This page intentionally left blank Principles of Econometrics F o u r t h E d i t i o n R.Carter Hill Louisiana State University William E. Griffiths University of Melbourne Guay C. Lim University of Melbourne John Wiley & Sons, Inc. VP & Publisher George Hoffman Acquisitions Editor Lacey Vitetta Project Editor Jennifer Manias Senior Editorial Assistant Emily McGee Content Manager Micheline Frederick Production Editor Amy Weintraub Creative Director Harry Nolan Designer Wendy Lai Senior Illustration Editor Anna Melhorn Associate Director of Marketing Amy Scholz Assistant Marketing Manager Diane Mars Executive Media Editor Allison Morris Media Editor Greg Chaput This book was set in 10/12 Times Roman by MPS Limited, a Macmillan Company, Chennai, India, and printed and bound by Donnelley/Von Hoffmann. The cover was printed by Lehigh-Phoenix. This book is printed on acid-free paper. * 1 Copyright # 2011 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748- 6008, website www.wiley.com/go/permissions. To order books or for customer service, please call 1-800-CALL WILEY (225-5945). Library of Congress Cataloging-in-Publication Data: Hill, R. Carter. Principles of econometrics / R. Carter Hill, William E. Griffiths, Guay C. Lim.—4th ed. p. cm. Includes index. ISBN 978-0-470-62673-3 (hardback) 1. Econometrics. I. Griffiths, William E. II. Lim, G. C. (Guay C.) III. Title. HB139.H548 2011 330.0105195—dc22 2010043316 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Carter Hill dedicates this work to his wife, Melissa Waters Bill Griffiths dedicates this work to JoAnn, Jill, David, Wendy, Nina, and Isabella Guay Lim dedicates this work to Tony Meagher Brief Contents Chapter 1 An Introduction to Econometrics Probability Primer Chapter 2 The Simple Linear Regression Model Chapter 3 Interval Estimation and Hypothesis Testing Chapter 4 Prediction, Goodness-of-Fit, and Modeling Issues Chapter 5 The Multiple Regression Model Chapter 6 Further Inference in the Multiple Regression Model Chapter 7 Using Indicator Variables Chapter 8 Heteroskedasticity Chapter 9 Regression with Time-Series Data: Stationary Variables Chapter 10 Random Regressors and Moment-Based Estimation Chapter 11 Simultaneous Equations Models Chapter 12 Regression with Time-Series Data: Nonstationary Variables Chapter 13 Vector Error Correction and Vector Autoregressive Models Chapter 14 Time-Varying Volatility and ARCH Models Chapter 15 Panel Data Models Chapter 16 Qualitative and Limited Dependent Variable Models Appendix A Mathematical Tools Appendix B Probability Concepts Appendix C Review of Statistical Inference Appendix D Tables Index Preface Principles of Econometrics, 4th edition, is an introductory book for undergraduate students in economics and finance, as well as for first-year graduate students in economics, finance, accounting, agricultural economics, marketing, public policy, sociology, law, and political science. It is assumed that students have taken courses in the principles of economics, and elementary statistics. Matrix algebra is not used, and calculus concepts are introduced and developed in the appendices. A brief explanation of the title is in order. This work is a revision of Principles of Econometrics, 3rd edition, by Hill, Griffiths, and Lim (Wiley, 2008), which was a revision of Undergraduate Econometrics, 2nd edition, by Hill, Griffiths, and Judge (Wiley, 2001). The earlier title was chosen to clearly differentiate the book from other more advanced books by the same authors. We made the title change because the book is appropriate not only for undergraduates, but also for first-year graduate students in many fields, as well as MBA students. Furthermore, naming it Principles of Econometrics emphasizes our belief that econometrics should be part of the economics curriculum, in the same way as the principles of microeconomics and the principles of macroeconomics. Those who have been studying and teaching econometrics as long as we have will remember that Principles of Econo- metrics was the title that Henri Theil used for his 1971 classic, which was also published by John Wiley and Sons. Our choice of the same title is not intended to signal that our book is similar in level and content. Theil’s work was, and remains, a unique treatise on advanced graduate level econometrics. Our book is an introductory-level econometrics text. Book Objectives Principles of Econometrics is designed to give students an understanding of why econo- metrics is necessary, and to provide them with a working knowledge of basic