Yves Croissant, Professor of Economics, CEMOI, Faculté de Droit et d'Economie, Université de La Réunion, France

Giovanni Millo, Senior Economist, Group Insurance Research, Assicurazioni Generali S.p.A., Trieste, Italy

Preface xiii

Acknowledgments xvii

About the CompanionWebsite xix

1 Introduction 1

1.1 Panel Data Econometrics: A Gentle Introduction 1

1.1.1 Eliminating Unobserved Components 2

1.1.1.1 Differencing Methods 2

1.1.1.2 LSDV Methods 2

1.1.1.3 Fixed Effects Methods 2

1.2 R for Econometric Computing 6

1.2.1 The Modus Operandi of R 7

1.2.2 Data Management 8

1.2.2.1 Outsourcing to Other Software 8

1.2.2.2 Data ManagementThrough Formulae 8

1.3 plm for the Casual R User 8

1.3.1 R for the Matrix Language User 9

1.3.2 R for the User of Econometric Packages 10

1.4 plm for the Proficient R User 11

1.4.1 Reproducible EconometricWork 12

1.4.2 Object-orientation for the User 13

1.5 plm for the R Developer 13

1.5.1 Object-orientation for Development 14

1.6 Notations 17

1.6.1 General Notation 18

1.6.2 Maximum Likelihood Notations 18

1.6.3 Index 18

1.6.4 The Two-way Error Component Model 18

1.6.5 Transformation for the One-way Error Component Model 19

1.6.6 Transformation for the Two-ways Error Component Model 20

1.6.7 Groups and Nested Models 20

1.6.8 Instrumental Variables 20

1.6.9 Systems of Equations 20

1.6.10 Time Series 21

1.6.11 Limited Dependent and Count Variables 21

1.6.12 Spatial Panels 21

2 The Error Component Model 23

2.1 Notations and Hypotheses 23

2.1.1 Notations 23

2.1.2 Some Useful Transformations 24

2.1.3 Hypotheses Concerning the Errors 25

2.2 Ordinary Least Squares Estimators 27

2.2.1 Ordinary Least Squares on the Raw Data: The Pooling Model 27

2.2.2 The between Estimator 28

2.2.3 The within Estimator 29

2.3 The Generalized Least Squares Estimator 33

2.3.1 Presentation of the GLS Estimator 34

2.3.2 Estimation of the Variances of the Components of the Error 35

2.4 Comparison of the Estimators 39

2.4.1 Relations between the Estimators 39

2.4.2 Comparison of the Variances 40

2.4.3 Fixed vs Random Effects 40

2.4.4 Some Simple Linear Model Examples 42

2.5 The Two-ways Error Components Model 47

2.5.1 Error Components in the Two-ways Model 47

2.5.2 Fixed and Random Effects Models 48

2.6 Estimation of a Wage Equation 49

3 Advanced Error Components Models 53

3.1 Unbalanced Panels 53

3.1.1 Individual Effects Model 53

3.1.2 Two-ways Error Component Model 54

3.1.2.1 Fixed Effects Model 55

3.1.2.2 Random Effects Model 56

3.1.3 Estimation of the Components of the Error Variance 57

3.2 Seemingly Unrelated Regression 64

3.2.1 Introduction 64

3.2.2 Constrained Least Squares 65

3.2.3 Inter-equations Correlation 66

3.2.4 SUR With Panel Data 67

3.3 The Maximum Likelihood Estimator 71

3.3.1 Derivation of the Likelihood Function 71

3.3.2 Computation of the Estimator 73

3.4 The Nested Error Components Model 74

3.4.1 Presentation of the Model 74

3.4.2 Estimation of the Variance of the Error Components 75

4 Tests on Error Component Models 83

4.1 Tests on Individual and/or Time Effects 83

4.1.1 F Tests 84

4.1.2 Breusch-Pagan Tests 84

4.2 Tests for Correlated Effects 88

4.2.1 The Mundlak Approach 89

4.2.2 Hausman Test 90

4.2.3 Chamberlain's Approach 90

4.2.3.1 Unconstrained Estimator 91

4.2.3.2 Constrained Estimator 93

4.2.3.3 Fixed Effects Models 93

4.3 Tests for Serial Correlation 95

4.3.1 Unobserved Effects Test 95

4.3.2 Score Test of Serial Correlation and/or Individual Effects 96

4.3.3 Likelihood Ratio Tests for AR(1) and Individual Effects 99

4.3.4 Applying Traditional Serial Correlation Tests to Panel Data 101

4.3.5 Wald Tests for Serial Correlation using within and First-differenced Estimators 102

4.3.5.1 Wooldridge's within-based Test 102

4.3.5.2 Wooldridge's First-difference-based Test 103

4.4 Tests for Cross-sectional Dependence 104

4.4.1 Pairwise Correlation Coefficients 104

4.4.2 CD-type Tests for Cross-sectional Dependence 105

4.4.3 Testing Cross-sectional Dependence in a pseries 107

5 Robust Inference and Estimation for Non-spherical Errors 109

5.1 Robust Inference 109

5.1.1 Robust Covariance Estimators 109

5.1.1.1 Cluster-robust Estimation in a Panel Setting 110

5.1.1.2 Double Clustering 115

5.1.1.3 Panel Newey-west and SCC 116

5.1.2 Generic Sandwich Estimators and Panel Models 120

5.1.2.1 Panel Corrected Standard Errors 122

