A Guide to Modern Econometrics PDF

 k Trim Size: 7in x 10in Verbeek ffirs.tex V1 - 05/13/2017 12:49 A.M. Page i k k k ...

 k Trim Size: 7in x 10in Verbeek ffirs.tex V1 - 05/13/2017 12:49 A.M. Page i k k k  k Trim Size: 7in x 10in Verbeek ffirs.tex V1 - 06/01/2017 1:12 P.M. Page i A Guide to Modern Econometrics k k Fifth Edition Marno Verbeek Rotterdam School of Management, Erasmus University, Rotterdam k  k Trim Size: 7in x 10in Verbeek ffirs.tex V1 - 06/01/2017 1:12 P.M. Page ii VP AND EDITORIAL DIRECTOR George Hoffman EDITORIAL DIRECTOR Veronica Visentin EXECUTIVE EDITOR Darren Lalonde SPONSORING EDITOR Jennifer Manias EDITORIAL MANAGER Gladys Soto CONTENT MANAGEMENT DIRECTOR Lisa Wojcik CONTENT MANAGER Nichole Urban SENIOR CONTENT SPECIALIST Nicole Repasky PRODUCTION EDITOR Annie Sophia Thapasumony COVER PHOTO CREDIT © Stuart Miles/Shutterstock This book was set in 10/12, TimesLTStd by SPi Global and printed and bound by Strategic Content Imaging. 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Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at: www.wiley.com/go/returnlabel. If you have chosen to adopt this textbook for use in your course, please accept this book as your complimentary desk copy. Outside of the United States, please contact your local sales representative. ISBN: 978-1-119-40115-5 (PBK) ISBN: 978-1-119-40119-3 (EVALC) Library of Congress Cataloging in Publication Data: Names: Verbeek, Marno, author. Title: A guide to modern econometrics / Marno Verbeek, Rotterdam School of Management, Erasmus University, Rotterdam. Description: 5th edition. | Hoboken, NJ : John Wiley & Sons, Inc., | Includes bibliographical references and index. | Identifiers: LCCN 2017015272 (print) | LCCN 2017019441 (ebook) | ISBN 9781119401100 (pdf) | ISBN 9781119401117 (epub) | ISBN 9781119401155 (pbk.) Subjects: LCSH: Econometrics. | Regression analysis. Classification: LCC HB139 (ebook) | LCC HB139.V465 2017 (print) | DDC 330.01/5195—dc23 LC record available at https://lccn.loc.gov/2017015272 The inside back cover will contain printing identification and country of origin if omitted from this page. In addition, if the ISBN on the back cover differs from the ISBN on this page, the one on the back cover is correct. k  k Trim Size: 7in x 10in Verbeek ftoc.tex V1 - 04/21/2017 3:53 P.M. Page iii Contents Preface xi 1 Introduction 1 1.1 About Econometrics 1 k 1.2 The Structure of This Book 3 k 1.3 Illustrations and Exercises 4 2 An Introduction to Linear Regression 6 2.1 Ordinary Least Squares as an Algebraic Tool 7 2.1.1 Ordinary Least Squares 7 2.1.2 Simple Linear Regression 9 2.1.3 Example: Individual Wages 11 2.1.4 Matrix Notation 11 2.2 The Linear Regression Model 12 2.3 Small Sample Properties of the OLS Estimator 15 2.3.1 The Gauss–Markov Assumptions 15 2.3.2 Properties of the OLS Estimator 16 2.3.3 Example: Individual Wages (Continued) 20 2.4 Goodness-of-Fit 20 2.5 Hypothesis Testing 23 2.5.1 A Simple t-Test 23 2.5.2 Example: Individual Wages (Continued) 25 2.5.3 Testing One Linear Restriction 25 2.5.4 A Joint Test of Significance of Regression Coefficients 26 2.5.5 Example: Individual Wages (Continued) 28 2.5.6 The General Case 29 2.5.7 Size, Power and p-Values 30 2.5.8 Reporting Regression Results 32 k  k Trim Size: 7in x 10in Verbeek ftoc.tex V1 - 04/21/2017 3:53 P.M. Page iv iv CONTENTS 2.6 Asymptotic Properties of the OLS Estimator 33 2.6.1 Consistency 33 2.6.2 Asymptotic Normality 35 2.6.3 Small Samples and Asymptotic Theory 37 2.7 Illustration: The Capital Asset Pricing Model 39 2.7.1 The CAPM as a Regression Model 40 2.7.2 Estimating and Testing the CAPM 41 2.7.3 The World’s Largest Hedge Fund 43 2.8 Multicollinearity 44 2.8.1 Example: Individual Wages (Continued) 47 2.9 Missing Data, Outliers and Influential Observations 48 2.9.1 Outliers and Influential Observations 48 2.9.2 Robust Estimation Methods 50 2.9.3 Missing Observations 51 2.10 Prediction 53 Wrap-up 54 Exercises 55 3 Interpreting and Comparing Regression Models 60 3.1 Interpreting the Linear Model 60 3.2 Selecting the Set of Regressors 65 3.2.1 Misspecifying the Set of Regressors 65 3.2.2 Selecting Regressors 66 k k 3.2.3 Comparing Non-nested Models 71 3.3 Misspecifying the Functional Form 73 3.3.1 Nonlinear Models 73 3.3.2 Testing the Functional Form 74 3.3.3 Testing for a Structural Break 74 3.4 Illustration: Explaining House Prices 76 3.5 Illustration: Predicting Stock Index Returns 79 3.5.1 Model Selection 80 3.5.2 Forecast Evaluation 82 3.6 Illustration: Explaining Individual Wages 85 3.6.1 Linear Models 85 3.6.2 Loglinear Models 88 3.6.3 The Effects of Gender 91 3.6.4 Some Words of Warning 92 Wrap-up 93 Exercises 94 4 Heteroskedasticity and Autocorrelation 97 4.1 Consequences for the OLS Estimator 98 4.2 Deriving an Alternative Estimator 99 4.3 Heteroskedasticity 100 4.3.1 Introduction 100 4.3.2 Estimator Properties and Hypothesis Testing 103 k  k Trim Size: 7in x 10in Verbeek ftoc.tex V1 - 04/21/2017 3:53 P.M. Page v CONTENTS v 4.3.3 When the Variances Are Unknown 104 4.3.4 Heteroskedasticity-consistent Standard Errors for OLS 105 4.3.5 Multiplicative Heteroskedasticity 106 4.3.6 Weighted Least Squares with Arbitrary Weights 107 4.4 Testing for Heteroskedasticity 108 4.4.1 Testing for Multiplicative Heteroskedasticity 108 4.4.2 The Breusch–Pagan Test 109 4.4.3 The White Test 109 4.4.4 Which Test? 110 4.5 Illustration: Explaining Labour Demand 110 4.6 Autocorrelation 114 4.6.1 First-order Autocorrelation 116 4.6.2 Unknown 𝜌 118 4.7 Testing for First-order Autocorrelation 119 4.7.1 Asymptotic Tests 119 4.7.2 The Durbin–Watson Test 120 4.8 Illustration: The Demand for Ice Cream 121 4.9 Alternative Autocorrelation Patterns 124 4.9.1 Higher-order Autocorrelation 124 4.9.2 Moving Average Errors 125 4.10 What to Do When You Find Autocorrelation? 