerned with using the collected data to test economic theories. Although mathematical statistics provides many of the tools employed in the trade, the econometrician often needs special methods because of the unique nature of most economic data, namely, that the data are not usually generated as the result of a controlled experiment. The econometrician, like the meteorologist, generally depends on data that cannot be controlled directly. Thus, data on consumption, income, investments, savings, prices, and so on, which are collected by public and private agencies, are nonexperimental in nature. The econometrician takes these data as given. This creates special problems not normally dealt with in mathematical statistics. Moreover, such data are likely to contain errors of measurement, of either omission or commission, and the econometrician may be called upon to develop special methods of analysis to deal with such errors of measurement. For students majoring in economics and business, there is a pragmatic reason for studying econometrics. After graduation, in their employment, they may be called upon to forecast sales, interest rates, and money supply or to estimate demand and supply functions or price elasticities for products. Quite often, economists appear as expert witnesses before federal and state regulatory agencies on behalf of their clients or the public at large. Thus, an economist appearing before a state regulatory commission that controls prices of gas and electricity may be required to assess the impact of a proposed price increase on the quantity demanded of electricity before the commission will approve the price increase. In situations like this, the economist may need to develop a demand function for electricity for this purpose. Such a demand function may enable the economist to estimate the price elasticity of demand, that is, the percentage change in the quantity demanded for a percentage change in the price. Knowledge of econometrics is very helpful in estimating such demand functions. It is fair to say that econometrics has become an integral part of training in economics and business. It may be added the technics and methods developed in econometrics have found uses in several other areas of social sciences, in politics and international relations, in agricultural and medical sciences, as some of the examples discussed in this book will reveal as we progress through the book.
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1.3 THE METHODOLOGY OF ECONOMETRICS How does one actually do an econometric study? Broadly speaking, econometric analysis proceeds along the following lines. 1. The object of research 2. Collecting data 3. Specifying the mathematical model of theory 4. Specifying the statistical, or econometric, model of theory 5. Estimating the parameters of the chosen econometric model 6. Checking for model adequacy: model specification testing 7. Testing hypotheses derived from the model 8. Using the model for prediction or forecasting To illustrate the methodology, consider this question: Do economic conditions affect people’s decisions to enter the labor force, that is, their willingness to work? As a measure of economic conditions, suppose we use the unemployment rate (UNR), and as a measure of labor force participation, we use the labor force participation rate (LFPR). Data on UNR and LFPR are regularly published by the government. So to answer the question, we proceed as follows.
1. The Object of Research The starting point is to find out what economic theory has to say on the subject you want to study. In labor economics, there are two rival hypotheses about the effect of economic conditions on people’s willingness to work. The discouraged-worker hypothesis (effect) states that when economic conditions worsen, as reflected in a higher unemployment rate, many unemployed workers give up hope of finding a job and drop out of the labor force. On the other hand, the added-worker hypothesis (effect) maintains that when economic conditions worsen, many secondary workers who are not currently in the labor market (e.g., mothers with children) may decide to join the labor force if the main breadwinner in the family loses his or her job. Even if the jobs these secondary workers get are low paying, the earnings will make up some of the loss in income suffered by the primary breadwinner. Whether, on balance, the labor force participation rate will increase or decrease will depend on the relative strengths of the added-worker and discouraged-worker effects. If the added-worker effect dominates, LFPR will increase even when the unemployment
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rate is high. Contrarily, if the discouraged-worker effect dominates, LFPR will decrease. How do we find this out? This now becomes our empirical question.
2. Collecting Data For empirical purposes, therefore, we need quantitative information on the two variables. There are three types of data that are generally available for empirical analysis. 1. Time series 2. Cross-sectional 3. Pooled (a combination of time series and cross-sectional) Times-series data are collected over a period of time, such as the data on GDP, employment, unemployment, money supply, or government deficits. Such data may be collected at regular intervals—daily (e.g., stock prices), weekly (e.g., money supply), monthly (e.g., the unemployment rate), quarterly (e.g., GDP), or annually (e.g., government budget). So-called high-frequency data are collected over an extremely short-period time, such as seconds and minutes. In flash trading in stock and foreign exchange markets, such high-frequency data have now become common. These data may be quantitative in nature (e.g., prices, income, money supply) or qualitative (e.g., male or female, employed or unemployed, married or unmarried, White or Black). As we will show, qualitative variables, also called dummy or categorical variables, can be every bit as important as quantitative variables. Since successive observations in time-series data may be correlated, they pose special problems for regressions involving time-series data, particularly the problem of autocorrelation, a topic we discuss at length in Chapter 10 with appropriate examples. Time-series data pose another problem, namely, that they may not be stationary. Loosely speaking, a time series is stationary if its mean and variance do not vary systematically over time. In Chapter 11 on time-series econometrics, we examine the nature of stationary and nonstationary time series and show the special statistical problems created by the latter. If we are dealing with time-series data, we will denote the observations subscript by t (e.g., Yt, Xt). Cross-sectional data are data on one or more variables collected at one point in time, such as the census of population conducted by the U.S. Census Bureau every 10 years (the most recent was on April 1, 2010; the results of the 2020 census are not yet available at the time of writing); the surveys of consumer expenditures conducted by the University of Michigan; and the opinion polls such as those conducted by Gallup, Harris, and other polling organizations. Like time-series data, cross-sectional
