Methods for Nonparametric and Semiparametric Regressions with Endogeneity: A Gentle Guide

This article reviews recent advances in estimation and inference for nonparametric and semiparametric models with endogeneity. It first describes methods of sieves and penalization for estimating unknown functions identified via conditional moment restrictions. Examples include nonparametric instrumental variables (NPIV) regression, nonparametric quantile IV regression, and many more semi/nonparametric structural models. Asymptotic properties of the sieve estimators and the sieve Wald, quasi-likelihood ratio hypothesis tests of functionals with nonparametric endogeneity are presented. For sieve NPIV estimation, the rate-adaptive data-driven choices of sieve regularization parameters and the sieve score bootstrap uniform confidence bands are described. Finally, simple sieve variance estimation and overidentification tests for the semiparametric two-step generalized method of moments are reviewed. Monte Carlo examples are also included.