Quantile Regression

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Paperback | 16 August 2005

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Quantile regression is gradually emerging as a unified statistical methodology for estimating models of conditional quantile functions. By complementing the exclusive focus of classical least squares regression on the conditional mean, quantile regression offers a systematic strategy for examining how covariates influence the location, scale and shape of the entire response distribution. This monograph is the first comprehensive treatment of the subject, encompassing models that are linear and nonlinear, parametric and nonparametric. The author has devoted more than 25 years of research to this topic.

The methods in the analysis are illustrated with a variety of applications from economics, biology, ecology and finance. The treatment will find its core audiences in econometrics, statistics, and applied mathematics in addition to the disciplines cited above.

About the Author

Roger Koenker is McKinley Professor of Economics and Professor of Statistics at the University of Illinois at Urbana-Champaign. From 1976 to 1983 he was a member of the technical staff at Bell Laboratories. He has held visiting positions at The University of Pennsylvania, Charles University, Prague, Nuffield College, Oxford, University College London and Australian National University. He is a Fellow of the Econometric Society.

Industry Reviews

'... well written and easy to read, with useful illustrations of important aspects of quantile regression. It is obvious that the author knows the subject inside out, giving an up-to-date, exhaustive account of the subject. ... The book is a valuable contribution to the statistical literature, and a must have for every statistician or econometrician interested in quantile regression methods.' Journal of the Royal Statistical Society 'It is well written and easy to read, with useful ilustrations of important aspects of quantile regression ... a valuable contribution to the statistical literature and is essential for every statistician or econometrician who is interested in quantile regression methods.' Andreas Karlsson, Uppsala University

