Chapter 52
THE BOOTSTRAP
JOEL L HOROWITZ
Department of Economics, Northwestern University, Evanston, IL, USA
Contents
Abstract 3160
Keywords 3160
1 Introduction 3161
2 The bootstrap sampling procedure and its consistency 3163
2.1 Consistency of the bootstrap 3166
2.2 Alternative resampling procedures 3169
3 Asymptotic refinements 3172
3.1 Bias reduction 3172
3.2 The distributions of statistics 3175
3.3 Bootstrap critical values for hypothesis tests 3179
3.4 Confidence intervals 3183
3.5 The importance of asymptotically pivotal statistics 3184
3.6 The parametric versus the nonparametric bootstrap 3185
3.7 Recentering 3186
4 Extensions 3188
4.1 Dependent data 3188
4.1 1 Methods for bootstrap sampling with dependent data 3188
4.1 2 Asymptotic refinements in GMM estimation with dependent data 3191
4.1 3 The bootstrap with non-stationary processes 3193
4.2 Kernel density and regression estimators 3195
4.2 1 Nonparametric density estimation 3196
4.2 2 Asymptotic bias and methods for controlling it 3197
4.2 3 Asymptotic refinements 3199
4.2 3 1 The error made by first-order asymptotics when nh+ does not converge
to 0 3201
4.2 4 Kernel nonparametric mean regression 3202
4.3 Non-smooth estimators 3205
4.3 1 The LAD estimator for a linear median-regression model 3205
4.3 2 The maximum score estimator for a binary-response model 3208
4.4 Bootstrap iteration 3210
4.5 Special problems 3212
4.6 The bootstrap when the null hypothesis is false 3212
5 Monte Carlo experiments 3213
5.1 The information-matrix test 3214
5.2 The t test in a heteroskedastic regression model 3215
5.3 The t test in a Box-Cox regression model 3217
5.4 Estimation of covariance structures 3218
6 Conclusions 3220
Acknowledgements 3221
Appendix A Informal derivation of Equation ( 3 27) 3221
References 3223

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