Identification and Estimation of Semiparametric Two Step Models

Let g0(X) be a function of some observable vector X that is identified and can be nonparametrically estimated. This paper provides new results on the identification and estimation of the function F and the vector β0 whenE(Y|X)=F[X⊤β0,g0(X)]. Many models fit this framework, including latent in… dex models with an endogenous regressor, and nonlinear models with sample selection. Our identification results show that identification based on functional form, without exclusions or instruments, extends to this semiparametric model. On estimation we provide a new uniform convergence result that allows for random weighting and data dependent bandwidths and trimming. We include Monte Carlo simulations and an empirical application.