econometric tools so that � They can apply these tools to modeling, estimation, inference, and forecasting in the context of real-world economic problems. � They can evaluate critically the results and conclusions from others who use basic econometric tools. � They have a foundation and understanding for further study of econometrics. � They have an appreciation of the range of more advanced techniques that exist and that may be covered in later econometric courses. The book is not an econometrics cookbook, nor is it in a theorem-proof format. It emphasizes motivation, understanding, and implementation. Motivation is achieved by introducing very simple economic models and asking economic questions that the student can answer. Understanding is aided by lucid description of techniques, clear interpretation, v and appropriate applications. Learning is reinforced by doing, with clear worked examples in the text and exercises at the end of each chapter. Overview of Contents This fourth edition retains the spirit and basic structure of the third edition. Chapter 1 introduces econometrics and gives general guidelines for writing an empirical research paper and for locating economic data sources. The Probability Primer preceding Chapter 2 summarizes essential properties of random variables and their probability distributions, and reviews summation notation. The simple linear regression model is covered in Chapters 2–4, while the multiple regression model is treated in Chapters 5–7. Chapters 8 and 9 introduce econometric problems that are unique to cross-sectional data (heteroskedasticity) and time-series data (dynamic models), respectively. Chapters 10 and 11 deal with random regressors, the failure of least squares when a regressor is endogenous, and instrumental variables estimation, first in the general case, and then in the simultaneous equations model. InChapter12theanalysisoftime-seriesdataisextendedtodiscussionsofnonstationarityand cointegration. Chapter 13 introduces econometric issues specific to two special time-series models, the vector error correction and vector autoregressive models, while Chapter 14 considers the analysis of volatility in data and the ARCH model. In Chapters 15 and 16 we introduce microeconometric models for panel data, and qualitative and limited dependent variables. In appendices A, B, and C we introduce math, probability, and statistical inference concepts that are used in the book. Summary of Changes and New Material This edition includes a great deal of new material, including new examples and exercises using real data, and some significant reorganizations. Important new features include: � Chapter 1 includes a discussion of data types, and sources of economic data on the Internet. Tips on writing a research paper are given up front so that students can form ideas for a paper as the course develops. � The Probability Primer precedes Chapter 2. This primer reviews the concepts of random variables, and how probabilities are calculated given probability density functions.Mathematicalexpectationandrulesofexpectedvaluesaresummarizedfor discrete random variables. These rules are applied to develop the concept of variance and covariance. Calculations of probabilities using the normal distribution are illustrated. � Chapter 2 is expanded to include brief introductions to nonlinear relationships and the concept of an indicator (or dummy) variable. A new section has been added on interpreting a standard error. An appendix has been added on Monte Carlo simulation and is used to illustrate the sampling properties of the least squares estimator. � Estimation and testing of linear combinations of parameters is now included in Chapter 3. An appendix is added using Monte Carlo simulation to illustrate the properties of interval estimators and hypothesis tests. Chapter 4 discusses in detail nonlinear relationships such as the log-log, log-linear, linear-log, and polynomial models. Model interpretations are discussed and examples given, along with an introduction to residual analysis. � The introductory chapter on multiple regression (Chapter 5) now includes material on standard errors for both linear and nonlinear functions of coefficients, and how they are used for interval estimation and hypothesis testing. The treatment of vi P RE F A CE polynomial and log-linear models given in Chapter 4 is extended to the multiple regression model; interaction variables are included and marginal effects are described. An appendix on large sample properties of estimators has been added. � Chapter 