Contains an overview of several technical topics of Quantile Regression
Volume two of Quantile Regression offers an important guide for applied researchers that draws on the same example-based approach adopted for the first volume. The text explores topics including robustness, expectiles, m-quantile, decomposition, time series, elemental sets and linear programming. Graphical representations are widely used to visually introduce several issues, and to illustrate each method. All the topics are treated theoretically and using real data examples. Designed as a practical resource, the book is thorough without getting too technical about the statistical background.
The authors cover a wide range of QR models useful in several fields. The software commands in R and Stata are available in the appendixes and featured on the accompanying website. The text:
- Provides an overview of several technical topics such as robustness of quantile regressions, bootstrap and elemental sets, treatment effect estimators
- Compares quantile regression with alternative estimators like expectiles, M-estimators and M-quantiles
- Offers a general introduction to linear programming focusing on the simplex method as solving method for the quantile regression problem
- Considers time-series issues like non-stationarity, spurious regressions, cointegration, conditional heteroskedasticity via quantile regression
- Offers an analysis that is both theoretically and practical
- Presents real data examples and graphical representations to explain the technical issues
Written for researchers and students in the fields of statistics, economics, econometrics, social and environmental science, this text offers guide to the theory and application of quantile regression models.
Table of Contents
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
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
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