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data have their particular problems, particularly the problem of heterogeneity. For example, if you collect data on executive salaries in a given industry at the same point in time, heterogeneity arises because the data may contain small-, medium-, and largesize companies with their own management style and policies. In Chapter 5, we show how the size or scale effect of heterogeneous companies can be taken into account. In pooled data, we have elements of both time-series and cross-sectional data. For example, if we collect data on the unemployment rate for 10 countries for a period of 20 years, the data will constitute an example of pooled data—data on the unemployment rate for each country for the 20-year period will form time-series data, whereas data on the unemployment rate for the 10 countries for any single year will be crosssectional data. In pooled data, we will have 200 observations—20 annual observations for each of the 10 countries. There is a special type of pooled data called panel data, also called longitudinal or micropanel data, in which the same cross-sectional unit, say, a family or firm, is surveyed over time. For example, the U.S. Department of Commerce conducts a census of housing at periodic intervals. At each periodic survey, the same household (or the people living at the same address) is interviewed to find out if there has been any change in the housing and financial conditions of that household since the last survey. The panel data that result from repeatedly interviewing the same household at periodic intervals provide very useful information on the dynamics of household behavior. We denote panel data by the double subscript it. Thus, Yit will denote the (crosssectional) observation for the ith unit at time t. Quality of the Data. The researcher must check carefully the reputation of the agency that collects the data, for very often the data contain errors of measurement, errors of omission of some observations, or errors of systematic rounding and the like. Data collected in public polls or in marketing surveys may be biased because of nonresponse or incomplete response from the participants. Sometimes the data are available only at a highly aggregated, or macro, level, which may not tell us much about the individual entities included in the aggregate. It should always be kept in mind that the results of research are only as good as the quality of the data. Since an individual researcher does not have the luxury of collecting data on their own, very often they have to depend on secondary sources. But every effort must be made to check the quality of the data used in empirical analysis. Data Revisions. Macro data on variables such as GDP, consumer price index (CPI), and other economic variables are often revised upward or downward as initially published data may be tentative. It behooves the researcher to keep track of the revised data.
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Not only that, macro and micro economic data are often “jolted” by unusual events, such as the great recession of 2008 and the following several years, which was triggered by the collapse of the housing market boon that was set in motion by the subpar loans that were given by real estate brokers and banks. This collapse spilled over into the stock market. The severe recession that started in the United States very quickly spread across the globe, so such unusual events should be taken into account in analyzing economic data. A startling example is the coronavirus disease 2019 (COVID-19) pandemic that started in one country in March 2019 and quickly spread to other countries, with devastating effects on their economies. In the United States, according to the U.S. Centers for Disease Control and Prevention, as of March 29, 2021, the total number of COVID-19 cases was 30,085,827 and the total number of deaths was 546,704. The long-term consequences of COVID-19 have yet to be assessed. So doing econometric analysis in such situations such as this is very challenging, to say the least. Sources of the Data. A word is in order regarding data sources. The success of any econometric study hinges on the quality as well as the quantity of data. Fortunately, the Internet has opened up a veritable wealth of data. In Appendix 1A, we give addresses of several websites that have all kinds of microeconomic and macroeconomic data. Students should be familiar with such sources of data, as well as how to access or download them. Of course, these data are continually updated so the reader should find the latest available data. Data From Statistical Packages. Statistical packages, such as EViews, Stata, Minitab, and SAS, have data sets for expository purposes. The Federal Reserve Bank of St. Louis has extensive data on several macroeconomic variables in Excel format that can be directly imported into Eviews (http://research.stlouisfed.org/ fred-addin), and FRED economic data are extremely useful for empirical research. Stata can also import FRED data in Stata format by issuing the command findit Freduse while you use Stata. For our analysis, we obtained the time-series data shown in Table 1-1 of the book's website. This table gives data on the civilian labor force participation rate (CLFPR) and the civilian unemployment rate (CUNR), defined as the number of civilians unemployed as a percentage of the civilian labor force, for the United States for the period 1980–2007.3 The data beyond this period are given in Problem 1.10 (see Table 1-2 found on the book’s website). 3We
consider here only the aggregate CLFPR and CUNR, but data are available by age, sex, and ethnic composition.
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Unlike physical sciences, most data collected in economics (e.g., GDP, money supply, Dow Jones index, car sales) are nonexperimental in that the data-collecting agency (e.g., government) may not have any direct control over the data. Thus, the data on labor force participation and unemployment are based on the information provided to the government by participants in the labor market. In a sense, the government is a passive collector of these data and may not be aware of the added- or discouragedworker hypotheses, or any other hypothesis, for that matter. Therefore, the collected data may be the result of several factors affecting the labor force participation decision made by the individual person. That is, the same data may be compatible with more than one theory.
3. Specifying the Mathematical Model of Labor Force Participation To see how CLFPR behaves in relation to CUNR, the first thing we should do is plot the data for these variables in a scatter diagram, or scattergram, as shown in Figure 1-1. The scattergram shows that CLFPR and CUNR are inversely related, perhaps suggesting that, on balance, the discouraged-worker effect is stronger than the addedworker effect.4 As a first approximation, we can draw a straight line through the scatter
FIGURE 1-1
R egression plot for civilian labor force participation rate (%) and civilian unemployment rate (%) Fitted Line Plot
67.5 67.0
CLFPR (%)
66.5 66.0 65.5 65.0 64.5 64.0 63.5 3.5
4.5
5.5
6.5 7.5 CUNR (%)
8.5
9.5
10.5
4On this, see Shelly Lundberg, “The Added Worker Effect,” Journal of Labor Economics, vol. 3, January 1985,
pp. 11–37.
