Identification and estimation of semiparametric two-step models

Let H 0 (X) be a function that can be nonparametrically estimated. Suppose E [ Y | X ]= F 0 [ X ß 0 H 0 (X) ] . Many models fit this framework, including latent in- dex models with an endogenous regressor and nonlinear models with sample se- lection. We show that the vector ß 0 and unknown function F 0 are generally point identified without exclusion restrictions or instruments, in contrast to the usual assumption that identification without instruments requires fully specified func- tional forms. We propose an estimator with asymptotic properties allowing for data dependent bandwidths and random trimming. A Monte Carlo experiment and an empirical application to migration decisions are also included.