This paper generalizes the notion of identi cation by functional form to semiparametric contexts. It considers identification and estimation of the function F and vector B where M(X) = F[X'B, G(X)], assuming the functions M and G are identified. Identification in these models is typically obtained by parametric restrictions or by instrument exclusion restrictions, e.g., assuming that an element of G(X) that does not appear in X'B. We show such models are generally identified without exclusion or functional form restrictions given either some monotonicity assumptions on F and G, or a nonlinearity assumption on G. We also provide semiparametric estimators for these models. Examples of models that fit this framework include selection models, double hurdle models, and control function endogenous regressor models. So, e.g., our results that the Blundell and Powell (2004) semiparametric binary choice model with an endogenous regressor is generally identified, and can be estimated using their estimator, without the exclusion restrictions they impose for identification and estimation
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