Quantile Regression

Preface ix

Acknowledgments xi

Introduction xii

Nomenclature xv

1 A visual introduction to quantile regression 1

Introduction 1

1.1 The essential toolkit 1

1.2 The simplest QR model: The case of the dummy regressor 8

1.3 A slightly more complex QR model: The case of a nominal regressor 13

1.4 A typical QR model: The case of a quantitative regressor 15

1.5 Summary of key points 20

References 21

2 Quantile regression: Understanding how and why 22

Introduction 22

2.1 How and why quantile regression works 22

2.2 A set of illustrative artificial data 33

2.3 How and why to work with QR 38

2.4 Summary of key points 60

References 62

3 Estimated coefficients and inference 64

Introduction 64

3.1 Empirical distribution of the quantile regression estimator 64

3.2 Inference in QR, the i.i.d. case 76

3.3 Wald, Lagrange multiplier, and likelihood ratio tests 84

3.4 Summary of key points 92

References 93

4 Additional tools for the interpretation and evaluation of the quantile regression model 94

Introduction 94

4.1 Data pre-processing 95

4.2 Response conditional density estimations 107

4.3 Validation of the model 117

4.4 Summary of key points 128

References 128

5 Models with dependent and with non-identically distributed data 131

Introduction 131

5.1 A closer look at the scale parameter, the independent and identically distributed case 131

5.2 The non-identically distributed case 137

5.3 The dependent data model 152

5.4 Summary of key points 158

References 158

Appendix 5.A Heteroskedasticity tests and weighted quantile regression, Stata and R codes 159

5.A.1 Koenker and Basset test for heteroskedasticity comparing two quantile regressions 159

5.A.2 Koenker and Basset test for heteroskedasticity comparing all quantile regressions 159

5.A.3 Quick tests for heteroskedasticity comparing quantile regressions 160

5.A.4 Compute the individual role of each explanatory variable to the dependent variable 161

5.A.5 R-codes for the Koenker and Basset test for heteroskedasticity 161

Appendix 5.B Dependent data 162

6 Additional models 163

Introduction 163

6.1 Nonparametric quantile regression 163

6.2 Nonlinear quantile regression 172

6.3 Censored quantile regression 175

6.4 Quantile regression with longitudinal data 183

6.5 Group effects through quantile regression 187

6.6 Binary quantile regression 195

6.7 Summary of key points 197

References 197

Appendix A Quantile regression and surroundings using R 201

Introduction 201

A.1 Loading data 202

A.2 Exploring data 205

A.3 Modeling data 211

A.4 Exporting figures and tables 217

References 218

Appendix B Quantile regression and surroundings using SAS 220

Introduction 220

B.1 Loading data 221

B.2 Exploring data 223

B.3 Modeling data 229

B.4 Exporting figures and tables 239

References 241

Appendix C Quantile regression and surroundings using Stata 242

Introduction 242

C.1 Loading data 243

C.2 Exploring data 245

C.3 Modeling data 249

C.4 Exporting figures and tables 255

References 256

Index 257