126 4.10.1 Misspecification 126 k k 4.10.2 Heteroskedasticity-and-autocorrelation-consistent Standard Errors for OLS 128 4.11 Illustration: Risk Premia in Foreign Exchange Markets 129 4.11.1 Notation 129 4.11.2 Tests for Risk Premia in the 1-Month Market 131 4.11.3 Tests for Risk Premia Using Overlapping Samples 134 Wrap-up 136 Exercises 136 5 Endogenous Regressors, Instrumental Variables and GMM 139 5.1 A Review of the Properties of the OLS Estimator 140 5.2 Cases Where the OLS Estimator Cannot Be Saved 143 5.2.1 Autocorrelation with a Lagged Dependent Variable 143 5.2.2 Measurement Error in an Explanatory Variable 144 5.2.3 Endogeneity and Omitted Variable Bias 146 5.2.4 Simultaneity and Reverse Causality 148 5.3 The Instrumental Variables Estimator 150 5.3.1 Estimation with a Single Endogenous Regressor and a Single Instrument 150 5.3.2 Back to the Keynesian Model 155 5.3.3 Back to the Measurement Error Problem 156 5.3.4 Multiple Endogenous Regressors 156 5.4 Illustration: Estimating the Returns to Schooling 157 k  k Trim Size: 7in x 10in Verbeek ftoc.tex V1 - 04/21/2017 3:53 P.M. Page vi vi CONTENTS 5.5 Alternative Approaches to Estimate Causal Effects 162 5.6 The Generalized Instrumental Variables Estimator 163 5.6.1 Multiple Endogenous Regressors with an Arbitrary Number of Instruments 163 5.6.2 Two-stage Least Squares and the Keynesian Model Again 167 5.6.3 Specification Tests 168 5.6.4 Weak Instruments 169 5.6.5 Implementing and Reporting Instrumental Variables Estimators 170 5.7 Institutions and Economic Development 171 5.8 The Generalized Method of Moments 175 5.8.1 Example 175 5.8.2 The Generalized Method of Moments 177 5.8.3 Some Simple Examples 179 5.8.4 Weak Identification 180 5.9 Illustration: Estimating Intertemporal Asset Pricing Models 181 Wrap-up 184 Exercises 185 6 Maximum Likelihood Estimation and Specification Tests 187 6.1 An Introduction to Maximum Likelihood 188 k 6.1.1 Some Examples 188 k 6.1.2 General Properties 191 6.1.3 An Example (Continued) 194 6.1.4 The Normal Linear Regression Model 195 6.1.5 The Stochastic Frontier Model 197 6.2 Specification Tests 198 6.2.1 Three Test Principles 198 6.2.2 Lagrange Multiplier Tests 200 6.2.3 An Example (Continued) 203 6.3 Tests in the Normal Linear Regression Model 204 6.3.1 Testing for Omitted Variables 204 6.3.2 Testing for Heteroskedasticity 206 6.3.3 Testing for Autocorrelation 207 6.4 Quasi-maximum Likelihood and Moment Conditions Tests 208 6.4.1 Quasi-maximum Likelihood 208 6.4.2 Conditional Moment Tests 210 6.4.3 Testing for Normality 211 Wrap-up 212 Exercises 212 7 Models with Limited Dependent Variables 215 7.1 Binary Choice Models 216 7.1.1 Using Linear Regression? 216 7.1.2 Introducing Binary Choice Models 216 7.1.3 An Underlying Latent Model 219 k  k Trim Size: 7in x 10in Verbeek ftoc.tex V1 - 04/21/2017 3:53 P.M. Page vii CONTENTS vii 7.1.4 Estimation 219 7.1.5 Goodness-of-Fit 221 7.1.6 Illustration: The Impact of Unemployment Benefits on Recipiency 223 7.1.7 Specification Tests in Binary Choice Models 226 7.1.8 Relaxing Some Assumptions in Binary Choice Models 228 7.2 Multiresponse Models 229 7.2.1 Ordered Response Models 230 7.2.2 About Normalization 231 7.2.3 Illustration: Explaining Firms’ Credit Ratings 231 7.2.4 Illustration: Willingness to Pay for Natural Areas 234 7.2.5 Multinomial Models 237 7.3 Models for Count Data 240 7.3.1 The Poisson and Negative Binomial Models 240 7.3.2 Illustration: Patents and R&D Expenditures 244 7.4 Tobit Models 246 7.4.1 The Standard Tobit Model 247 7.4.2 Estimation 249 7.4.3 Illustration: Expenditures on Alcohol and Tobacco (Part 1) 250 7.4.4 Specification Tests in the Tobit Model 253 7.5 Extensions of Tobit Models 256 k k 7.5.1 The Tobit II Model 256 7.5.2 Estimation 259 7.5.3 Further Extensions 261 7.5.4 Illustration: Expenditures on Alcohol and Tobacco (Part 2) 262 7.6 Sample Selection Bias 265 7.6.1 The Nature of the Selection Problem 266 7.6.2 Semi-parametric Estimation of the Sample Selection Model 268 7.7 Estimating Treatment Effects 269 7.7.1 Regression-based Estimators 271 7.7.2 Regression Discontinuity Design 274 7.7.3 Weighting and Matching 276 7.8 Duration Models 278 7.8.1 Hazard Rates and Survival Functions 278 7.8.2 Samples and Model Estimation 281 7.8.3 Illustration: Duration of Bank Relationships 283 Wrap-up 284 Exercises 285 8 Univariate Time Series Models 288 8.1 Introduction 289 8.1.1 Some Examples 289 8.1.2 Stationarity and the Autocorrelation Function 291 k  k Trim Size: 7in x 10in Verbeek ftoc.tex V1 - 04/21/2017 3:53 P.M. Page viii viii CONTENTS 8.2 General ARMA Processes 294 8.2.1 Formulating ARMA Processes 294 8.2.2 Invertibility of Lag Polynomials 297 8.2.3 Common Roots 298 8.3 Stationarity and Unit Roots 299 8.4 Testing for Unit Roots 301 8.4.1 Testing for Unit Roots in a First-order Autoregressive Model 301 8.4.2 Testing for Unit Roots in Higher-Order Autoregressive Models 304 8.4.3 Extensions 306 8.4.4 Illustration: Stock Prices and Earnings 307 8.5 Illustration: Long-run Purchasing Power Parity (Part 1) 309 8.6 Estimation of ARMA Models 313 8.6.1 Least Squares 314 8.6.2 Maximum Likelihood 315 8.7 Choosing a Model 316 8.7.1 The Autocorrelation Function 316 8.7.2 The Partial Autocorrelation Function 318 8.7.3 Diagnostic Checking 319 8.7.4 Criteria for Model Selection 319 8.8 Illustration: The Persistence of Inflation 320 k 8.9 Forecasting with ARMA Models 324 k 8.9.1 The Optimal Forecast 324 8.9.2 Forecast Accuracy 327 8.9.3 Evaluating Forecasts 329 8.10 Illustration: The Expectations Theory of the Term Structure 330 8.11 Autoregressive Conditional Heteroskedasticity 335 8.11.1 ARCH and GARCH Models 335 8.11.2 Estimation and Prediction 338 8.11.3 Illustration: Volatility in Daily Exchange Rates 340 8.12 What about Multivariate Models? 