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points and write the relationship between CLFPR and CUNR by the following simple mathematical model:
CLFPR = B1 + B2 CUNR (1.1)
Equation (1.1) states that CLFPR is linearly related to CUNR. B1 and B2 are known as the parameters of the linear function.5 B1 is also known as the intercept; it gives the value of CLFPR when CUNR is zero.6 B2 is known as the slope. The slope measures the rate of change in CLFPR for a unit change in CUNR or, more generally, the rate of change in the value of the variable on the left-hand side of the equation for a unit change in the value of the variable on the right-hand side. The slope coefficient B2 can be positive (if the added-worker effect dominates the discouraged-worker effect) or negative (if the discouraged-worker effect dominates the added-worker effect). Figure 1-1 suggests that in the present case, it is negative.
4. Specifying the Statistical, or Econometric, Model of Labor Force Participation The purely mathematical model of the relationship between CLFPR and CUNR given in Equation (1.1), although of prime interest to the mathematical economist, is of limited appeal to the econometrician, for such a model assumes an exact, or deterministic, relationship between the two variables; that is, for a given CUNR, there is a unique value of CLFPR. In reality, one rarely finds such neat relationships between economic variables. Most often, the relationships are inexact, or statistical, in nature. This is seen clearly from the scattergram given in Figure 1-1. Although the two variables are inversely related, the relationship between them is not perfectly or exactly linear, for if we draw a straight line through the 28 data points, not all the data points will lie exactly on that straight line. Recall that to draw a straight line, we need only two points.7 Why don’t the 28 data points lie exactly on the straight line specified by the mathematical model, Equation (1.1)? Remember that our data on labor force and unemployment are nonexperimentally collected. Therefore, as noted earlier, besides the added- and discouraged-worker hypotheses, there may be other forces affecting labor force participation decisions. As a result, the observed relationship between CLFPR and CUNR is likely to be imprecise. 5Broadly speaking, a parameter is an unknown quantity that may vary over a certain set of values. In statistics, a probability distribution function (PDF) of a random variable is often characterized by its parameters, such as its mean and variance. This topic is discussed in greater detail in Appendixes A and B. 6In Chapter 2, we give a more precise interpretation of the intercept in the context of regression analysis. 7 We even tried to fit a parabola to the scatter points given in Figure 1-1, but the results were not materially different from the linear specification.
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Let us allow for the influence of all other variables affecting CLFPR in a catchall variable u and write Equation (1.2) as follows:
CLFPR = B1 + B2CUNR + u (1.2)
where u represents the random error term, or simply the error term.8 We let u represent all those forces (besides CUNR) that affect CLFPR but are not explicitly introduced in the model, as well as purely random forces. As we will see in Part II, the error term distinguishes econometrics from purely mathematical economics. Equation (1.2) is an example of a statistical, or empirical or econometric, model. More precisely, it is an example of what is known as a linear regression model, which is a prime subject of this book. In such a model, the variable appearing on the left-hand side of the equation is called the dependent variable, and the variable on the right-hand side is called the independent, or explanatory, variable. In linear regression analysis, our primary objective is to explain the behavior of one variable (the dependent variable) in relation to the behavior of one or more other variables (the explanatory variables), allowing for the fact that the relationship between them is inexact. Notice that the econometric model, Equation (1.2), is derived from the mathematical model, Equation (1.1), which shows that mathematical economics and econometrics are mutually complementary disciplines. This is clearly reflected in the definition of econometrics given at the outset. Before proceeding further, a warning regarding causation is in order. In the regression model, Equation (1.2), we have stated that CLFPR is the dependent variable and CUNR is the independent, or explanatory, variable. Does that mean that the two variables are causally related; that is, is CUNR the cause and CLFPR the effect? In other words, does regression imply causation? Not necessarily. As Kendall and Stuart note, “A statistical relationship, however strong and however suggestive, can never establish causal connection: our ideas of causation must come from outside statistics, ultimately from some theory or other.”9 In our example, it is up to economic theory (e.g., the discouraged-worker hypothesis) to establish the cause-and-effect relationship, if any, between the dependent and explanatory variables. If causality cannot be established, it is better to call the relationship, Equation (1.2), a predictive relationship: Given CUNR, can we predict CLFPR?
8In statistical lingo, the random error term is known as the stochastic error term. 9M.
