A Guide to Modern Econometrics


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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

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Library of Congress Cataloging-in-Publication Data
Verbeek, Marno. A guide to modern econometrics / Marno Verbeek. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0-470-85773-0 (pbk. : alk. paper) 1. Econometrics. 2. Regression analysis. I. Title. 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 Endogeneity, 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 Models 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

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A Guide to Modern Econometrics