342 Wrap-up 343 Exercises 344 9 Multivariate Time Series Models 348 9.1 Dynamic Models with Stationary Variables 349 9.2 Models with Nonstationary Variables 352 9.2.1 Spurious Regressions 352 9.2.2 Cointegration 353 9.2.3 Cointegration and Error-correction Mechanisms 356 9.3 Illustration: Long-run Purchasing Power Parity (Part 2) 358 9.4 Vector Autoregressive Models 360 9.5 Cointegration: the Multivariate Case 364 9.5.1 Cointegration in a VAR 364 9.5.2 Example: Cointegration in a Bivariate VAR 366 9.5.3 Testing for Cointegration 367 k  k Trim Size: 7in x 10in Verbeek ftoc.tex V1 - 04/21/2017 3:53 P.M. Page ix CONTENTS ix 9.5.4 Illustration: Long-run Purchasing Power Parity (Part 3) 370 9.6 Illustration: Money Demand and Inflation 372 Wrap-up 378 Exercises 379 10 Models Based on Panel Data 382 10.1 Introduction to Panel Data Modelling 383 10.1.1 Efficiency of Parameter Estimators 384 10.1.2 Identification of Parameters 385 10.2 The Static Linear Model 386 10.2.1 The Fixed Effects Model 386 10.2.2 The First-difference Estimator 388 10.2.3 The Random Effects Model 390 10.2.4 Fixed Effects or Random Effects? 394 10.2.5 Goodness-of-Fit 395 10.2.6 Alternative Instrumental Variables Estimators 396 10.2.7 Robust Inference 398 10.2.8 Testing for Heteroskedasticity and Autocorrelation 400 10.2.9 The Fama–MacBeth Approach 402 10.3 Illustration: Explaining Individual Wages 403 10.4 Dynamic Linear Models 405 k 10.4.1 An Autoregressive Panel Data Model 406 k 10.4.2 Dynamic Models with Exogenous Variables 411 10.4.3 Too Many Instruments 412 10.5 Illustration: Explaining Capital Structure 414 10.6 Panel Time Series 419 10.6.1 Heterogeneity 420 10.6.2 First Generation Panel Unit Root Tests 421 10.6.3 Second Generation Panel Unit Root Tests 424 10.6.4 Panel Cointegration Tests 425 10.7 Models with Limited Dependent Variables 426 10.7.1 Binary Choice Models 427 10.7.2 The Fixed Effects Logit Model 428 10.7.3 The Random Effects Probit Model 429 10.7.4 Tobit Models 431 10.7.5 Dynamics and the Problem of Initial Conditions 431 10.7.6 Semi-parametric Alternatives 433 10.8 Incomplete Panels and Selection Bias 433 10.8.1 Estimation with Randomly Missing Data 434 10.8.2 Selection Bias and Some Simple Tests 436 10.8.3 Estimation with Nonrandomly Missing Data 438 10.9 Pseudo Panels and Repeated Cross-sections 439 10.9.1 The Fixed Effects Model 440 10.9.2 An Instrumental Variables Interpretation 441 10.9.3 Dynamic Models 442 Wrap-up 444 Exercises 445 k  k Trim Size: 7in x 10in Verbeek ftoc.tex V1 - 04/21/2017 3:53 P.M. Page x x CONTENTS A Vectors and Matrices 450 A.1 Terminology 450 A.2 Matrix Manipulations 451 A.3 Properties of Matrices and Vectors 452 A.4 Inverse Matrices 453 A.5 Idempotent Matrices 454 A.6 Eigenvalues and Eigenvectors 454 A.7 Differentiation 455 A.8 Some Least Squares Manipulations 456 B Statistical and Distribution Theory 458 B.1 Discrete Random Variables 458 B.2 Continuous Random Variables 459 B.3 Expectations and Moments 460 B.4 Multivariate Distributions 461 B.5 Conditional Distributions 462 B.6 The Normal Distribution 463 B.7 Related Distributions 466 Bibliography 468 Index 488 k k k  Trim Size: 7in x 10in k Verbeek fpref.tex V2 - 05/04/2017 4:58 P.M. Page xi Preface Emperor Joseph II: “Your work is ingenious. It’s quality work. And there are simply too many notes, that’s all. Just cut a few and it will be perfect.” Wolfgang Amadeus Mozart: “Which few did you have in mind, Majesty?” from the movie Amadeus, 1984 (directed by Milos Forman) The field of econometrics has developed rapidly in the last three decades, while the k use of up-to-date econometric techniques has become more and more standard prac- k tice in empirical work in many fields of economics. Typical topics include unit root tests, cointegration, estimation by the generalized method of moments, heteroskedasticity and autocorrelation consistent standard errors, modelling conditional heteroskedasticity, causal inference and the estimation of treatment effects, models based on panel data, models with limited dependent variables, endogenous regressors and sample selection. At the same time econometrics software has become more and more user friendly and up-to-date. As a consequence, users are able to implement fairly advanced techniques even without a basic understanding of the underlying theory and without realizing poten- tial drawbacks or dangers. In contrast, many introductory econometrics textbooks pay a disproportionate amount of attention to the standard linear regression model under the strongest set of assumptions. Needless to say that these assumptions are hardly satisfied in practice (but not really needed either). On the other hand, the more advanced economet- rics textbooks are often too technical or too detailed for the average economist to grasp the essential ideas and to extract the information that is needed. This book tries to fill this gap. The goal of this book is to familiarize the reader with a wide range of topics in modern econometrics, focusing on what is important for doing and understanding empirical work. This means that the text is a guide to (rather than an overview of) alternative techniques. Consequently, it does not concentrate on the formulae behind each technique (although the necessary ones are given) nor on formal proofs, but on the intuition behind the approaches and their practical relevance. The book covers a wide range of topics that is usually not found in textbooks at this level. In particular, attention is paid to cointegration, the generalized method of moments, models with limited dependent variables and panel data models. As a result, the book discusses developments in time series analysis, cross-sectional methods as well as panel data modelling. More than 25 full-scale empirical illustrations are provided in separate sections and subsections, taken from fields like labour economics, finance, international economics, consumer behaviour, environmental economics and macro-economics. These illustrations carefully k  Trim Size: 7in x 10in k Verbeek fpref.tex V2 - 05/04/2017 4:58 P.M. Page xii xii PREFACE discuss and interpret econometric analyses of relevant economic problems, and each of them covers between two and nine pages of the text. As before, data sets are available through the supporting website of this book. In addition, a number of exercises are of an empirical nature and require the use of actual data. This fifth edition builds upon the success of its predecessors. The text has been carefully checked