G. Kendall and A. Stuart, The Advanced Theory of Statistics, Charles Griffin, New York, 1961, vol. 2, chap. 26, p. 279.
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5. Estimating the Parameters of the Chosen Econometric Model Given the data on CLFPR and CUNR, such as that in Table 1-1, how do we estimate the parameters of the model, Equation (1.2), namely, B1 and B2? That is, how do we find the numerical values (i.e., estimates) of these parameters? This will be the focus of our attention in Part II, where we develop the appropriate methods of computation, especially the method of ordinary least squares (OLS). Using OLS and the data given in Table 1-1, we obtained the following results:
= 69.4620 − 0.5814CUNR (1.3) CLFPR
Note that we have put the symbol Λ on CLFPR (read as “CLFPR hat”) to remind us that Equation (1.3) is an estimate of Equation (1.2). The estimated regression line is shown in Figure 1-1, along with the actual data points. As Equation (1.3) shows, the estimated value of B1 is ≈ 69.5 and that of B2 is ≈ –0.58, where the symbol ≈ means approximately. Thus, if the unemployment rate goes up by one unit (i.e., one percentage point), ceteris paribus, CLFPR is expected to decrease on the average by about 0.58 percentage points; that is, as economic conditions worsen, on average, there is a net decrease in the labor force participation rate of about 0.58 percentage points, perhaps suggesting that the discouraged-worker effect dominates. We say “on the average” because the presence of the error term u, as noted earlier, is likely to make the relationship somewhat imprecise. This is vividly seen in Figure 1-1, where the points not on the estimated regression line are the actual participation rates and the (vertical) distance between them and the points on the regression line are the estimated us. As we will see in Chapter 2, the estimated us are called residuals. In short, the estimated regression line, Equation (1.3), gives the relationship between average CLFPR and CUNR, that is, on average, how CLFPR responds to a unit change in CUNR. The value of about 69.5 suggests that the average value of CLFPR will be about 69.5% if the CUNR is zero; that is, about 69.5% of the civilian working-age population will participate in the labor force if there is full employment (i.e., zero unemployment).10
6. Checking for Model Adequacy: Model Specification Testing How adequate is our model, Equation (1.3)? It is true that a person will take into account labor market conditions as measured by, say, the unemployment rate before entering the labor market. For example, in 1982 (a recession year), the civilian unemployment rate was about 9.7%. Compared to that, in 2001, it was only 4.7%. A person 10This is, however, a mechanical interpretation of the intercept. We will see in Chapter 2 how to interpret the
intercept term meaningfully in a given context.
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is more likely to be discouraged from entering the labor market when the unemployment rate is more than 9% than when it is 5%. But other factors also enter into labor force participation decisions. For example, hourly wages, or earnings, prevailing in the labor market also will be an important decision variable. In the short run at least, a higher wage may attract more workers to the labor market, other things remaining the same (ceteris paribus). To see its importance, in Table 1-1, we have also given data on real average hourly earnings (AHE82), where real earnings are measured in 1982 dollars. To take into account the influence of AHE82, we now consider the following model:
= B + B CUNR + B AHE82 + u (1.4) CLFPR 1 2 3
Equation (1.4) is an example of a multiple linear regression model, in contrast to Equation (1.2), which is an example of a simple (two-variable or bivariate) linear regression model. In the two-variable model, there is a single explanatory variable, whereas in a multiple regression, there are several, or multiple, explanatory variables. Notice that in the multiple regression, Equation (1.4), we also have included the error term, u, for no matter how many explanatory variables one introduces in the model, one cannot fully explain the behavior of the dependent variable. How many variables one introduces in the multiple regression is a decision that the researcher will have to make in a given situation. Of course, the underlying economic theory will often tell what these variables might be. However, keep in mind the warning given earlier that regression does not mean causation; the relevant theory must determine whether one or more explanatory variables are related to the dependent variable. How do we estimate the parameters of the multiple regression, Equation (1.4)? We cover this topic in Chapter 4, after we discuss the two-variable model in Chapters 2 and 3. We consider the two-variable case first because it is the building block of the multiple regression model. As we shall see in Chapter 4, the multiple regression model is in many ways a straightforward extension of the two-variable model. For our illustrative example, the empirical counterpart of Equation (1.4) is as follows (these results are based on OLS):
CLFPR = 81.2267 − 0.6384CUNR − 1.4449 AHE 82 (1.5)
These results are interesting because both the slope coefficients are negative. The negative coefficient of CUNR suggests that, ceteris paribus (i.e., holding the influence of AHE82 constant), a one-percentage-point increase in the unemployment rate leads, on average, to about a 0.64-percentage-point decrease in CLFPR, perhaps once again supporting the discouraged-worker hypothesis. On the other hand, holding the
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influence of CUNR constant, an increase in real average hourly earnings of one dollar, on average, leads to about a 1.44-percentage-point decline in CLFPR.11 Does the negative coefficient for AHE82 make economic sense? Would one not expect a positive coefficient—the higher the hourly earnings, the higher the attraction of the labor market? However, one could justify the negative coefficient by recalling the twin concepts of microeconomics, namely, the income effect and the substitution effect.12 Which model do we choose, Equation (1.3) or Equation (1.5)? Since Equation (1.5) encompasses Equation (1.3) and adds an additional dimension (earnings) to the analysis, we may choose Equation (1.5). After all, Equation (1.2) was based implicitly on the assumption that variables other than the unemployment rate were held constant. But where do we stop? For example, labor force participation may also depend on family wealth, number of children under age 6 (this is especially critical for married women thinking of joining the labor market), availability of daycare centers for young children, religious beliefs, availability of welfare benefits, unemployment insurance, and so on. Even if data on these variables are available, we may not want to introduce them all in the model because the purpose of developing an econometric model is not to capture total reality but just its salient features. If we decide to include every conceivable variable in the regression model, the model will be so unwieldy that it will be of little practical use. The model ultimately chosen should be a reasonably good replica of the underlying reality, but keeping in mind the principle of parsimony or Ockham’s razor. William Ockham (1285–1349), an English philosopher, held that complicated explanation should not be accepted without good reason, or as he put it, “It is vain to do with more what can be done with less.” In Chapter 7, we will discuss this question further and find out how one can go about developing a model.