6 contains a new section on model selection criteria and a reorganization of material on the F-test for joint hypotheses. � Chapter 7 now deals exclusively with indicator variables. In addition to the standard material, we introduce the linear probability model and treatment effect models, including difference and difference-in-difference estimators. � Chapter 8 has been reorganized so that testing for heteroskedasticity precedes estimation with heteroskedastic errors. A section on heteroskedasticity in the linear probability model has been added. � Chapter 9 on regression with stationary time series data has been restructured to emphasize autoregressive distributed lag models and their special cases: finite distributed lags, autoregressive models, and the AR(1) error model. Testing for serial correlation using the correlogram and Lagrange multiplier tests now precedes estimation. Two new macroeconomic examples, Okun’s law and the Phillips curve, are used to illustrate the various models. Sections on exponential smoothing and model selection criteria have been added, and the section on multiplier analysis has been expanded. � Chapter 10 on endogeneity problems has been streamlined, using real data examples in the body of the chapter as illustrations. New material on assessing instrument strength has been added. An appendix on testing for weak instruments introduces the Stock-Yogo critical values for the Cragg-Donald F-test. A Monte Carlo experiment is included to demonstrate the properties of instrumental variables estimators. � Chapter 11 now includes an appendix describing two alternatives to two-stage least squares: the limited information maximum likelihood and the k-class estimators. The Stock-Yogo critical values for LIML and k-class estimator are provided. Monte Carlo results illustrate the properties of LIML and the k-class estimator. � Chapter 12 now contains a section on the derivation of the short-run error correction model. � Chapter 13 now contains an example and exercise using data which includes the recent global financial crisis. � Chapter 14 now contains a revised introduction to the ARCH model. � Chapter 15 has been restructured to give more prominence to the fixed effects and random effects models. New sections on cluster-robust standard errors and the Hausman-Taylor estimator have been added. � Chapter 16 includes more on post-estimation analysis within choice models. The average marginal effect is explained and illustrated. The ‘‘delta method’’ is used to create standard errors of estimated marginal effects and predictions. An appendix gives algebraic detail on the ‘‘delta method.’’ � Appendix A now introduces the concepts of derivatives and integrals. Rules for derivatives are given, and the Taylor series approximation explained. Both derivatives and integrals are explained intuitively using graphs and algebra, with each in separate sections. � Appendix B includes a discussion and illustration of the properties of both discrete and continuous random variables. Extensive examples are given, including integration techniques for continuous random variables. The change-of-variable technique for deriving the probability density function of a function of a continuous random variable is discussed. The method of inversion for drawing P R E F A C E vii random values is discussed and illustrated. Linear congruential generators for uniform random numbers are described. � Appendix C now includes a section on kernel density estimation. � Brief answers to selected problems, along with all data files, will now be included on the book website at www.wiley.com/college/hill. Computer Supplement Books The following books are offered by John Wiley and Sons as computer supplements to Principles of Econometrics: � Using EViews for Principles of Econometrics, 4th edition, by Griffiths, Hill and Lim [ISBN 978-1-11803207-7 or at www.coursesmart.com]. This supple- mentary book presents the EViews 7.1 [www.eviews.com] software commands required for the examples in Principles of Econometrics in a clear and concise way. It includes many illustrations that are student friendly. It is useful not only for students and instructors who will be using this software as part of their econometrics course, but also for those who wish to learn how to use EViews. � Using Stata for Principles of Econometrics, 4th edition, by Adkins and Hill [ISBN 978-1-11803208-4 or at www.coursesmart.com]. This supplementary book presents the Stata 11.1 [www.stata.com] software commands required for the examples in Principles