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Let H0(X) be a function that can be nonparametrically estimated. Suppose E [Y|X]=F0[X¿ß0, 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 ß0 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 and random trimming. A Monte Carlo experiment and an empirical application to migration decisions are also included.
identification by functional form; double index models; two-step estimators; semiparametric regression; control function estimators; sample selection models; empirical process theory; limited dependent variables; migration