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Preface	p. xiii
Introduction	p. 1
Means and Ends	p. 1
The First Regression: A Historical Prelude	p. 2
Quantiles, Ranks, and Optimization	p. 5
Preview of Quantile Regression	p. 9
Three Examples	p. 13
Salaries versus Experience	p. 13
Student Course Evaluations and Class Size	p. 17
Infant Birth Weight	p. 20
Conclusion	p. 25
Fundamentals of Quantile Regression	p. 26
Quantile Treatment Effects	p. 26
How Does Quantile Regression Work?	p. 32
Regression Quantiles Interpolate p Observations	p. 33
The Subgradient Condition	p. 34
Equivariance	p. 38
Censoring	p. 40
Robustness	p. 42
The Influence Function	p. 42
The Breakdown Point	p. 45
Interpreting Quantile Regression Models	p. 47
Some Examples	p. 48
Caution: Quantile Crossing	p. 55
A Random Coefficient Interpretation	p. 59
Inequality Measures and Their Decomposition	p. 62
Expectiles and Other Variations	p. 63
Interpreting Misspecified Quantile Regressions	p. 65
Problems	p. 66
Inference for Quantile Regression	p. 68
The Finite-Sample Distribution of Regression Quantiles	p. 68
A Heuristic Introduction to Quantile Regression Asymptotics	p. 71
Confidence Intervals for the Sample Quantiles	p. 72
Quantile Regression Asymptotics with IID Errors	p. 73
Quantile Regression Asymptotics in Non-IID Settings	p. 74
Wald Tests	p. 75
Two-Sample Tests of Location Shift	p. 75
General Linear Hypotheses	p. 76
Estimation of Asymptotic Covariance Matrices	p. 77
Scalar Sparsity Estimation	p. 77
Covariance Matrix Estimation in Non-IID Settings	p. 79
Rank-Based Inference	p. 81
Rank Tests for Two-Sample Location Shift	p. 81
Linear Rank Statistics	p. 84
Asymptotics of Linear Rank Statistics	p. 85
Rank Tests Based on Regression Rankscores	p. 87
Confidence Intervals Based on Regression Rankscores	p. 91
Quantile Likelihood Ratio Tests	p. 92
Inference on the Quantile Regression Process	p. 95
Wald Processes	p. 97
Quantile Likelihood Ratio Processes	p. 98
The Regression Rankscore Process Revisited	p. 98
Tests of the Location-Scale Hypothesis	p. 98
Resampling Methods and the Bootstrap	p. 105
Bootstrap Refinements, Smoothing, and Subsampling	p. 107
Resampling on the Subgradient Condition	p. 108
Monte Carlo Comparison of Methods	p. 110
Model 1: A Location-Shift Model	p. 111
Model 2: A Location-Scale-Shift Model	p. 112
Problems	p. 113
Asymptotic Theory of Quantile Regression	p. 116
Consistency	p. 117
Univariate Sample Quantiles	p. 117
Linear Quantile Regression	p. 118
Rates of Convergence	p. 120
Bahadur Representation	p. 122
Nonlinear Quantile Regression	p. 123
The Quantile Regression Rankscore Process	p. 124
Quantile Regression Asymptotics under Dependent Conditions	p. 126
Autoregression	p. 126
ARMA Models	p. 128
ARCH-like Models	p. 129
Extremal Quantile Regression	p. 130
The Method of Quantiles	p. 131
Model Selection, Penalties, and Large-p Asymptotics	p. 133
Model Selection	p. 134
Penalty Methods	p. 135
Asymptotics for Inference	p. 138
Scalar Sparsity Estimation	p. 139
Covariance Matrix Estimation	p. 141
Resampling Schemes and the Bootstrap	p. 141
Asymptotics for the Quantile Regression Process	p. 142
The Durbin Problem	p. 142
Khmaladization of the Parametric Empirical Process	p. 144
The Parametric Quantile Process	p. 145
The Parametric Quantile Regression Process	p. 146
Problems	p. 149
L-Statistics and Weighted Quantile Regression	p. 151
L-Statistics for the Linear Model	p. 151
Optimal L-Estimators of Location and Scale	p. 153
L-Estimation for the Linear Model	p. 155
Kernel Smoothing for Quantile Regression	p. 158
Kernel Smoothing of the [rho subscript tau]-Function	p. 160
Weighted Quantile Regression	p. 160
Weighted Linear Quantile Regression	p. 160
Estimating Weights	p. 161
Quantile Regression for Location-Scale Models	p. 164
Weighted Sums of [rho subscript tau]-Functions	p. 168
Problems	p. 170
Computational Aspects of Quantile Regression	p. 173
Introduction to Linear Programming	p. 173
Vertices	p. 174
Directions of Descent	p. 176
Conditions for Optimality	p. 177
Complementary Slackness	p. 178
Duality	p. 180
Simplex Methods for Quantile Regression	p. 181
Parametric Programming for Quantile Regression	p. 185
Parametric Programming for Regression Rank Tests	p. 188
Interior Point Methods for Canonical LPs	p. 190
Newton to the Max: An Elementary Example	p. 193
Interior Point Methods for Quantile Regression	p. 199
Interior vs. Exterior: A Computational Comparison	p. 202
Computational Complexity	p. 204
Preprocessing for Quantile Regression	p. 206
"Selecting" Univariate Quantiles	p. 207
Implementation	p. 207
Confidence Bands	p. 208
Choosing m	p. 209
Nonlinear Quantile Regression	p. 211
Inequality Constraints	p. 213
Weighted Sums of [rho subscript tau]-Functions	p. 214
Sparsity	p. 216
Conclusion	p. 220
Problems	p. 220
Nonparametric Quantile Regression	p. 222
Locally Polynomial Quantile Regression	p. 222
Average Derivative Estimation	p. 226
Additive Models	p. 228
Penalty Methods for Univariate Smoothing	p. 229
Univariate Roughness Penalties	p. 229
Total Variation Roughness Penalties	p. 230
Penalty Methods for Bivariate Smoothing	p. 235
Bivariate Total Variation Roughness Penalties	p. 235
Total Variation Penalties for Triograms	p. 236
Penalized Triogram Estimation as a Linear Program	p. 240
On Triangulation	p. 241
On Sparsity	p. 242
Automatic [lambda] Selection	p. 242
Boundary and Qualitative Constraints	p. 243
A Model of Chicago Land Values	p. 243
Taut Strings and Edge Detection	p. 246
Additive Models and the Role of Sparsity	p. 248
Twilight Zone of Quantile Regression	p. 250
Quantile Regression for Survival Data	p. 250
Quantile Functions or Hazard Functions?	p. 252
Censoring	p. 253
Discrete Response Models	p. 255
Binary Response	p. 255
Count Data	p. 259
Quantile Autoregression	p. 260
Quantile Autoregression and Comonotonicity	p. 261
Copula Functions and Nonlinear Quantile Regression	p. 265
Copula Functions	p. 265
High-Breakdown Alternatives to Quantile Regression	p. 268
Multivariate Quantiles	p. 272
The Oja Median and Its Extensions	p. 273
Half-Space Depth and Directional Quantile Regression	p. 275
Penalty Methods for Longitudinal Data	p. 276
Classical Random Effects as Penalized Least Squares	p. 276
Quantile Regression with Penalized Fixed Effects	p. 278
Causal Effects and Structural Models	p. 281
Structural Equation Models	p. 281
Chesher's Causal Chain Model	p. 283
Interpretation of Structural Quantile Effects	p. 284
Estimation and Inference	p. 285
Choquet Utility, Risk, and Pessimistic Portfolios	p. 287
Choquet Expected Utility	p. 287
Choquet Risk Assessment	p. 289
Pessimistic Portfolios	p. 291
Conclusion	p. 293
Quantile Regression in R: A Vignette	p. 295
Introduction	p. 295
What Is a Vignette?	p. 296
Getting Started	p. 296
Object Orientation	p. 298
Formal Inference	p. 299
More on Testing	p. 305
Inference on the Quantile Regression Process	p. 307
Nonlinear Quantile Regression	p. 308
Nonparametric Quantile Regression	p. 310
Conclusion	p. 316
Asymptotic Critical Values	p. 317
References	p. 319
Name Index	p. 337
Subject Index	p. 342
Table of Contents provided by Ingram. All Rights Reserved.

Quantile Regression

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