A Guide to Modern Econometrics 2nd edition Marno Verbeek Erasmus University Rotterdam A Guide to Modern Econometrics A Guide to Modern Econometrics 2nd edition Marno Verbeek Erasmus University Rotterdam Copyright  2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com 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 under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. 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HB139.V465 2004 330  .01  5195 – dc22 2004004222 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-85773-0 Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. Contents Preface xiii 1 Introduction 1 1.1 About Econometrics 1 1.2 The Structure of this Book 3 1.3 Illustrations and Exercises 4 2 An Introduction to Linear Regression 7 2.1 Ordinary Least Squares as an Algebraic Tool 8 2.1.1 Ordinary Least Squares 8 2.1.2 Simple Linear Regression 10 2.1.3 Example: Individual Wages 12 2.1.4 Matrix Notation 12 2.2 The Linear Regression Model 14 2.3 Small Sample Properties of the OLS Estimator 16 2.3.1 The Gauss–Markov Assumptions 16 2.3.2 Properties of the OLS Estimator 17 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 27 2.5.5 Example: Individual Wages (Continued) 28 2.5.6 The General Case 30 2.5.7 Size, Power and p-Values 31 vi CONTENTS 2.6 Asymptotic Properties of the OLS Estimator 32 2.6.1 Consistency 32 2.6.2 Asymptotic Normality 34 2.6.3 Small Samples and Asymptotic Theory 36 2.7 Illustration: The Capital Asset Pricing Model 38 2.7.1 The CAPM as a Regression Model 38 2.7.2 Estimating and Testing the CAPM 39 2.8 Multicollinearity 42 2.8.1 Example: Individual Wages (Continued) 44 2.9 Prediction 44 Exercises 46 3 Interpreting and Comparing Regression Models 51 3.1 Interpreting the Linear Model 51 3.2 Selecting the Set of Regressors 55 3.2.1 Misspecifying the Set of Regressors 55 3.2.2 Selecting Regressors 56 3.2.3 Comparing Non-nested Models 59 3.3 Misspecifying the Functional Form 62 3.3.1 Nonlinear Models 62 3.3.2 Testing the Functional Form 63 3.3.3 Testing for a Structural Break 63 3.4 Illustration: Explaining House Prices 65 3.5 Illustration: Explaining Individual Wages 68 3.5.1 Linear Models 68 3.5.2 Loglinear Models 71 3.5.3 The Effects of Gender 74 3.5.4 Some Words of Warning 76 Exercises 77 4 Heteroskedasticity and Autocorrelation 79 4.1 Consequences for the OLS Estimator 79 4.2 Deriving an Alternative Estimator 81 4.3 Heteroskedasticity 82 4.3.1 Introduction 82 4.3.2 Estimator Properties and Hypothesis Testing 84 4.3.3 When the Variances are Unknown 85 4.3.4 Heteroskedasticity-consistent Standard Errors for OLS 87 4.3.5 A Model with Two Unknown Variances 88 4.3.6 Multiplicative Heteroskedasticity 89 4.4 Testing for Heteroskedasticity 90 4.4.1 Testing Equality of Two Unknown Variances 90 4.4.2 Testing for Multiplicative Heteroskedasticity 91 4.4.3 The Breusch–Pagan Test 91 4.4.4 The White Test 92 4.4.5 Which Test? 92 CONTENTS vii 4.5 Illustration: Explaining Labour Demand 92 4.6 Autocorrelation 97 4.6.1 First Order Autocorrelation 98 4.6.2 Unknown ρ 100 4.7 Testing for First Order Autocorrelation 101 4.7.1 Asymptotic Tests 101 4.7.2 The Durbin–Watson Test 102 4.8 Illustration: The Demand for Ice Cream 103 4.9 Alternative Autocorrelation Patterns 106 4.9.1 Higher Order Autocorrelation 106 4.9.2 Moving Average Errors 107 4.10 What to do When you Find Autocorrelation? 108 4.10.1 Misspecification 108 4.10.2 Heteroskedasticity-and-autocorrelation-consistent Standard Errors for OLS 110 4.11 Illustration: Risk Premia in Foreign Exchange Markets 112 4.11.1 Notation 112 4.11.2 Tests for Risk Premia in the One-month Market 113 4.11.3 Tests for Risk Premia Using Overlapping Samples 116 Exercises 119 5 E ndogeneity, Instrumental Variables and GMM 121 5.1 A Review of the Properties of the OLS Estimator 122 5.2 Cases Where the OLS Estimator Cannot be Saved 125 5.2.1 Autocorrelation with a Lagged Dependent Variable 126 5.2.2 An Example with Measurement Error 127 5.2.3 Simultaneity: the Keynesian Model 129 5.3 The Instrumental Variables Estimator 131 5.3.1 Estimation with a Single Endogenous Regressor and a Single Instrument 131 5.3.2 Back to the Keynesian Model 135 5.3.3 Back to the Measurement Error Problem 136 5.3.4 Multiple Endogenous Regressors 136 5.4 Illustration: Estimating the Returns to Schooling 137 5.5 The Generalized Instrumental Variables Estimator 142 5.5.1 Multiple Endogenous Regressors with an Arbitrary Number of Instruments 142 5.5.2 Two-stage Least Squares and the Keynesian Model Again 145 5.5.3 Specification Tests 146 5.5.4 Weak Instruments 147 5.6 The Generalized Method of Moments 148 5.6.1 Example 149 5.6.2 The Generalized Method of Moments 150 5.6.3 Some Simple Examples 153 5.7 Illustration: Estimating Intertemporal Asset Pricing Models 154 viii CONTENTS 5.8 Concluding Remarks 157 Exercises 158 6 Maximum Likelihood Estimation and Specification Tests 161 6.1 An Introduction to Maximum Likelihood 162 6.1.1 Some Examples 162 6.1.2 General Properties 166 6.1.3 An Example (Continued) 169 6.1.4 The Normal Linear Regression Model 170 6.2 Specification Tests 171 6.2.1 Three Test Principles 171 6.2.2 Lagrange Multiplier Tests 173 6.2.3 An Example (Continued) 177 6.3 Tests in the Normal Linear Regression Model 178 6.3.1 Testing for Omitted Variables 178 6.3.2 Testing for Heteroskedasticity 179 6.3.3 Testing for Autocorrelation 181 6.4 Quasi-maximum Likelihood and Moment Conditions Tests 182 6.4.1 Quasi-maximum Likelihood 182 6.4.2 Conditional Moment Tests 184 6.4.3 Testing for Normality 185 Exercises 186 7 M odels with Limited Dependent Variables 189 7.1 Binary Choice Models 190 7.1.1 Using Linear Regression? 