and updated, taking into account recent developments and insights. It includes new material on causal inference, the use and limitations of p-values, instrumental vari- ables estimation and its implementation, regression discontinuity design, standardized coefficients, and the presentation of estimation results. Several empirical illustrations are new or updated. For example, Section 5.7 is added containing a new illustration on the causal effect of institutions on economic development, to illustrate the use of instrumental variables. Overall, the presentation is meant to be concise and intuitive, providing refer- ences to primary sources wherever possible. Where relevant, I pay particular attention to implementation concerns, for example, relating to identification issues. A large number of new references has been added in this edition to reflect the changes in the text. Increas- ingly, the literature provides critical surveys and practical guides on how more advanced econometric techniques, like robust standard errors, sample selection models or causal inference methods, are used in specific areas, and I have tried to refer to them in the text too. This text originates from lecture notes used for courses in Applied Econometrics in the M.Sc. programmes in Economics at K. U. Leuven and Tilburg University. It is written for an intended audience of economists and economics students that would like to become familiar with up-to-date econometric approaches and techniques, important for doing, k understanding and evaluating empirical work. It is very well suited for courses in applied k econometrics at the master’s or graduate level. At some schools this book will be suited for one or more courses at the undergraduate level, provided students have a sufficient background in statistics. Some of the later chapters can be used in more advanced courses covering particular topics, for example, panel data, limited dependent variable models or time series analysis. In addition, this book can serve as a guide for managers, research economists and practitioners who want to update their insufficient or outdated knowledge of econometrics. Throughout, the use of matrix algebra is limited. I am very much indebted to Arie Kapteyn, Bertrand Melenberg, Theo Nijman and Arthur van Soest, who all have contributed to my understanding of econometrics and have shaped my way of thinking about many issues. The fact that some of their ideas have materialized in this text is a tribute to their efforts. I also owe many thanks to several generations of students who helped me to shape this text into its current form. I am very grateful to a large number of people who read through parts of the manuscript and provided me with comments and suggestions on the basis of the first three editions. In particular, I wish to thank Niklas Ahlgren, Sascha Becker, Peter Boswijk, Bart Capéau, Geert Dhaene, Tom Doan, Peter de Goeij, Joop Huij, Ben Jacobsen, Jan Kiviet, Wim Koevoets, Erik Kole, Marco Lyrio, Konstantijn Maes, Wessel Marquering, Bertrand Melenberg, Paulo Nunes, Anatoly Peresetsky, Francesco Ravazzolo, Regina Riphahn, Max van de Sande Bakhuyzen, Erik Schokkaert, Peter Sephton, Arthur van Soest, Ben Tims, Frederic Vermeulen, Patrick Verwijmeren, Guglielmo Weber, Olivier Wolthoorn, Kuo-chun Yeh and a number of anonymous reviewers. Of course I retain sole responsibility for any remaining errors. Special thanks go to Jean-Francois Flechet for his help with many empirical illustrations and his constructive comments on many early drafts. Finally, I want to thank my wife Marcella and our three children, Timo, Thalia and Tamara, for their patience and understanding for all the times that my mind was with this book when it should have been with them. k  Trim Size: 7in x 10in k Verbeek c01.tex V2 - 04/22/2017 6:44 A.M. Page 1 1 Introduction 1.1 About Econometrics Economists are frequently interested in relationships between different quantities, for example between individual wages and the level of schooling. The most important job of econometrics is to quantify these relationships on the basis of available data and using statistical techniques, and to interpret, use or exploit the resulting outcomes appropriately. k Consequently, econometrics is the interaction of economic theory, observed data and sta- k tistical methods. It is the interaction of these three that makes econometrics interesting, challenging and, perhaps, difficult. In the words of a seminar speaker, several years ago: ‘Econometrics is much easier without data’. Traditionally econometrics has focused upon aggregate economic relationships. Macro-economic models consisting of several up to many hundreds of equations were specified, estimated and used for policy evaluation and forecasting. The recent theoretical developments in this area, most importantly the concept of cointegration, have generated increased attention to the modelling of macro-economic relationships and their dynamics, although typically focusing on particular aspects of the economy. Since the 1970s econometric methods have increasingly been employed in micro- economic models describing individual, household or firm behaviour, stimulated by the development of appropriate econometric models and estimators that take into account problems like discrete dependent variables and sample selection, by the availability of large survey data sets and by the increasing computational possibilities. More recently, the empirical analysis of financial markets has required and stimulated many theoretical developments in econometrics. Currently econometrics plays a major role in empirical work in all fields of economics, almost without exception, and in most cases it is no longer sufficient to be able to run a few regressions and interpret the results. As a result, introductory econometrics textbooks usually provide insufficient coverage for applied researchers. On the other hand, the more advanced econometrics textbooks are often too technical or too detailed for