Quantile Regression

Marilena Furno, Department of Agriculture, University of Naples Federico II, Italy Domenico Vistocco, Department of Economics and Law, University of Cassino, Italy

• Preface xiAcknowledgements xiiiIntroduction xvAbout the companion website xix1 Robust regression 1Introduction 11.1 The Anscombe data and OLS 11.2 The Ancombe data and quantile regression 81.2.1 Real data examples: the French data 121.2.2 The Netherlands example 141.3 The influence function and the diagnostic tools 171.3.1 Diagnostic in the French and the Dutch data 221.3.2 Example with error contamination 221.4 A summary of key points 26References 26Appendix: computer codes in Stata 272 Quantile regression and related methods 29Introduction 292.1 Expectiles 302.1.1 Expectiles and contaminated errors 392.1.2 French data: influential outlier in the dependent variable 392.1.3 The Netherlands example: outlier in the explanatoryvariable 452.2 M-estimators 492.2.1 M-estimators with error contamination 542.2.2 The French data 582.2.3 The Netherlands example 592.3 M-quantiles 602.3.1 M-quantiles estimates in the error-contaminated model 642.3.2 M-quantiles in the French and Dutch examples 642.3.3 Further applications: small-area estimation 702.4 A summary of key points 72References 73Appendix: computer codes 74[1][1] [1][1]viii CONTENTS3 Resampling, subsampling, and quantile regression 81Introduction 813.1 Elemental sets 813.2 Bootstrap and elemental sets 893.3 Bootstrap for extremal quantiles 943.3.1 The French data set 973.3.2 The Dutch data set 983.4 Asymptotics for central-order quantiles 1003.5 Treatment effect and decomposition 1013.5.1 Quantile treatment effect and decomposition 1073.6 A summary of key points 117References 118Appendix: computer codes 1204 A not so short introduction to linear programming 127Introduction 1274.1 The linear programming problem 1274.1.1 The standard form of a linear programming problem 1294.1.2 Assumptions of a linear programming problem 1314.1.3 The geometry of linear programming 1324.2 The simplex algorithm 1414.2.1 Basic solutions 1414.2.2 Optimality test 1474.2.3 Change of the basis: entering variable and leaving variable 1484.2.4 The canonical form of a linear programming problem 1504.2.5 The simplex algorithm 1534.2.6 The tableau version of the simplex algorithm 1594.3 The two–phase method 1684.4 Convergence and degeneration of the simplex algorithm 1764.5 The revised simplex algorithm 1814.6 A summary of key points 190References 1905 Linear programming for quantile regression 191Introduction 1915.1 LP formulation of the L1 simple regression problem 1915.1.1 A first formulation of the L1 regression problem 1935.1.2 A more convenient formulation of the L1 regressionproblem 2045.1.3 The Barrodale–Roberts algorithm for L1 regression 2105.2 LP formulation of the quantile regression problem 2175.3 Geometric interpretation of the median and quantile regressionproblem: the dual plot 2185.4 A summary of key points 228References 229[1][1] [1][1]CONTENTS ix6 Correlation 233Introduction 2336.1 Autoregressive models 2336.2 Non-stationarity 2426.2.1 Examples of non-stationary series 2436.3 Inference in the unit root model 2486.3.1 Related tests for unit root 2526.4 Spurious regression 2546.5 Cointegration 2596.5.1 Example of cointegrated variables 2606.5.2 Cointegration tests 2616.6 Tests of changing coefficients 2626.6.1 Examples of changing coefficients 2656.7 Conditionally heteroskedastic models 2696.7.1 Example of a conditional heteroskedastic model 2726.8 A summary of key points 274References 274Appendix: Stata computer codes 275Index 283