190 7.1.2 Introducing Binary Choice Models 190 7.1.3 An Underlying Latent Model 192 7.1.4 Estimation 193 7.1.5 Goodness-of-fit 194 7.1.6 Illustration: the Impact of Unemployment Benefits on Recipiency 197 7.1.7 Specification Tests in Binary Choice Models 199 7.1.8 Relaxing Some Assumptions in Binary Choice Models 201 7.2 Multi-response Models 202 7.2.1 Ordered Response Models 203 7.2.2 About Normalization 204 7.2.3 Illustration: Willingness to Pay for Natural Areas 205 7.2.4 Multinomial Models 208 7.3 Models for Count Data 211 7.3.1 The Poisson and Negative Binomial Models 211 7.3.2 Illustration: Patents and R&D Expenditures 215 7.4 Tobit Models 218 7.4.1 The Standard Tobit Model 218 7.4.2 Estimation 220 [...]... 5 Now that our β coefficients have a meaning, we can try to use the sample (yi , xi ), i = 1, , N to say something about it The rule which says how a given sample is translated into an approximate value for β is referred to as an estimator The result for a given sample is called an estimate The estimator is a vector of random variables, because the sample may change The estimate is a vector of numbers... to alternative states of the world In such a case we will need to make some assumptions about the way the data are generated (rather than the way the data are sampled) It is important to realize that without additional restrictions the statistical model in (2.25) is a tautology: for any value for β one can always define a set of εi s such that (2.25) holds exactly for each observation We thus need to. .. while we only observe a sample of N observations We shall consider this sample as one realization of all potential samples of size N that could have been drawn from the same population In this way we can view yi and εi (and often xi ) as random variables Each observation corresponds to a realization of these random variables Again we can use matrix notation and stack all observations to write y = Xβ +... sample covariance between x and y and the sample variance of x From (2.15), the intercept is determined so as to make the average approximation error (residual) equal to zero AN INTRODUCTION TO LINEAR REGRESSION 12 2.1.3 Example: Individual Wages An example that will appear frequently in this chapter is based on a sample of individual wages with background characteristics, like gender, race and years... that are close to the ones reported I do not advocate the use of any particular software package For the linear regression 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 else than what’s... use a subsample of the US National Longitudinal Survey (NLS) that relates to 1987 and we have a sample of 3294 young working individuals, of which 1569 are female.3 The average hourly wage rate in this sample equals $6.31 for males and $5.15 for females Now suppose we try to approximate wages by a linear combination of a constant and a 0–1 variable denoting whether the individual is male or not That... Note that we did not use assumptions (A3 ) and (A4 ) in the proof This shows that the OLS estimator is unbiased as long as the error terms are mean zero and independent of all explanatory variables, even if heteroskedasticity or autocorrelation are present We shall come back to this issue in Chapter 4 In addition to knowing that we are, on average, correct, we would also like to make statements about... 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 (for example households, firms or countries) The last decade the use of panel data has become important in many areas of economics Micro-economic panels of households and firms are readily available and, given the increase in computing... Needless to say that these assumptions are hardly satisfied in practice (but not really needed either) 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 This book tries to fill this gap The goal of this book is to familiarize the reader with a wide range of topics... Moreover, Chapter 2 now includes a subsection on Monte Carlo simulation At several places, I pay more attention to the possibility that small sample distributions of estimators and test statistics may differ from their asymptotic approximations Several new tests have been added to Chapters 3 and 5, and the presentation in Chapters 6 and 8 has been improved At a number of places, empirical illustrations have . that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Verbeek, Marno. A guide to modern econometrics. Chapters 6 and 8 has been improved. At a number of places, empirical illustrations have been updated or added. As before, (almost) all data sets are available