7. Testing Hypotheses Derived From the Model Having finally settled on a model, we may want to perform hypothesis testing. That is, we may want to find out whether the estimated model makes economic sense and whether the results obtained conform with the underlying economic theory. For example, the discouraged-worker hypothesis postulates a negative relationship between labor force participation and the unemployment rate. Is this hypothesis borne out by our results? Our statistical results seem to be in conformity with this hypothesis because the estimated coefficient of CUNR is negative. 11 As
we will discuss in Chapter 4, the coefficients of CUNR and AHE82 given in Equation (1.5) are known as partial regression coefficients. In that chapter, we will discuss the precise meaning of partial regression coefficients.
12Consult
any standard textbook on microeconomics. One intuitive justification of this result is as follows. Suppose both spouses are in the labor force and the earnings of one spouse rise substantially. This may prompt the other spouse to withdraw from the labor force without substantially affecting the family income.
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However, hypothesis testing can be complicated. In our illustrative example, suppose someone told us that in a prior study, the coefficient of CUNR was found to be about –1. Are our results in agreement? If we rely on the model, Equation (1.3), we might get one answer, but if we rely on Equation (1.5), we might get another answer. How do we resolve this question? Although we will develop the necessary tools to answer such questions, we should keep in mind that the answer to a particular hypothesis may depend on the model we finally choose. The point worth remembering is that in regression analysis, we may be interested not only in estimating the parameters of the regression model but also in testing certain hypotheses suggested by economic theory and/or prior empirical experience. Although the basic principles of hypothesis testing are covered in a basic course in statistics, Appendix D discusses this topic at some length for the benefit of the reader as a refresher course.
8. Using the Model for Prediction or Forecasting Having gone through this multistage procedure, you can legitimately ask the following question: What do we do with the estimated model, such as Equation (1.5)? Quite naturally, we would like to use it for prediction, or forecasting. For instance, suppose we have 2008 data on the CUNR and AHE82. Assume these values are 6.0 and 10, respectively. If we put these values in Equation (1.5), we obtain 62.9473% as the predicted value of CLFPR for 2008. That is, if the unemployment rate in 2008 were 6.0% and the real hourly earnings were $10, the civilian labor force participation rate for 2008 would be about 63%. Of course, when data on CLFPR for 2008 actually become
TABLE 1-3
Summary of the Steps Involved in Econometric Analysis
Step
Example
1. Statement of theory
The added/discouraged-worker hypothesis
2. Collection of data
Table 1-1
3. Mathematical model of theory
CLFPR = B1 + B2CUNR
4. Econometric model of theory
CLFPR = B1 + B2CUNR + u
5. Parameter estimation
CLFPR = 69.462 – 0.5814CUNR
6. Model adequacy check
CLFPR = 81.3 – 0.638CUNR – 1.445AHE82
7. Hypothesis test
B2 < 0 or B2 > 0
8. Prediction
What is CLFPR, given values of CUNR and AHE82?
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The Nature and Scope of Econometrics
available, we can compare the predicted value with the actual value (see Problem 1.10). The discrepancy between the two will represent the prediction error. Naturally, we would like to keep the prediction error as small as possible. Although we examined econometric methodology using an example from labor economics, we should point out that a similar procedure can be employed to analyze quantitative relationships between variables in any field of knowledge. As a matter of fact, regression analysis has been used in politics, international relations, psychology, sociology, meteorology, and many other disciplines. As an example, see Problem 1.9.
1.4 THE ROAD AHEAD Now that we have provided a glimpse of the nature and scope of econometrics, let us see what lies ahead. The book is divided into four parts. Part I introduces the reader to the bread-and-butter tool of econometrics, namely, the classical linear regression model (CLRM). A thorough understanding of CLRM is a must in order to follow research in the general areas of economics and business. Part II considers the practical aspects of regression analysis and discusses a variety of problems that the practitioner will have to tackle when one or more assumptions of the CLRM do not hold. Part III discusses two comparatively advanced topics, time-series econometrics and panel data regression models. Part IV, consisting of Appendixes A, B, C, and D, reviews the basics of probability and statistics for the benefit of those readers whose knowledge of statistics has become rusty. The reader should have some previous background in introductory statistics. This book keeps the needs of the beginner in mind. The discussion of most topics is straightforward and unencumbered with mathematical proofs, derivations, and so on.13 I firmly believe that the apparently forbidding subject of econometrics can be taught to beginners in such a way that they can see the value of the subject without getting bogged down in mathematical and statistical minutiae. The student should keep in mind that an introductory econometrics course is just like the introductory statistics course he or she has already taken. As in statistics, econometrics is primarily about estimation and hypothesis testing. What is different, and generally much more interesting and useful, is that the parameters being estimated or tested are not
13Some
of the proofs and derivations are presented in our Basic Econometrics, 5th ed., McGraw-Hill, New York, 2009. A more mathematical treatment is given in Damodar N. Gujarati, Linear Regression: A Mathematical Introduction, SAGE, Los Angeles, 2018.
