Quantile Regression

Preface xi

Acknowledgements xiii

Introduction xv

About the companion website xix

1 Robust regression 1

Introduction 1

1.1 The Anscombe data and OLS 1

1.2 The Ancombe data and quantile regression 8

1.2.1 Real data examples: the French data 12

1.2.2 The Netherlands example 14

1.3 The influence function and the diagnostic tools 17

1.3.1 Diagnostic in the French and the Dutch data 22

1.3.2 Example with error contamination 22

1.4 A summary of key points 26

References 26

Appendix: computer codes in Stata 27

2 Quantile regression and related methods 29

Introduction 29

2.1 Expectiles 30

2.1.1 Expectiles and contaminated errors 39

2.1.2 French data: influential outlier in the dependent variable 39

2.1.3 The Netherlands example: outlier in the explanatory

variable 45

2.2 M-estimators 49

2.2.1 M-estimators with error contamination 54

2.2.2 The French data 58

2.2.3 The Netherlands example 59

2.3 M-quantiles 60

2.3.1 M-quantiles estimates in the error-contaminated model 64

2.3.2 M-quantiles in the French and Dutch examples 64

2.3.3 Further applications: small-area estimation 70

2.4 A summary of key points 72

References 73

Appendix: computer codes 74

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viii CONTENTS

3 Resampling, subsampling, and quantile regression 81

Introduction 81

3.1 Elemental sets 81

3.2 Bootstrap and elemental sets 89

3.3 Bootstrap for extremal quantiles 94

3.3.1 The French data set 97

3.3.2 The Dutch data set 98

3.4 Asymptotics for central-order quantiles 100

3.5 Treatment effect and decomposition 101

3.5.1 Quantile treatment effect and decomposition 107

3.6 A summary of key points 117

References 118

Appendix: computer codes 120

4 A not so short introduction to linear programming 127

Introduction 127

4.1 The linear programming problem 127

4.1.1 The standard form of a linear programming problem 129

4.1.2 Assumptions of a linear programming problem 131

4.1.3 The geometry of linear programming 132

4.2 The simplex algorithm 141

4.2.1 Basic solutions 141

4.2.2 Optimality test 147

4.2.3 Change of the basis: entering variable and leaving variable 148

4.2.4 The canonical form of a linear programming problem 150

4.2.5 The simplex algorithm 153

4.2.6 The tableau version of the simplex algorithm 159

4.3 The two-phase method 168

4.4 Convergence and degeneration of the simplex algorithm 176

4.5 The revised simplex algorithm 181

4.6 A summary of key points 190

References 190

5 Linear programming for quantile regression 191

Introduction 191

5.1 LP formulation of the L1 simple regression problem 191

5.1.1 A first formulation of the L1 regression problem 193

5.1.2 A more convenient formulation of the L1 regression

problem 204

5.1.3 The Barrodale-Roberts algorithm for L1 regression 210

5.2 LP formulation of the quantile regression problem 217

5.3 Geometric interpretation of the median and quantile regression

problem: the dual plot 218

5.4 A summary of key points 228

References 229

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CONTENTS ix

6 Correlation 233

Introduction 233

6.1 Autoregressive models 233

6.2 Non-stationarity 242

6.2.1 Examples of non-stationary series 243

6.3 Inference in the unit root model 248

6.3.1 Related tests for unit root 252

6.4 Spurious regression 254

6.5 Cointegration 259

6.5.1 Example of cointegrated variables 260

6.5.2 Cointegration tests 261

6.6 Tests of changing coefficients 262

6.6.1 Examples of changing coefficients 265

6.7 Conditionally heteroskedastic models 269

6.7.1 Example of a conditional heteroskedastic model 272

6.8 A summary of key points 274

References 274

Appendix: Stata computer codes 275

Index 283