the average economist to grasp the essential ideas and to extract the information that is needed. Thus there is a need for an accessible textbook that discusses the recent and relatively more advanced developments. k  Trim Size: 7in x 10in k Verbeek c01.tex V2 - 04/22/2017 6:44 A.M. Page 2 2 INTRODUCTION The relationships that economists are interested in are formally specified in mathemat- ical terms, which lead to econometric or statistical models. In such models there is room for deviations from the strict theoretical relationships owing to, for example, measure- ment errors, unpredictable behaviour, optimization errors or unexpected events. Broadly, econometric models can be classified into a number of categories. A first class of models describes relationships between present and past. For example, how does the short-term interest rate depend on its own history? This type of model, typ- ically referred to as a time series model, usually lacks any economic theory and is mainly built to get forecasts for future values and the corresponding uncertainty or volatility. A second type of model considers relationships between economic quantities over a certain time period. These relationships give us information on how (aggregate) economic quantities fluctuate over time in relation to other quantities. For example, what happens to the long-term interest rate if the monetary authority adjusts the short-term one? These models often give insight into the economic processes that are operating. Thirdly, there are models that describe relationships between different variables mea- sured at a given point in time for different units (e.g. households or firms). Most of the time, this type of relationship is meant to explain why these units are different or behave differently. For example, one can analyse to what extent differences in household savings can be attributed to differences in household income. Under particular conditions, these cross-sectional relationships can be used to analyse ‘what if’ questions. For example, how much more would a given household, or the average household, save if income were to increase by 1%? Finally, one can consider relationships between different variables measured for differ- k ent units over a longer time span (at least two periods). These relationships simultane- k ously describe differences between different individuals (why does person 1 save much more than person 2?), and differences in behaviour of a given individual over time (why does person 1 save more in 1992 than in 1990?). This type of model usually requires panel data, repeated observations over the same units. They are ideally suited for analysing pol- icy changes on an individual level, provided that it can be assumed that the structure of the model is constant into the (near) future. The job of econometrics is to specify and quantify these relationships. That is, econo- metricians formulate a statistical model, usually based on economic theory, confront it with the data and try to come up with a specification that meets the required goals. The unknown elements in the specification, the parameters, are estimated from a sample of available data. Another job of the econometrician is to judge whether the resulting model is ‘appropriate’. That is, to check whether the assumptions made to motivate the estima- tors (and their properties) are correct, and to check whether the model can be used for its intended purpose. For example, can it be used for prediction or analysing policy changes? Often, economic theory implies that certain restrictions apply to the model that is esti- mated. For example, the efficient market hypothesis implies that stock market returns are not predictable from their own past. An important goal of econometrics is to formulate such hypotheses in terms of the parameters in the model and to test their validity. The number of econometric techniques that can be used is numerous, and their valid- ity often depends crucially upon the validity of the underlying assumptions. This book attempts to guide the reader through this forest of estimation and testing procedures, not by describing the beauty of all possible trees, but by walking through this forest in a structured way, skipping unnecessary side-paths, stressing the similarity of the different species that are encountered and pointing out dangerous pitfalls. The resulting walk is hopefully enjoyable and prevents the reader from getting lost in the econometric forest. k  Trim Size: 7in x 10in k Verbeek c01.tex V2 - 04/22/2017 6:44 A.M. Page 3 THE STRUCTURE OF THIS BOOK 3 1.2 The Structure of This Book The first part of this book consists of Chapters 2, 3 and 4. Like most textbooks, it starts with discussing the linear regression model and the OLS estimation method. Chapter 2 presents the basics of this important estimation method, with some emphasis on its valid- ity under fairly weak conditions, while Chapter 3 focuses on the interpretation of the models and the comparison of alternative specifications. Chapter 4 considers two partic- ular deviations from the standard assumptions of the linear model: autocorrelation and heteroskedasticity of the error terms. It is discussed how one can test for these phenom- ena, how they affect the validity of the OLS estimator and how this can be corrected. This includes a critical inspection of the model specification, the use of adjusted standard errors for the OLS estimator and the use of alternative (GLS) estimators. These three chapters are essential for the remaining part of this book and should be the starting point in any course. In Chapter 5 another deviation from the standard assumptions of the linear model is discussed, which is, however, fatal for the OLS estimator. As soon as the error term in the model is correlated with one or more of the explanatory variables, all good properties of the OLS estimator disappear, and we necessarily have to use alternative approaches. This raises the challenge of identifying causal effects with