Quantile Regression
Roger Koenker
University of Illinois
Contents
viii Contents
Inference for Quantile Regression 68
3.1 The Finite-Sample Distribution of Regression Quantiles 68
3.2 Heuristic Introduction to Quantile Regression
Asymptotics 71
3.2.1 Confidence Intervals for the Sample Quantiles 72
3.2.2 Quantile Regression Asymptotics with IID Errors 73
3.2.3 Quantile Regression Asymptotics in Non-IID
Settings 74
3.3 Wald Tests 75
3.3.1 Two-Sample Tests of Location Shift 75
3.3.2 General Linear Hypotheses 76
3.4 Estimation of Asymptotic Covariance Matrices 77
3.4.1 Scalar Sparsity Estimation 77
3.4.2 Covariance Matrix Estimation in Non-IID Settings 79
3.5 Rank-Based Inference 81
3.5.1 Rank Tests for Two-Sample Location Shift 81
3.5.2 Linear Rank Statistics 84
3.5.3 Asymptotics of Linear Rank Statistics 85
3.5.4 Rank Tests Based on Regression Rankscores 87
3.5.5 Confidence Intervals Based on Regression
Rankscores 91
3.6 Quantile Likelihood Ratio Tests 92
3.7 Inference on the Quantile Regression Process 95
3.7.1 Wald Processes 97
3.7.2 Quantile Likelihood Ratio Processes 98
3.7.3 The Regression Rankscore Process Revisited 98
3.8 Tests of the Location-Scale Hypothesis 98
3.9 Resampling Methods and the Bootstrap 105
3.9.1 Bootstrap Refinements, Smoothing, and
Subsampling 107
3.9.2 Resampling on the Subgradient Condition 108
3.10 Monte Carlo Comparison of Methods 110
3.10.1 Model 1: Location-Shift Model 111
3.10.2 Model 2: Location–Scale-Shift Model 112
3.11 Problems 113
Asymptotic Theory of Quantile Regression 116
4.1 Consistency 117
4.1.1 Univariate Sample Quantiles 117
4.1.2 Linear Quantile Regression 118
4.2 Rates of Convergence 120
4.3 Bahadur Representation 122
4.4 Nonlinear Quantile Regression 123
4.5 The Quantile Regression Rankscore Process 124
4.6 Quantile Regression Asymptotics under Dependent
Conditions 126
Contents
6.4.3 Interior vs. Exterior: Computational
Comparison 202
6.4.4 Computational Complexity 204
6.5 Preprocessing for Quantile Regression 206
6.5.1 “Selecting” Univariate Quantiles 207
6.5.2 Implementation 207
6.5.3 Confidence Bands 208
6.5.4 Choosing 209
6.6 Nonlinear Quantile Regression 211
6.7 Inequality Constraints 213
6.8 Weighted Sums of ρτ -Functions 214
6.9 Sparsity 216
6.10 Conclusion 220
6.11 Problems 220
Nonparametric Quantile Regression 222
7.1 Locally Polynomial Quantile Regression 222
7.1.1 Average Derivative Estimation 226
7.1.2 Additive Models 228
7.2 Penalty Methods for Univariate Smoothing 229
7.2.1 Univariate Roughness Penalties 229
7.2.2 Total Variation Roughness Penalties 230
7.3 Penalty Methods for Bivariate Smoothing 235
7.3.1 Bivariate Total Variation Roughness Penalties 235
7.3.2 Total Variation Penalties for Triograms 236
7.3.3 Penalized Triogram Estimation as Linear
Program 240
7.3.4 On Triangulation 241
7.3.5 On Sparsity 242
7.3.6 Automatic Selection 242
7.3.7 Boundary and Qualitative Constraints 243
7.3.8 Model of Chicago Land Values 243
7.3.9 Taut Strings and Edge Detection 246
7.4 Additive Models and the Role of Sparsity 248
Twilight Zone of Quantile Regression 250
8.1 Quantile Regression for Survival Data 250
8.1.1 Quantile Functions or Hazard Functions? 252
8.1.2 Censoring 253
8.2 Discrete Response Models 255
8.2.1 Binary Response 255
8.2.2 Count Data 259
8.3 Quantile Autoregression 260
8.3.1 Quantile Autoregression and Comonotonicity 261
8.4 Copula Functions and Nonlinear Quantile Regression 265
8.4.1 Copula Functions 265
Contents xi
8.5 High-Breakdown Alternatives to Quantile Regression 268
8.6 Multivariate Quantiles 272
8.6.1 The Oja Median and Its Extensions 273
8.6.2 Half-Space Depth and Directional Quantile
Regression 275
8.7 Penalty Methods for Longitudinal Data 276
8.7.1 Classical Random Effects as Penalized
Least Squares 276
8.7.2 Quantile Regression with Penalized Fixed Effects 278
8.8 Causal Effects and Structural Models 281
8.8.1 Structural Equation Models 281
8.8.2 Chesher’s Causal Chain Model 283
8.8.3 Interpretation of Structural Quantile Effects 284
8.8.4 Estimation and Inference 285
8.9 Choquet Utility, Risk, and Pessimistic Portfolios 287
8.9.1 Choquet Expected Utility 287
8.9.2 Choquet Risk Assessment 289
8.9.3 Pessimistic Portfolios 291
Conclusion 293
Quantile Regression in R: Vignette 295
A.Introduction 295
A.What Is Vignette? 296
A.Getting Started 296
A.Object Orientation 298
A.Formal Inference 299
A.More on Testing 305
A.Inference on the Quantile Regression Process 307
A.Nonlinear Quantile Regression 308
A.Nonparametric Quantile Regression 310
A.10 Conclusion 316
Asymptotic Critical Values 317
References 319
Name Index 337
Subject Index 342
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