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just means and variances but relationships between variables, which is what much of economics and other social sciences is all about. A final word: The availability of comparatively inexpensive computer software packages has now made econometrics readily accessible to beginners. In this book, we will largely use four software packages: EViews, Excel, STATA, and MINITAB. These packages are readily available and widely used. Once students get used to such packages, they will soon realize that learning econometrics is really great fun, and they will have a better appreciation of the much maligned “dismal” science of economics.
KEY TERMS AND CONCEPTS The key terms and concepts introduced in this chapter, and page numbers where they are referenced, are as follows: Econometrics 1 Mathematical economics 2 Discouraged-worker hypothesis (effect) 4 Added-worker hypothesis (effect) 4 Time-series data: Quantitative and qualitative 5 High-frequency data 5 Flash trading 5 Autocorrelation 5 Stationary 5
Cross-sectional data 5 Heterogeneity 6 Size or scale effect 6 Pooled data 6 Panel (or longitudinal or micropanel data) 6 Scatter diagram (scattergram) 8 Parameters: Intercept and slopes 9 Random error term (error term) 10
Linear regression model: Dependent variable, independent (or explanatory) variable 10 Causation 10 Parameter estimates 11 Principle of parsimony or Ockham’s razor 13 Hypothesis testing 16 Prediction (forecasting) 16
QUESTIONS 1.1. Suppose a local government decides to increase the tax rate on residential properties under its jurisdiction. What will be the effect of this on the prices of residential houses? Follow the eightstep procedure discussed in the text to answer this question. 1.2. How do you perceive the role of econometrics in decision making in business and government?
1.3. Suppose you are an economic adviser to the chairman of the Federal Reserve Board (the Fed), and he asks you whether it is advisable to increase the money supply to bolster the economy. What factors would you take into account in your advice? How would you use econometrics in your advice? 1.4. To reduce the dependence on foreign oil supplies, the government is thinking of
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increasing the federal taxes on gasoline. Suppose the Ford Motor Company has hired you to assess the impact of the tax increase on the demand for its cars. How would you go about advising the company? 1.5. President Joe Biden plans to propose to the U.S. Congress an infrastructure investment plan (highways, bridges, tunnels, etc.) at
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The Nature and Scope of Econometrics
a cost of about $2 trillion. To pay for this, he also plans to increase the tax rate on high-income earners as well as private corporations, although the details are yet to be worked out. How would you design an econometric study to assess the economic consequences, both short term and long term, of his proposal?
PROBLEMS 1.6. Table 1-4 on the book's website gives monthly data on the closing prices of the Dow Jones Industrial Average and the Standard & Poor's 500 stock market indexes. The data are from Yahoo Finance's historical stock quotations page. a. Plot these data with time on the horizontal axis and the two variables on the vertical axis. If you prefer, you may use a separate figure for each variable. b. What relationships do you expect to find between the two indexes? Why? c. For each variable, “eyeball” a regression line from the scattergram. d. Obtain monthly data for the two variables for the period from January 2012 to December 2020 and repeat questions a, b, and c and find out if there are any changes in the results. If so, what might account for the change? 1.7. Table 1-5 on the book's website gives data on the exchange rate between the U.K. pound and the U.S. dollar (number of U.K. pounds per U.S. dollar), as well as the consumer price indexes in the two countries for the period 1985–2007.
a. Plot the exchange rate (ER) and the two consumer price indexes against time, measured in years. b. Divide the U.S. CPI by the U.K. CPI and call it the relative price ratio (RPR). c. Plot ER against RPR. d. Visually sketch a regression line through the scatter points. e. Update the data in Table 1-5 to year 2020. Repeat questions a, b, c, and d and find out if there is any changes in the results. What accounts for the change, if any, in the results? 1.8. Table 1-6 on the textbook website contains data on 1,247 cars for 2008.14 To find out if there is there a relationship between a car’s MPG (miles per gallon) and the number of cylinders it has: a. Create a scatterplot of the combined MPG for the vehicles based on the number of cylinders. b. Sketch a line that seems to fit the data. c. What type of relationship is indicated by the plot?
14Data were collected from the U.S. Department of Energy website at http://www.fueleconomy.gov/.