nonexperimental data. The chapter discusses instrumental variable (IV) estimators and, more generally, the gen- eralized method of moments (GMM). This chapter, at least its earlier sections, is also recommended as an essential part of any econometrics course. Chapter 6 is mainly theoretical and discusses maximum likelihood (ML) estimation. k Because in empirical work maximum likelihood is often criticized for its dependence k upon distributional assumptions, it is not discussed in the earlier chapters where alter- natives are readily available that are either more robust than maximum likelihood or (asymptotically) equivalent to it. Particular emphasis in Chapter 6 is on misspecification tests based upon the Lagrange multiplier principle. While many empirical studies tend to take the distributional assumptions for granted, their validity is crucial for consistency of the estimators that are employed and should therefore be tested. Often these tests are relatively easy to perform, although most software does not routinely provide them (yet). Chapter 6 is crucial for understanding Chapter 7 on limited dependent variable models and for a small number of sections in Chapters 8 to 10. The last part of this book contains four chapters. Chapter 7 presents models that are typically (though not exclusively) used in micro-economics, where the dependent vari- able is discrete (e.g. zero or one), partly discrete (e.g. zero or positive) or a duration. This chapter covers probit, logit and tobit models and their extensions, as well as models for count data and duration models. It also includes a critical discussion of the sample selec- tion problem. Particular attention is paid to alternative approaches to estimate the causal impact of a treatment upon an outcome variable in case the treatment is not randomly assigned (‘treatment effects’). Chapters 8 and 9 discuss time series modelling including unit roots, cointegration and error-correction models. These chapters can be read immediately after Chapter 4 or 5, with the exception of a few parts that relate to maximum likelihood estimation. The theoretical developments in this area over the last three decades have been substantial, and many recent textbooks seem to focus upon it almost exclusively. Univariate time series models are covered in Chapter 8. In this case, models are developed that explain an economic variable from its own past. These include ARIMA models, as well as GARCH models for the conditional variance of a series. Multivariate time series models that k  Trim Size: 7in x 10in k Verbeek c01.tex V2 - 04/22/2017 6:44 A.M. Page 4 4 INTRODUCTION consider several variables simultaneously are discussed in Chapter 9. These include vector autoregressive models, cointegration and error-correction models. Finally, Chapter 10 covers models based on panel data. Panel data are available if we have repeated observations of the same units (e.g. households, firms or countries). Over recent decades the use of panel data has become important in many areas of eco- nomics. Micro-economic panels of households and firms are readily available and, given the increase in computing resources, more manageable than in the past. In addition, it has become increasingly common to pool time series of several countries. One of the reasons for this may be that researchers believe that a cross-sectional comparison of countries provides interesting information, in addition to a historical comparison of a country with its own past. This chapter also discusses the recent developments on unit roots and coin- tegration in a panel data setting. Furthermore, a separate section is devoted to repeated cross-sections and pseudo panel data. At the end of the book the reader will find two short appendices discussing mathemati- cal and statistical results that are used in several places in the book. This includes a discus- sion of some relevant matrix algebra and distribution theory. In particular, a discussion of properties of the (bivariate) normal distribution, including conditional expectations, variances and truncation, is provided. In my experience the material in this book is too much to be covered in a single course. Different courses can be scheduled on the basis of the chapters that follow. For example, a typical graduate course in applied econometrics would cover Chapters 2, 3, 4 and parts of Chapter 5, and then continue with selected parts of Chapters 8 and 9 if the focus is k on time series analysis, or continue with Section 6.1 and Chapter 7 if the focus is on k cross-sectional models. A more advanced undergraduate or graduate course may focus attention on the time series chapters (Chapters 8 and 9), the micro-econometric chapters (Chapters 6 and 7) or panel data (Chapter 10 with some selected parts from Chapters 6 and 7). Given the focus and length of this book, I had to make many choices concerning which material to present or not. As a general rule I did not want to bother the reader with details that I considered not essential or not to have empirical relevance. The main goal was to give a general and comprehensive overview of the different methodologies and approaches, focusing on what is relevant for doing and understanding empirical work. Some topics are only very briefly mentioned, and no attempt is made to discuss them at any length. To compensate for this I have tried to give references in appropriate places to other sources, including specialized textbooks, survey articles and chapters, and guides with advice for practitioners. 