Semiparametric and Nonparametric Methods in Econometrics (eBook)

Preface 6
Contents 8
1 Introduction 12
1.1 The Goals of This Book 12
1.2 Dimension Reduction 14
1.3 Are Semiparametric and Nonparametric Methods Really Different from Parametric Ones? 17
2 Single-Index Models 18
2.1 Definition of a Single-Index Model of a Conditional Mean Function 18
2.2 Multiple-Index Models 21
2.3 Identification of Single-Index Models 23
2.3.1 Conditions for Identification of ß and G 23
2.3.2 Identification Analysis When X Is Discrete 26
2.4 Estimating G in a Single-Index Model 28
2.5 Optimization Estimators of 30
2.5.1 Nonlinear Least Squares 31
2.5.2 Choosing the Weight Function 36
2.5.3 Semiparametric Maximum-Likelihood Estimation of Binary-Response Models 38
2.5.4 Semiparametric Maximum-Likelihood Estimation of Other Single-Index Models 40
2.5.5 Semiparametric Rank Estimators 40
2.6 Direct Semiparametric Estimators 41
2.6.1 Average-Derivative Estimators 42
2.6.2 An Improved Average-Derivative Estimator 46
2.6.3 Direct Estimation with Discrete Covariates 48
2.6.4 One-Step Asymptotically Efficient Estimators 53
2.7 Bandwidth Selection 55
2.8 An Empirical Example 57
2.9 Single-Index Models of Conditional Quantile Functions 58
3 Nonparametric Additive Models and Semiparametric Partially Linear Models 63
3.1 Nonparametric Additive Models with Identity Link Functions 65
3.1.1 Marginal Integration 65
3.1.2 Backfitting 73
3.1.3 Two-Step, Oracle-Efficient Estimation 74
3.1.3.1 Informal Description of the Estimator 75
3.1.3.2 Asymptotic Properties of the Two-Stage Estimator 76
3.2 Estimation with a Nonidentity Link Function 80
3.2.1 Estimation 81
3.2.2 Bandwidth Selection 85
3.3 Estimation with an Unknown Link Function 87
3.4 Estimation of a Conditional Quantile Function 90
3.5 An Empirical Example 93
3.6 The Partially Linear Model 95
3.6.1 Identification 95
3.6.2 Estimation of 96
3.6.3 Partially Linear Models of Conditional Quantiles 100
3.6.4 Empirical Applications 101
4 Binary-Response Models 104
4.1 Random-Coefficients Models 104
4.2 Identification 105
4.2.1 Identification Analysis When X Has Bounded Support 109
4.2.2 Identification When X Is Discrete 110
4.3 Estimation 112
4.3.1 Estimating 113
4.3.2 Estimating ß: The Maximum-Score Estimator 114
4.3.3 Estimating ß: The Smoothed Maximum-Score Estimator 117
4.4 Extensions of the Maximum-Score and Smoothed Maximum-Score Estimators 127
4.4.1 Choice-Based Samples 128
4.4.2 Panel Data 132
4.4.3 Ordered-Response Models 137
4.5 Other Estimators for Heteroskedastic Binary-Response Models 140
4.6 An Empirical Example 141
5 Statistical Inverse Problems 143
5.1 Deconvolution in a Model of Measurement Error 144
5.1.1 Rate of Convergence of the Density Estimator 146
5.1.2 Why Deconvolution Estimators Converge Slowly 149
5.1.3 Asymptotic Normality of the Density Estimator 151
5.1.4 A Monte Carlo Experiment 152
5.2 Models for Panel Data 152
5.2.1 Estimating fU and fe 154
5.2.2 Large Sample Properties of fne and fnU 156
5.2.3 Estimating First-Passage Times 159
5.2.4 Bias Reduction 160
5.2.5 Monte Carlo Experiments 162
5.3 Nonparametric Instrumental-Variables Estimation 164
5.3.1 Regularization Methods 172
5.3.1.1 Tikhonov Regularization 173
5.3.1.2 Regularization by Series Truncation 177
5.4 Nonparametric Instrumental-Variables Estimation When T Is Unknown 179
5.4.1 Estimation by Tikhonov Regularization When T Is Unknown 179
5.4.1.1 Computation of ˆg 185
5.4.2 Estimation by Series Truncation When T Is Unknown 186
5.4.2.1 Computation of ˆg 192
5.5 Other Approaches to Nonparametric Instrumental-Variables Estimation 193
5.5.1 Nonparametric Quantile IV 193
5.5.2 Control Functions 194
5.6 An Empirical Example 195
6 Transformation Models 197
6.1 Estimation with Parametric T and Nonparametric F 198
6.1.1 Choosing the Instruments 201
6.1.2 The Box--Cox Regression Model 202
6.1.3 The Weibull Hazard Model with Unobserved Heterogeneity 204
6.2 Estimation with Nonparametric T and Parametric F 209
6.2.1 The Proportional Hazards Model 209
6.2.2 The Proportional Hazards Model with Unobserved Heterogeneity 212
6.2.3 The Case of Discrete Observations of Y 216
6.2.4 Estimating 217
6.2.5 Other Models in Which F Is Known 221
6.3 Estimation When Both T and F Are Nonparametric 223
6.3.1 Derivation of Horowitz’s Estimators of T and F 224
6.3.2 Asymptotic Properties of Tn and Fn 227
6.3.3 Chen’s Estimator of T 229
6.3.4 The Proportional Hazards Model with Unobserved Heterogeneity 231
6.4 Predicting Y Conditional on X 238
6.5 An Empirical Example 238
Appendix: Nonparametric Density Estimationand Nonparametric Regression 241
6.1 Nonparametric Density Estimation 241
A.1.1 Density Estimation When X Is Multidimensional 245
A.1.2 Estimating Derivatives of a Density 247
6.2 Nonparametric Mean Regression 248
A.2.1 The Nadaraya--Watson Kernel Estimator 248
A.2.2 Local-Linear Mean Regression 250
A.2.3 Series Estimation of a Conditional Mean Function 253
A.2.3.1 Hilbert Spaces 253
A.2.3.2 Nonparametric Regression 256
6.3 Nonparametric Quantile Regression 257
A.3.1 A Kernel-Type Estimator of qa(x) 258
A.3.2 Local-Linear Estimation of qa(x) 259
A.3.3 Series Estimation of qa(x) 261
References 264
Index 274