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1.9. Table 1-7 on the book’s website gives data on Corruption Perception Index and GDP per worker. a. Plot Corruption Perception Index against GDP per worker. b. A priori, what kind of relationship do you expect between the two variables? c. Does the scattergram suggest that the relationship between the two variables is linear (i.e., a straight line)? If so, sketch the regression line. 1.10. Table 1-2 on the website updates the data given in Table 1-1 for the years 2001–2016. For the years 2001–2007, the CLFR and CUNR figures are the same as those shown in Table 1-1. However, the AHE82 figures differ in the two periods. As pointed out in the text, the differences are usually due to data revisions. a. Plot CLFR and CUNR as in Figure 1-1. What difference due you see in the two scattergrams? b. Is the relationship between the variables linear as in Figure 1-1? If so, visually sketch a regression line through the scatterplot. c. Is there a “break” in the data in the sense that after a certain date, the relationship between the two variables has changed? Can you spot that break point? d. Based on the data in Table 1-2, the regression results corresponding to Equation (1.3) are as follows: ∧
CLFR = 66.5245 − 0.2465CUNR How does t his regression differ from the one shown in Equation (1.3)? What may be the reason for the difference? e. The regression results corresponding to Equation (1.5) using the data in Table 1-2 are as follows: ∧
CLF PR = 1121761+ 0.0150CUNR − 5.4385 AHE 82 15Paul
How does this regression differ from the one shown in Equation (1.5)? What might explain the difference between the two regression results? Note: The full results of the preceding two regressions will be discussed in Chapter 3 after we discuss the theory behind regression analysis. 1.11. Table 1-8 on the book's website gives quarterly data on real personal consumption expenditure (RPCE) and real personal disposable (after-tax) income (RPDI) for the years 2014–2019. a. Plot RPCE and RPDI on the same graph. What is your impression about the two time series? b. Graph RPCE against RPDI. What does the scattergram show? c. Visually sketch a regression line through the scatter points. What does it show? d. Save the data for further analysis in subsequent chapters. 1.12. Based on the data for 1947–2002, Kellsted and Whitten obtained the following regression:15 Mt = 74.00 – 2.71GDPt where M = percentage of households in which a married couple is present and GDP = gross domestic product. a. Does this result make sense? b. How would you interpret the regression? c. Is there a cause-and-effect relationship between the two variables? d. The regression results give above may be an example of what is called spurious or nonsense regression. We may have more to say about it in a later chapter. 1.13. Table 1.9 on the book's website gives data on the following variables for 99 countries
M. Kellstedt and Guy D. Whitten, The Fundamentals of Political Science Research, Cambridge University Press, 2nd ed., New York, 2013, p. 262.
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The Nature and Scope of Econometrics
obtained from the Human Development Report for 1994.
a. Plot life expectancy against each of the other variables in separate graphs.
LifeExp = 1992 life expectancy at birth
b. A priori, what do you expect the relationship is between LifeExp and each of the other variables: positive, negative, or no relationship?
TV = Televisions per 100 people PopDoc = Population per doctor (1990) GDP = real GDP per person adjusted for PPP (purchasing power parity)
SUGGESTIONS FOR FURTHER READING “The Usefulness of Applied Econometrics to the Policy Maker,” Address by R. Frances, President, Federal Bank of St. Louis, at the National Association of Business Economist Seminar, Chicago, Illinois, April 4, 1973, Federal Bank of St. Louis, May 1973.
“What Is Econometrics?” International Monetary Fund, Finance and Development, December 2011, vol. 48, No. 4 (https://www.imf.org/extenal/pubs/ft/ famd/2011/12/basics.htm). On corruption, read https://ourworldindata.org/ corruption.
APPENDIX 1A: Economic Data on the World Wide Web16 Economic Statistics Briefing Room: An excellent source of data on output, income, employment, unemployment, earnings, production and business activity, prices and money, credits and security markets, and international statistics.
http://www.nber.org
http://www.whitehouse.gov/fsbr/esbr.htm
http://www.umich.edu/-psid
Federal Reserve System Beige Book: Gives a summary of current economic conditions by the Federal Reserve District. There are 12 Federal Reserve Districts.
The Federal Web Locator: Provides information on almost every sector of the federal government; has international links.
www.federalreserve.gov/FOMC/BeigeBook/2008
www.lib.auburn.edu/madd/docs/fedloc.html
National Bureau of Economic Research (NBER) Home Page: This highly regarded private economic research institute has extensive data on asset prices, labor, productivity, money supply, business cycle indicators, and so on. NBER has many links to other websites.
WebEC: Resources in Economics: A most comprehensive library of economic facts and figures.
Panel Study: Provides data on longitudinal survey of representative sample of U.S. individuals and families. These data have been collected annually since 1968.
www.helsinki.fi/WebEc American Stock Exchange: Information on some 700 companies listed on the second largest stock market.
16It should be noted that this list is by no means exhaustive. The sources listed here are updated continually.
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http://www.amex.com/
http://www.census.gov/
Bureau of Economic Analysis (BEA) Home Page: This agency of the U.S. Department of Commerce, which publishes the Survey of Current Business, is an excellent source of data on all kinds of economic activities.
General Social Survey: Annual personal interview survey data on U.S. households that began in 1972. More than 35,000 have responded to some 2,500 different questions covering a variety of data.
www.bea.gov
www.norc.org/GCS+Website
Business Cycle Indicators: You will find data on about 256 economic time series.
Institute for Research on Poverty: Data collected by nonpartisan and nonprofit university-based research center on a variety of questions relating to poverty and social inequality.
http://www.globalexposure.com/bci.html CIA Publication: You will find the World Fact Book (annual). www.cia.gov/library/publications Energy Information Administration (Department of Energy [DOE]): Economic information and data on each fuel category. http://www.eia.doe.gov/ FRED Database: Federal Reserve Bank of St. Louis publishes historical economic and social data, which include interest rates, monetary and business indicators, exchange rates, and so on.
http://www.ssc.wisc.edu/irp/ Social Security Administration: The official website of the Social Security Administration with a variety of data. http://www.ssa.gov Federal Deposit Insurance Corporation, Bank Data and Statistics http://www.fdic.gov/bank/statistical/ Federal Reserve Board, Economic Research and Data
http://www.stls.frb.org/fred/
http://www.federalreserve.gov/econresdata
International Trade Administration: Offers many web links to trade statistics, cross-country programs, and so on.