1.3 Illustrations and Exercises In most chapters a variety of empirical illustrations are provided in separate sections or subsections. While it is possible to skip these illustrations essentially without losing continuity, these sections do provide important aspects concerning the implementation of the methodology discussed in the preceding text. In addition, I have attempted to provide illustrations that are of economic interest in themselves, using data that are typical of current empirical work and cover a wide range of different areas. This means that most data sets are used in recently published empirical work and are fairly large, both in terms k  Trim Size: 7in x 10in k Verbeek c01.tex V2 - 04/22/2017 6:44 A.M. Page 5 ILLUSTRATIONS AND EXERCISES 5 of number of observations and in terms of number of variables. Given the current state of computing facilities, it is usually not a problem to handle such large data sets empirically. Learning econometrics is not just a matter of studying a textbook. Hands-on experience is crucial in the process of understanding the different methods and how and when to implement them. Therefore, readers are strongly encouraged to get their hands dirty and to estimate a number of models using appropriate or inappropriate methods, and to perform a number of alternative specification tests. With modern software becoming more and more user friendly, the actual computation of even the more complicated estimators and test statistics is often surprisingly simple, sometimes dangerously simple. That is, even with the wrong data, the wrong model and the wrong methodology, programmes may come up with results that are seemingly all right. At least some expertise is required to prevent the practitioner from such situations, and this book plays an important role in this. To stimulate the reader to use actual data and estimate some models, almost all data sets used in this text are available through the website www.wileyeurope.com/college/ verbeek. Readers are encouraged to re-estimate the models reported in this text and check whether their results are the same, as well as to experiment with alternative specifications or methods. Some of the exercises make use of the same or additional data sets and pro- vide a number of specific issues to consider. It should be stressed that, for estimation methods that require numerical optimization, alternative programmes, algorithms or set- tings may give slightly different outcomes. However, you should get results that are close to the ones reported. I do not advocate the use of any particular software package. For the linear regression k k model any package will do, while for the more advanced techniques each package has its particular advantages and disadvantages. There is typically a trade-off between user- friendliness and flexibility. Menu-driven packages often do not allow you to compute anything other than what’s on the menu, but, if the menu is sufficiently rich, that may not be a problem. Command-driven packages require somewhat more input from the user, but are typically quite flexible. For the illustrations in the text, I made use of Eviews, RATS and Stata. Several alternative econometrics programmes are available, including MicroFit, PcGive, TSP and SHAZAM; for more advanced or tailored methods, econo- metricians make use of GAUSS, Matlab, Ox, S-Plus and many other programmes, as well as specialized software for specific methods or types of model. Journals like the Journal of Applied Econometrics and the Journal of Economic Surveys regularly publish software reviews. The exercises included at the end of each chapter consist of a number of questions that are primarily intended to check whether the reader has grasped the most important concepts. Therefore, they typically do not go into technical details or ask for derivations or proofs. In addition, several exercises are of an empirical nature and require the reader to use actual data, made available through the book’s website. k  Trim Size: 7in x 10in k Verbeek c02.tex V3 - 04/21/2017 3:58 P.M. Page 6 2 An Introduction to Linear Regression The linear regression model in combination with the method of ordinary least squares (OLS) is one of the cornerstones of econometrics. In the first part of this book we shall review the linear regression model with its assumptions, how it can be estimated, k evaluated and interpreted and how it can be used for generating predictions and for k testing economic hypotheses. This chapter starts by introducing the ordinary least squares method as an algebraic tool, rather than a statistical one. This is because OLS has the attractive property of providing a best linear approximation, irrespective of the way in which the data are generated, or any assumptions imposed. The linear regression model is then introduced in Section 2.2, while Section 2.3 discusses the properties of the OLS estimator in this model under the so-called Gauss–Markov assumptions. Section 2.4 discusses goodness-of-fit measures for the linear model, and hypothesis testing is treated in Section 2.5. In Section 2.6, we move to cases where the Gauss–Markov conditions are not necessarily satisfied and the small sample properties of the OLS estimator are unknown. In such cases, the limiting behaviour of the OLS estimator when – hypothetically – the sample size becomes infinitely large is commonly used to approximate its small sample properties. An empirical example concerning the capital asset pricing model (CAPM) is provided in Section 2.7. Sections 2.8 and 2.9 discuss data problems related to multicollinearity, outliers and missing observations, while Section 2.10 pays attention to prediction using a linear regression model. Throughout, an empirical example concerning individual wages is used to illustrate the main issues. Additional discussion on how to interpret the coefficients in the linear model, how to test some of the model’s assumptions and how to compare alternative models is provided in Chapter 3, which also contains three extensive empirical illustrations. k  Trim Size: 7in x 10in k Verbeek c02.tex V3 - 04/21/2017 3:58 P.M. Page 7 ORDINARY LEAST SQUARES AS AN ALGEBRAIC TOOL 7 2.1 Ordinary Least Squares as an Algebraic Tool 2.1.1 Ordinary Least Squares Suppose we have a sample with N observations on individual wages and a number of background characteristics, like gender, years of education and experience. Our main interest lies in the question as to how in this sample wages are related to the