Identification and Estimation of Semiparametric Two Step Models∗

LetH0(X) be a function that can be nonparametrically estimated. Suppose E [Y |X ] = F0[X⊤β0, H0(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 β0 and unknown function F0 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, random trimming, and slow rates of convergence for first stageH0 estimates. A Monte Carlo experiment and an empirical application to migration decisions are also included.