Methods for Nonparametric and Semiparametric Regressions with Endogeneity: a Gentle Guide,

  This paper 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 regression (NPIV), nonparametric quantile IV regression and many more semi-nonparametric structural models. Asymptotic properties of the sieve estimators and the sieve Wald, quasi-likelihood ratio (QLR) 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 over-identification test for semiparametric two-step GMM are reviewed. Monte Carlo examples are included.   Keywords: Conditional moment restrictions containing unknown functions, (Quantile) Instrumental variables, Linear and nonlinear functionals, Sieve minimum distance, Sieve GMM, Sieve Wald, QLR, Bootstrap, Semiparametric two-step GMM, Numerical equivalence   JEL Classificaion: C12, C14, C32