of Econometrics. It is useful not only for students and instructors who will be using this software as part of their econometrics course, but also for those who wish to learn how to use Stata. Screen shots illustrate the use of Stata’s drop-down menus. Stata commands are explained and the use of ‘‘do-files’’ illustrated. � Using SAS for Econometrics by Hill and Campbell [ISBN 978-1-11803209-1 or at www.coursesmart.com]. This stand-alone book gives SAS 9.2 [www.sas. com] software commands for econometric tasks, following the general outline of Principles of Econometrics. It includes enough background material on econometrics so that instructors using any textbook can easily use this book as a supplement. The volume spans several levels of econometrics. It is suitable for undergraduate students who will use ‘‘canned’’ SAS statistical procedures, and for graduate students who will use advanced procedures as well as direct programming in SAS’s matrix language; the latter is discussed in chapter appendices. � Using Excel for Principles of Econometrics, 4th edition, by Briand and Hill [ISBN 978-1-11803210-7 or at www.coursesmart.com]. This supplement explains how to use Excel to reproduce most of the examples in Principles of Econometrics. Detailed instructions and screen shots are provided explaining both the computations and clarifying the operations of Excel. Templates are developed for common tasks. � Using GRETL for Principles of Econometrics, 4th edition, by Adkins. This free supplement, readable using Adobe Acrobat, explains how to use the freely available statistical software GRETL (download from http://gretl .sourceforge.net). Professor Adkins explains in detail, using screen shots, how to use GRETL to replicate the examples in Principles of Econometrics. The manual is freely available at www.learneconometrics.com/gretl.html. viii P R E F A C E Resources for Students Available at both the book website, www.wiley.com/college/hill, and at the author website, principlesofeconometrics.com, are � Data files � Answers to selected exercises Data Files Data files for the book are provided in a variety of formats at the book website www.wiley .com/college/hill. These include � ASCII format (*.dat). These are text files containing only data. � Definition files (*.def). These are text files describing the data file contents, with a listing of variable names, variable definitions, and summary statistics. � EViews (*.wf1) workfiles for each data file � Excel 2007 (*.xlsx) workbooks for each data file, including variable names in the first row � Stata (*.dta) data files � SAS (*.sas7bdat) data files � GRETL (*.gdt) data files Resources for Instructors For instructors, also available at the website www.wiley.com/college/hill are � An Instructor’s Resources Guide with complete solutions, in both Microsoft Word and *.pdf formats, to all exercises in the text � PowerPoint Presentation Slides � Supplementary exercises with solutions Author Website The authors’ website—principlesofeconometrics.com—includes � Individual data files in each format, as well as Zip files containing data in compressed format � Book errata � Links to other useful websites, including RATS and SHAZAM computer resources for Principles of Econometrics, and tips on writing research papers � Answers to selected exercises � Hints and resources for writing Acknowledgments Several colleagues have helped us improve our book. We owe very special thanks to Genevieve Briand and Gawon Yoon, who have provided detailed and helpful comments on every part of the book. Also, we have benefited from comments made by Christian Kleiber, Daniel Case, Eric Hillebrand, Silvia Golem, Leandro M. Magnusson, Tom Means, Tong Zeng, Michael Rabbitt, Chris Skeels, Robert Dixon, Robert Brooks, Shuang Zhu, Jill Wright, and the many reviewers who have contributed feedback and suggestions over the P R E FA C E ix years. Individuals who have pointed out errors of one sort or another are recognized in the errata listed at principlesofeconometrics.com. Finally, authors Hill and Griffiths want to acknowledge the gifts given to them over the past 40 years by mentor, friend, and colleague George Judge. Neither this book, nor any of the other books in whose writing we have shared, would have ever seen the light of day without his vision and inspiration. R. Carter Hill William E. Griffiths Guay C. Lim x P R E F A C E Contents Preface v Chapter 1 An Introduction to Econometrics 1 1.1 Why Study Econometrics? 1 1.2 What Is Econometrics About? 