《Koenker(2005,BOOK)  Quantile_regression》

Preface pagexiii
1 Introduction 1
1.1 MeansandEnds 1
1.2 The FirstRegression:AHistorical Prelude 2
1.3 Quantiles,Ranks, and Optimization 5
1.4 PreviewofQuantileRegression 9
1.5 ThreeExamples 13
1.5.1 Salaries versus Experience 13
1.5.2 Student Course EvaluationsandClass Size 17
1.5.3 InfantBirthWeight 20
1.6 Conclusion 25
2 Fundamentalsof QuantileRegression 26
2.1 Quantile Treatment Effects 26
2.2 HowDoes Quantile Regression Work? 32
2.2.1 Regression QuantilesInterpolate p Observations 33
2.2.2 TheSubgradient Condition 34
2.2.3 Equivariance 38
2.2.4 Censoring 40
2.3 Robustness 42
2.3.1 TheInfluence Function 42
2.3.2 TheBreakdownPoint 45
2.4 InterpretingQuantile Regression Models 47
2.4.1 Some Examples 48
2.5 Caution:QuantileCrossing 55
2.6 ARandomCoefficient Interpretation 59
2.7 Inequality Measures and Their Decomposition 62
2.8 ExpectilesandOther Variations 63
2.9 InterpretingMisspecifiedQuantile Regressions 65
2.10 Problems 66
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viii Contents
3 Inferencefor Quantile Regression 68
3.1 The Finite-Sample Distribution of Regression Quantiles 68
3.2 AHeuristicIntroduction toQuantileRegression
Asymptotics 71
3.2.1 Confidence Intervals fortheSampleQuantiles 72
3.2.2 Quantile RegressionAsymptoticswith IID Errors 73
3.2.3 Quantile RegressionAsymptoticsin Non-IID
Settings 74
3.3 WaldTests 75
3.3.1 Two-SampleTests of LocationShift 75
3.3.2 General Linear Hypotheses 76
3.4 EstimationofAsymptotic CovarianceMatrices 77
3.4.1 Scalar Sparsity Estimation 77
3.4.2 CovarianceMatrixEstimationinNon-IIDSettings 79
3.5 Rank-BasedInference 81
3.5.1 Rank Tests for Two-SampleLocationShift 81
3.5.2 LinearRank Statistics 84
3.5.3 Asymptotics ofLinearRankStatistics 85
3.5.4 Rank Tests Basedon Regression Rankscores 87
3.5.5 Confidence Intervals Based onRegression
Rankscores 91
3.6 Quantile LikelihoodRatio Tests 92
3.7 Inference ontheQuantileRegression Process 95
3.7.1 WaldProcesses 97
3.7.2 Quantile LikelihoodRatio Processes 98
3.7.3 TheRegression Rankscore Process Revisited 98
3.8 TestsoftheLocation-Scale Hypothesis 98
3.9 ResamplingMethodsandtheBootstrap 105
3.9.1 Bootstrap Refinements,Smoothing, and
Subsampling 107
3.9.2 Resampling on the Subgradient Condition 108
3.10 MonteCarloComparisonofMethods 110
3.10.1 Model1:A Location-Shift Model 111
3.10.2 Model2:A Location–Scale-Shift Model 112
3.11 Problems 113
4 AsymptoticTheory of Quantile Regression 116
4.1 Consistency 117
4.1.1 Univariate SampleQuantiles 117
4.1.2 LinearQuantile Regression 118
4.2 RatesofConvergence 120
4.3 BahadurRepresentation 122
4.4 NonlinearQuantile Regression 123
4.5 The QuantileRegression Rankscore Process 124
4.6 Quantile RegressionAsymptotics under Dependent
Conditions 126