other observ- ables. Let us denote wages by y (the regressand) and the other K − 1 characteristics by x2 ,... , xK (the regressors). It will become clear below why this numbering of variables is convenient. Now we may ask the question: which linear combination of x2 ,... , xK and a constant gives a good approximation of y? To answer this question, first consider an arbitrary linear combination, including a constant, which can be written as 𝛽̃1 + 𝛽̃2 x2 + · · · + 𝛽̃K xK , (2.1) where 𝛽̃1 ,... , 𝛽̃K are constants to be chosen. Let us index the observations by i such that i = 1,... , N. Now, the difference between an observed value yi and its linear approximation is yi − [𝛽̃1 + 𝛽̃2 xi2 + · · · + 𝛽̃K xiK ]. (2.2) To simplify the derivations we shall introduce some shorthand notation. Appendix A provides additional details for readers unfamiliar with the use of vector notation. The special case of K = 2 is discussed in the next subsection. For general K we collect the x-values for individual i in a vector xi , which includes the constant. That is, k k xi = (1 xi2 xi3... xiK ) where  is used to denote a transpose. Collecting the 𝛽̃ coefficients in a K-dimensional vector 𝛽̃ = (𝛽̃1... 𝛽̃K ) , we can briefly write (2.2) as ̃ yi − xi 𝛽. (2.3) Clearly, we would like to choose values for 𝛽̃1 ,... , 𝛽̃K such that these differences are small. Although different measures can be used to define what we mean by ‘small’, the most common approach is to choose 𝛽̃ such that the sum of squared differences is as small as possible. In this case we determine 𝛽̃ to minimize the following objective function: ∑N ̃ S(𝛽) ≡ ̃ 2. (yi − xi 𝛽) (2.4) i=1 That is, we minimize the sum of squared approximation errors. This approach is referred to as the ordinary least squares or OLS approach. Taking squares makes sure that pos- itive and negative deviations do not cancel out when taking the summation. To solve the minimization problem, we consider the first-order conditions, obtained ̃ with respect to the vector 𝛽. by differentiating S(𝛽) ̃ (Appendix A discusses some rules on how to differentiate a scalar expression, like (2.4), with respect to a vector.) k  Trim Size: 7in x 10in k Verbeek c02.tex V3 - 04/21/2017 3:58 P.M. Page 8 8 AN INTRODUCTION TO LINEAR REGRESSION This gives the following system of K conditions: ∑ N −2 ̃ =0 xi (yi − xi 𝛽) (2.5) i=1 or (N ) ∑ ∑ N xi xi 𝛽̃ = xi yi. (2.6) i=1 i=1 These equations are sometimes referred to as normal equations. As this system has K unknowns, ∑N one can obtain a unique solution for 𝛽̃ provided that the symmetric matrix  i=1 xi xi , which contains sums of squares and cross-products of the regressors xi , can be inverted. For the moment, we shall assume that this is the case. The solution to the minimization problem, which we shall denote by b, is then given by (N )−1 N ∑  ∑ b= xi x i xi yi. (2.7) i=1 i=1 By checking the second-order conditions, it is easily verified that b indeed corresponds to a minimum of (2.4). The resulting linear combination of xi is thus given by ŷ i = xi b, k k which is the best linear approximation of y from x2 ,... , xK and a constant. The phrase ‘best’ refers to the fact that the sum of squared differences between the observed values yi and fitted values ŷ i is minimal for the least squares solution b. In deriving the linear approximation, we have not used any economic or statistical theory. It is simply an algebraic tool, and it holds irrespective of the way the data are generated. That is, given a set of variables we can always determine the best linear approximation of one variable using the other variables. The only assumption that we ∑N had to make (which is directly checked from the data) is that the K × K matrix i=1 xi xi is invertible. This says that none of the xik s is an exact linear combination of the other ones and thus redundant. This is usually referred to as the no-multicollinearity assumption. It should be stressed that the linear approximation is an in-sample result (i.e. in principle it does not give information about observations (individuals) that are not included in the sample) and, in general, there is no direct interpretation of the coefficients. Despite these limitations, the algebraic results on the least squares method are very use- ful. Defining a residual ei as the difference between the observed and the approximated value, ei = yi − ŷ i = yi − xi b, we can decompose the observed yi as yi = ŷ i + ei = xi b + ei. (2.8) This allows us to write the minimum value for the objective function as ∑ N S(b) = e2i , (2.9) i=1 k  Trim Size: 7in x 10in k Verbeek c02.tex V3 - 04/21/2017 3:58 P.M. Page 9 ORDINARY LEAST SQUARES AS AN ALGEBRAIC TOOL 9 which is referred to as the residual sum of squares. It can be shown that the approximated value xi b and the residual ei satisfy certain properties by construction. For example, if we rewrite (2.5), substituting the OLS solution b, we obtain ∑ N ∑ N xi (yi − xi b) = xi ei = 0. (2.10) i=1 i=1 This means that the vector e = (e1 ,... , eN ) is orthogonal1 to each vector of observa- ∑ tions on an x-variable. For example, if xi contains a constant, it implies that Ni=1 ei = 0. That is, the average residual is zero. This is an intuitively appealing result. If the average residual were nonzero, this would mean that we could improve upon the approximation by adding or subtracting the same constant for each observation, that is, by changing b1. Consequently, for the average observation it follows that ȳ = x̄  b, (2.11) ∑N ∑N where ȳ = (1∕N) i=1 yi and x̄ = (1∕N) i=1 xi , a K-dimensional vector of sample means. This shows that for the average observation there is no approximation error. Sim- ilar interpretations hold for the other regressors: if the derivative ∑ of the sum of squared approximation errors with respect to 𝛽̃k is positive, that is if Ni=1 xik ei > 0, it means that we can improve the objective function in (2.4) by decreasing 𝛽̃k. Equation (2.8) thus decomposes the observed value of yi into two orthogonal components: the fitted value (relat