2 1.2.1 Some Examples 3 1.3 The Econometric Model 4 1.4 How Are Data Generated? 5 1.4.1 Experimental Data 5 1.4.2 Nonexperimental Data 6 1.5 Economic Data Types 6 1.5.1 Time-Series Data 7 1.5.2 Cross-Section Data 8 1.5.3 Panel or Longitudinal Data 8 1.6 The Research Process 9 1.7 Writing An Empirical Research Paper 11 1.7.1 Writing a Research Proposal 11 1.7.2 A Format for Writing a Research Report 11 1.8 Sources of Economic Data 13 1.8.1 Links to Economic Data on the Internet 13 1.8.2 Interpreting Economic Data 14 1.8.3 Obtaining the Data 14 Probability Primer 17 Learning Objectives 17 Keywords 18 P.1 Random Variables 18 P.2 Probability Distributions 19 P.3 Joint, Marginal, and Conditional Probabilities 21 P.3.1 Marginal Distributions 22 P.3.2 Conditional Probability 22 P.3.3 Statistical Independence 23 P.4 A Digression: Summation Notation 24 P.5 Properties of Probability Distributions 26 P.5.1 Expected Value of a Random Variable 26 P.5.2 Conditional Expectation 27 P.5.3 Rules for Expected Values 27 xi P.5.4 Variance of a Random Variable 28 P.5.5 Expected Values of Several Random Variables 30 P.5.6 Covariance Between Two Random Variables 30 P.6 The Normal Distribution 32 P.7 Exercises 34 Chapter 2 The Simple Linear Regression Model 39 Learning Objectives 39 Keywords 40 2.1 An Economic Model 40 2.2 An Econometric Model 43 2.2.1 Introducing the Error Term 46 2.3 Estimating the Regression Parameters 49 2.3.1 The Least Squares Principle 51 2.3.2 Estimates for the Food Expenditure Function 53 2.3.3 Interpreting the Estimates 53 2.3.3a Elasticities 54 2.3.3b Prediction 55 2.3.3c Computer Output 55 2.3.4 Other Economic Models 56 2.4 Assessing the Least Squares Estimators 56 2.4.1 The Estimator b2 57 2.4.2 The Expected Values of b1 and b2 58 2.4.3 Repeated Sampling 59 2.4.4 The Variances and Covariance of b1 and b2 60 2.5 The Gauss-Markov Theorem 62 2.6 The Probability Distributions of the Least Squares Estimators 63 2.7 Estimating the Variance of the Error Term 64 2.7.1 Estimating the Variances and Covariance of the Least Squares Estimators 65 2.7.2 Calculations for the Food Expenditure Data 65 2.7.3 Interpreting the Standard Errors 67 2.8 Estimating Nonlinear Relationships 68 2.8.1 Quadratic Functions 69 2.8.2 Using a Quadratic Model 69 2.8.3 A Log-Linear Function 70 2.8.4 Using a Log-Linear Model 71 2.8.5 Choosing a Functional Form 73 2.9 Regression with Indicator Variables 74 2.10 Exercises 75 2.10.1 Problems 75 2.10.2 Computer Exercises 78 Appendix 2A Derivation of the Least Squares Estimates 83 Appendix 2B Deviation from the Mean Form of b2 84 Appendix 2C b2 Is a Linear Estimator 85 Appendix 2D Derivation of Theoretical Expression for b2 85 Appendix 2E Deriving the Variance of b2 86 Appendix 2F Proof of the Gauss-Markov Theorem 87 xii C O N T E N T S Appendix 2G Monte Carlo Simulation 88 2G.1 The Regression Function 88 2G.2 The Random Error 89 2G.3 Theoretically True Values 90 2G.4 Creating a Sample of Data 91 2G.5 Monte Carlo Objectives 92 2G.6 Monte Carlo Results 92 Chapter 3 Interval Estimation and Hypothesis Testing 94 Learning Objectives 94 Keywords 94 3.1 Interval Estimation 95 3.1.1 The t-Distribution 95 3.1.2 Obtaining Interval Estimates 97 3.1.3 An Illustration 98 3.1.4 The Repeated Sampling Context 99 3.2 Hypothesis Tests 100 3.2.1 The Null Hypothesis 101 3.2.2 The Alternative Hypothesis 101 3.2.3 The Test Statistic 101 3.2.4 The Rejection Region 101 3.2.5 A Conclusion 102 3.3 Rejection Regions for Specific Alternatives 102 3.3.1 One-Tail Tests with Alternative ‘‘Greater Than’’ (>) 102 3.3.2 One-Tail Tests with Alternative ‘‘Less Than’’ (<) 103 3.3.3 Two-Tail Tests with Alternative ‘‘Not Equal To’’ (6¼) 104 3.4 Examples of Hypothesis Tests 105 3.4.1 Right-Tail Tests 105 3.4.1a One-Tail Test of Significance 105 3.4.1b One-Tail Test of an Economic Hypothesis 106 3.4.2 Left-Tail Tests 107 3.4.3 Two-Tail Tests 108 3.4.3a Two-Tail Test of an Economic Hypothesis 108 3.4.3b Two-Tail Test of Significance 109 3.5 The p-Value 110 3.5.1 p-Value for a Right-Tail Test 111 3.5.2 p-Value for a Left-Tail Test 112 3.5.3 p-Value for a Two-Tail Test 112 3.5.4 p-Value for a Two-Tail Test of Significance 113 3.6 Linear Combinations of Parameters 114 3.6.1 Estimating Expected Food Expenditure 115 3.6.2 An Interval Estimate of Expected Food Expenditure 115 3.6.3 Testing a Linear Combination of Parameters 116 3.6.4 Testing Expected Food Expenditure 117 3.7 Exercises 118 3.7.1 Problems 118 3.7.2 Computer Exercises 120 Appendix 3A Derivation of the t-Distribution 125 Appendix 3B Distribution of the t-Statistic under H1 126 C O NT E NT S xiii