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Contents ix
4.6.1 Autoregression 126
4.6.2 ARMA Models 128
4.6.3 ARCH-like Models 129
4.7 ExtremalQuantile Regression 130
4.8 The Method ofQuantiles 131
4.9 ModelSelection, Penalties, and Large-p Asymptotics 133
4.9.1 ModelSelection 134
4.9.2 Penalty Methods 135
4.10 Asymptoticsfor Inference 138
4.10.1 Scalar SparsityEstimation 139
4.10.2 Covariance Matrix Estimation 141
4.11 ResamplingSchemes and the Bootstrap 141
4.12 Asymptoticsfor theQuantileRegression Process 142
4.12.1 TheDurbin Problem 142
4.12.2 KhmaladizationoftheParametric Empirical
Process 144
4.12.3 TheParametric Quantile Process 145
4.12.4 TheParametric Quantile Regression Process 146
4.13 Problems 149
5 L-Statisticsand Weighted Quantile Regression 151
5.1 L-Statisticsforthe Linear Model 151
5.1.1 OptimalL-Estimatorsof LocationandScale 153
5.1.2 L-Estimation fortheLinearModel 155
5.2 Kernel Smoothing forQuantileRegression 158
5.2.1 Kernel Smoothing oftheρτ-Function 160
5.3 Weighted Quantile Regression 160
5.3.1 WeightedLinearQuantile Regression 160
5.3.2 EstimatingWeights 161
5.4 Quantile Regression forLocation–ScaleModels 164
5.5 Weighted Sumsofρτ-Functions 168
5.6 Problems 170
6 Computational Aspectsof QuantileRegression 173
6.1 IntroductiontoLinear Programming 173
6.1.1 Vertices 174
6.1.2 Directionsof Descent 176
6.1.3 Conditions for Optimality 177
6.1.4 Complementary Slackness 178
6.1.5 Duality 180
6.2 SimplexMethodsfor Quantile Regression 181
6.3 ParametricProgramming forQuantileRegression 185
6.3.1 ParametricProgrammingfor Regression
RankTests 188
6.4 InteriorPointMethods forCanonical LPs 190
6.4.1 Newtonto the Max: An Elementary Example 193
6.4.2 InteriorPoint Methods for QuantileRegression 199

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6.4.3 Interiorvs. Exterior: A Computational
Comparison 202
6.4.4 ComputationalComplexity 204
6.5 Preprocessing for Quantile Regression 206
6.5.1 “Selecting” UnivariateQuantiles 207
6.5.2 Implementation 207
6.5.3 Confidence Bands 208
6.5.4 Choosing m 209
6.6 NonlinearQuantile Regression 211
6.7 Inequality Constraints 213
6.8 Weighted Sumsofρτ-Functions 214
6.9 Sparsity 216
6.10 Conclusion 220
6.11 Problems 220
7 NonparametricQuantile Regression 222
7.1 Locally Polynomial Quantile Regression 222
7.1.1 Average Derivative Estimation 226
7.1.2 AdditiveModels 228
7.2 PenaltyMethods forUnivariate Smoothing 229
7.2.1 Univariate Roughness Penalties 229
7.2.2 TotalVariation Roughness Penalties 230
7.3 PenaltyMethods forBivariate Smoothing 235
7.3.1 BivariateTotalVariation Roughness Penalties 235
7.3.2 TotalVariation PenaltiesforTriograms 236
7.3.3 Penalized TriogramEstimation as a Linear
Program 240
7.3.4 On Triangulation 241
7.3.5 On Sparsity 242
7.3.6 Automatic λSelection 242
7.3.7 Boundary and Qualitative Constraints 243
7.3.8 AModelof Chicago Land Values 243
7.3.9 TautStringsandEdge Detection 246
7.4 AdditiveModels andtheRole of Sparsity 248
8 TwilightZoneofQuantile Regression 250
8.1 Quantile Regression forSurvival Data 250
8.1.1 QuantileFunctions or Hazard Functions? 252
8.1.2 Censoring 253
8.2 DiscreteResponseModels 255
8.2.1 BinaryResponse 255
8.2.2 CountData 259
8.3 Quantile Autoregression 260
8.3.1 QuantileAutoregression and Comonotonicity 261
8.4 CopulaFunctions and Nonlinear Quantile Regression 265
8.4.1 CopulaFunctions 265

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Contents xi
8.5 High-BreakdownAlternativesto QuantileRegression 268
8.6 Multivariate Quantiles 272
8.6.1 TheOjaMedian and Its Extensions 273
8.6.2 Half-Space Depth and Directional Quantile
Regression 275
8.7 PenaltyMethodsforLongitudinal Data 276
8.7.1 ClassicalRandom Effects as Penalized
LeastSquares 276
8.7.2 QuantileRegression withPenalized FixedEffects 278
8.8 CausalEffects and Structural Models 281
8.8.1 StructuralEquationModels 281
8.8.2 Chesher’sCausalChain Model 283
8.8.3 Interpretation of StructuralQuantileEffects 284
8.8.4 EstimationandInference 285
8.9 Choquet Utility,Risk, and Pessimistic Portfolios 287
8.9.1 ChoquetExpectedUtility 287
8.9.2 ChoquetRiskAssessment 289
8.9.3 PessimisticPortfolios 291
9 Conclusion 293
A QuantileRegressionin R: A Vignette 295
A.1 Introduction 295
A.2 What IsaVignette? 296
A.3 Getting Started 296
A.4 ObjectOrientation 298
A.5 FormalInference 299
A.6 MoreonTesting 305
A.7 InferenceontheQuantile RegressionProcess 307
A.8 NonlinearQuantileRegression 308
A.9 NonparametricQuantile Regression 310
A.10 Conclusion 316
B Asymptotic Critical Values 317
References 319
NameIndex 337
SubjectIndex 342