U.S. Census Bureau, Home Page
http://www.ita.doc.gov/ STAT-USA Databases: The National Trade Data Bank provides the most comprehensive source of international trade data and export promotion information. It also contains extensive data on demographic, political, and socioeconomic conditions for several countries. http://www.stat-usa.gov/ Bureau of Labor Statistics: The home page contains data related to various aspects of employment, unemployment, and earnings and provides links to other statistical websites. http://stats.bls.gov U.S. Census Bureau Home Page: Prime source of social, demographic, and economic data on income, employment, income distribution, and poverty.
http://www.census.gov U.S. Department of Energy, Energy Information Administration www.eia.doe.gov/overview_hd.html U.S. Department of Health and Human Services, National Center for Health Statistics http://www.cdc.gov/nchs U.S. Department of Housing and Urban Development, Data Sets http://www.huduser.org/datasets/pdrdatas.html U.S. Department of Labor, Bureau of Labor Statistics http://www.bls.gov U.S. Department of Transportation, TranStats http://www.transtats.bts.gov U.S. Department of the Treasury, Internal Revenue Service, Tax Statistics
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The Nature and Scope of Econometrics
http://www.irs.gov/taxstats
http://psidonline.isr.umich.edu
Rockefeller Institute of Government, State and Local Fiscal Data
National Longitudinal Surveys
www.rockinst.org/research/sl_finance
National Association of Home Builders, Economic and Housing Data
American Economic Association, Resources for Economists
http://www.bls.gov/nls/
http://www.rfe.org
http://www.nahb.org/page.aspx/category/ sectionID=113
American Statistical Association, Business and Economic Statistics
National Science Foundation, Division of Science Resources Statistics
www.amstat.org/publications/jbes
http://www.nsf.gov/sbe/srs/
American Statistical Association, Statistics in Sports
Economic Report of the President
http://www.amstat.org/sections/sis/
http://www.gpoaccess.gov/eop/
European Central Bank, Statistics
Various Economic Data Sets
http://www.ecb.int/stats
http://www.economy.com/freelunch/
World Bank, Data and Statistics
The Economist Market Indicators
http://www.worldbank.org/data
http://www.economist.com/markets/indicators
International Monetary Fund, Statistical Topics
Statistical Resources on the Military
http://www.imf.org/external/np/sta/
http://www.lib.umich.edu/govdocs/stmil.html
Penn World Tables
World Economic Indicators
http://pwt.econ.upenn.edu
http://devdata.worldbank.org/
Current Population Survey
Economic Time Series Data
http://www.bls.census.gov/cps/
http://www.economagic.com/
Consumer Expenditure Survey
United Nations Population Division's Annual Estimates and Projections
http://www.bls.gov/cex/ Survey of Consumer Finances
http://unstats.un.org/unsd/default.htm
http://www.federalreserve.gov/pubs/oss/
United Nations Statistics Division-UNdata
City and County Data Book
http://data.un.org/Default.aspx
http://www.census.gov/statab/www/ccdb.html
World Bank Data
Panel Study of Income Dynamics
http://databank.worldbank.org/
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T
he objective of Part I, which consists of five chapters, is to introduce the reader to the “bread-and-butter” tool of econometrics, namely, the linear regression model.
PART I
THE LINEAR REGRESSION MODEL Chapter 2 discusses the basic ideas of linear regression in terms of the simplest possible linear regression model, in particular, the two-variable model. We make an important distinction between the population regression model and the sample regression model and estimate the former from the latter. This estimation is done using the method of least squares, one of the popular methods of estimation.1 Chapter 3 considers hypothesis testing. As in any hypothesis testing in statistics, we try to find out whether the estimated values of the parameters of the regression model are compatible with the hypothesized values of the parameters. We do this hypothesis testing in the context of the classical linear regression model (CLRM). We discuss why the CLRM is used and point out that the CLRM is a useful starting point. In Part II, we will reexamine the assumptions of the CLRM to see what happens to the CLRM if one or more of its assumptions are not fulfilled. Chapter 4 extends the idea of the two-variable linear regression model developed in the previous two chapters to multiple regression models, that is, models having more than one explanatory variable. Although in many ways the multiple regression model is an extension of the two-variable model, there are differences when it comes to interpreting the coefficients of the model and in the hypothesis-testing procedure. The linear regression model, whether two-variable or multivariable, only requires that the parameters of the model be linear; the variables entering the model need not themselves be linear.
1An alternative is the method of maximum likelihood (ML), which we do not discuss in this text because
it is mathematically a bit complex. For an introduction to ML, see Damodar Gujarati, Econometrics by Example, 2nd ed., Palgrave-Macmillan, London, 2015, pp. 25−26.
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Chapter 5 considers a variety of models that are linear in the parameters (or can be made so) but are not necessarily linear in the variables. With several illustrative examples, we point out how and where such models can be used. Often the explanatory variables entering into a regression model are qualitative in nature, such as sex, race, and religion. Chapter 6 shows how such variables can be measured and how they enrich the linear regression model by taking into account the influence of variables that otherwise cannot be quantified. This chapter also considers briefly models in which the dependent variable is also dummy or qualitative. Part I makes an effort to “wed” practice to theory. The availability of user-friendly regression packages allows you to estimate a regression model without knowing much theory, but remember the adage that “a little knowledge is a dangerous thing.” So even though theory may be boring, it is absolutely essential in understanding and interpreting regression results. Besides, by omitting all mathematical derivations, we have made the theory “less boring.”
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