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 specification is sufficiently general to include structural models, triangular simultaneous equations and certain models of measurement error. One motivation of the model specification is program evaluation problems, which arise frequently in empirical policy applications. 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 identified. 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 verifiable condition for identification, 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 efficiency gains in imposing additivity. The first 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. Nonparametric estimation of the instruments permits bypassing the specification of conditional distributions, but is heuristic, and does not affect the subsequent large sample results of the estimator. Linear functionals of the estimator are shown to be asymptotically normal, including root-n-consistent when certain regularity conditions hold.