Identification and estimation of semiparametric two-step models

Let H-0(X) be a function that can be nonparametrically estimated. Suppose E[Y vertical bar X] = F-0[X-inverted perpendicular beta(0), H-0(X)]. Many models fit this framework, including latent index models with an endogenous regressor and nonlinear models with sample selection. We show that the vector beta(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 functional 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.