Nonparametric and Semiparametric Econometrics Lecture Notes for Econ 221. Yixiao Sun Department of Economics, University of California, San Diego

Transcription

1 Nonparametric and Semiparametric Econometrics Lecture Notes for Econ 221 Yixiao Sun Department of Economics, University of California, San Diego Winter 2007

2 Contents Preface ix 1 Kernel Smoothing: Density Estimation Introduction Kernel Estimator Bias and Variance Optimal Bandwidth Choice: Plug-in Approach Optimal Bandwidth Choice: Cross Validation Optimal Kernel Bias Reduction: High Order Kernels High-order Kernels Flat-top Kernels Asymptotic Normality Uniform Consistency Multivariate Density Estimation Conditional Density Estimation References Kernel Smoothing: Regression Estimation Introduction Kernel Estimators: Local Smoothing Nadaraya-Watson Estimator k-nearest Neighbor Estimators Local Polynomial Estimators Robust Smoothing Asymptotic Properties: Local Constant Estimator Consistency Asymptotic Normality Bandwidth Selection xxxii

3 lxxxv MSE-optimal Bandwidth Plug-in Implementation Cross Validation Computation and GCV Model Selection Perspective Mallows' C p Criterion A Theoretical Development of AIC (Digression) A Shrinkage Interpretation Uniform Consistency Uniform Condence Intervals (Optional) Asymptotic Properties of Local Linear Estimator Asymptotic Variance of Local Linear Estimators Asymptotic Bias of Local Linear Estimators References Sieve Estimation Examples and Motivations Sieve Spaces Holder Class and Finite Dimensional Linear Sieves L p and Finite Dimensional Linear Sieves Other Smoothness Classes and Finite Dimensional Nonlinear Sieves Innite Dimensional Sieves Conditional Moment Estimation Smoothing Spline: Innite Dimensional Sieves Series Estimators: Finite Dimensional Sieves Density Estimation Penalized Likelihood Estimation: Innite Dimensional Sieves Series Estimation: Finite Dimensional Sieves The General Sieve Extremum Estimation Consistency of Sieve Extremum Estimators Convergence Rates of Sieve M-estimators Bibliographical Remarks References Examples of Semiparametric Models Introduction Regression Models Partially Linear Model The Single Index Model Selectivity Models

4 cxxxvii 4.4 Ecient and Adaptive Estimation Ecient Estimation in the Presence of Heteroskedasticity Adaptive Estimation Bibiographical Remarks References Case Study: Partially Linear Model Introduction A Semiparametric Estimator Asymptotics via Stochastic Equicontinuity References Semiparametric Methods: A Unied Framework The Framework Limit Theorem Preliminaries Smoothness Assumptions Stochastic Approximations Consistency Root-n Consistency Asymptotic Normality Some Details Orthogonality Conditions Asymptotic Normality of Objective Function Case Studies Partial Linear Model Single Index Model Sample Selection Model Nonparametric WLS Non-dierentiable Objective Function Consistency Asymptotic Normality References Empirical Applications of Semiparametric Models Structural Estimation of Production Functions Model Estimation More Examples References

5 cxxxviii 8 Semiparametric Eciency Bounds Hilbert Space for Random Vectors RAL Estimators in Parametric Models Cramer-Rao Bound Hodges Estimator RAL Estimators Properties of IFs for RAL Estimators Characterization of IFs Ecient Inuence Function Summary of the Results Semiparametric Eciency Bounds Denition Characterization of Semiparametric Eciency Bounds Gradient Function Tangent Space for Nonparametric Models Example: estimating the mean functional Example: estimating the mean functional using the gradient function Semiparametric Conditional Moment Model The model Nuisance Parameter Tangent Space Computing the Projection Optimal Instruments Semiparametric Partially Linear Model References