Instrumental Variables Estimation of Nonparametric Models with Discrete Endogenous Regressors

This paper presents new instrumental variables estimators for nonparametric models with discrete endogenous regressors. The model speci cation is su ciently general to include structural models, triangular simultaneous equations and certain models of measurement error. Restricting the analysis to discrete endogenous regressors is an integral component of the analysis since a similar model with continuously distributed endogenous regressors is ill-posed and cannot be identi ed. The central contribution of this paper is a consistent two-step nonparametric instrumental variables estimator of the model. Large sample results, including global convergence rates and asmptotic normality are also provided. Discreteness of the regressors is shown to produce an additive representation of the model which leads to a simple veri able condition for identi cation, and a restriction that is imposed in estimation. The proposed nonparametric two-step IV estimator is based on series estimation which is particularly amenable to additive models, and yields e ciency gains in imposing additivity. The rst step constitutes nonparametric estimation of the instrument, while the second step constructs the IV estimator from a linear combination of an instrument matrix and a matrix of the regression covariates. Linear functionals of the estimator are shown to be asymptotically normal, including p n-consistent when certain regularity conditions hold.