5.1.3 Robust Testing of Linear Hypotheses 123

5.1.3.1 An Application: Robust Hausman Testing 125

5.2 Unrestricted Generalized Least Squares 127

5.2.1 General Feasible Generalized Least Squares 128

5.2.1.1 Pooled GGLS 129

5.2.1.2 Fixed Effects GLS 130

5.2.1.3 First Difference GLS 132

5.2.2 Applied Examples 133

6 Endogeneity 139

6.1 Introduction 139

6.2 The Instrumental Variables Estimator 140

6.2.1 Generalities about the Instrumental Variables Estimator 140

6.2.2 The within Instrumental Variables Estimator 141

6.3 Error Components Instrumental Variables Estimator 143

6.3.1 The General Model 143

6.3.2 Special Cases of the General Model 145

6.3.2.1 The within Model 145

6.3.2.2 Error Components Two Stage Least Squares 146

6.3.2.3 The Hausman and Taylor Model 146

6.3.2.4 The Amemiya-Macurdy Estimator 147

6.3.2.5 The Breusch, Mizon and Schmidt's Estimator 147

6.3.2.6 Balestra and Varadharajan-Krishnakumar Estimator 147

6.4 Estimation of a System of Equations 154

6.4.1 TheThree Stage Least Squares Estimator 155

6.4.2 The Error Components Three Stage Least Squares Estimator 156

6.5 More Empirical Examples 158

7 Estimation of a Dynamic Model 161

7.1 Dynamic Model and Endogeneity 163

7.1.1 The Bias of the ols Estimator 163

7.1.2 The within Estimator 164

7.1.3 Consistent Estimation Methods for Dynamic Models 165

7.2 GMM Estimation of the Differenced Model 168

7.2.1 Instrumental Variables and Generalized Method of Moments 168

7.2.2 One-step Estimator 169

7.2.3 Two-steps Estimator 171

7.2.4 The Proliferation of Instruments in the Generalized Method of Moments Difference Estimator 172

7.3 Generalized Method of Moments Estimator in Differences and Levels 174

7.3.1 Weak Instruments 174

7.3.2 Moment Conditions on the Levels Model 175

7.3.3 The System GMM Estimator 177

7.4 Inference 178

7.4.1 Robust Estimation of the Coefficients' Covariance 178

7.4.2 Overidentification Tests 179

7.4.3 Error Serial Correlation Test 181

7.5 More Empirical Examples 182

8 Panel Time Series 185

8.1 Introduction 185

8.2 Heterogeneous Coefficients 186

8.2.1 Fixed Coefficients 186

8.2.2 Random Coefficients 187

8.2.2.1 The Swamy Estimator 187

8.2.2.2 The Mean Groups Estimator 190

8.2.3 Testing for Poolability 192

8.3 Cross-sectional Dependence and Common Factors 194

8.3.1 The Common Factor Model 195

8.3.2 Common Correlated Effects Augmentation 196

8.3.2.1 cce Mean Groups vs. cce Pooled 198

8.3.2.2 Computing the ccep Variance 199

8.4 Nonstationarity and Cointegration 200

8.4.1 Unit Root Testing: Generalities 201

8.4.2 First Generation Unit Root Testing 204

8.4.2.1 Preliminary Results 204

8.4.2.2 Levin-Lin-Chu Test 205

8.4.2.3 Im, Pesaran and Shin Test 205

8.4.2.4 The Maddala and Wu Test 206

8.4.3 Second Generation Unit Root Testing 207

9 Count Data and Limited Dependent Variables 211

9.1 Binomial and Ordinal Models 213

9.1.1 Introduction 213

9.1.1.1 The Binomial Model 213

9.1.1.2 Ordered Models 214

9.1.2 The Random Effects Model 214

9.1.2.1 The Binomial Model 214

9.1.2.2 Ordered Models 217

9.1.3 The Conditional Logit Model 219

9.2 Censored or Truncated Dependent Variable 223

9.2.1 Introduction 223

9.2.2 The Ordinary Least Squares Estimator 223

9.2.3 The Symmetrical Trimmed Estimator 225

9.2.3.1 Truncated Sample 225

9.2.3.2 Censored Sample 226

9.2.4 The Maximum Likelihood Estimator 226

9.2.4.1 Truncated Sample 226

9.2.4.2 Censored Sample 227

9.2.5 Fixed Effects Model 227

9.2.5.1 Truncated Sample 227

9.2.5.2 Censored Sample 229

9.2.6 The Random Effects Model 233

9.2.6.1 Truncated Sample 233

9.2.6.2 Censored Sample 234

9.3 Count Data 236

9.3.1 Introduction 236

9.3.1.1 The Poisson Model 236

9.3.1.2 The NegBin Model 237

9.3.2 Fixed Effects Model 237

9.3.2.1 The Poisson Model 237

9.3.2.2 Negbin Model 239

9.3.3 Random Effects Models 239

9.3.3.1 The Poisson Model 239

9.3.3.2 The NegBin Model 240

9.4 More Empirical Examples 243

10 Spatial Panels 245

10.1 Spatial Correlation 245

10.1.1 Visual Assessment 245

10.1.2 Testing for Spatial Dependence 246

10.1.2.1 CD p Tests for Local Cross-sectional Dependence 247

10.1.2.2 The Randomized W Test 247

10.2 Spatial Lags 250

10.2.1 Spatially Lagged Regressors 251

10.2.2 Spatially Lagged Dependent Variables 253

10.2.2.1 Spatial OLS 254

10.2.2.2 ML Estimation of the sar Model 254

10.2.3 Spatially Correlated...