Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions

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Abstract

We propose an estimation method for models of conditional moment restrictions, which contain finite dimensional unknown parameters (θ) and infinite dimensional unknown functions (h). Our proposal is to approximate h with a sieve and to estimate θ and the sieve parameters jointly by applying the method of minimum distance. We show that: (i) the sieve estimator of h is consistent with a rate faster than n-1/4 under certain metric; (ii) the estimator of θ is √n consistent and asymptotically normally distributed; (iii) the estimator for the asymptotic covariance of the θ estimator is consistent and easy to compute; and (iv) the optimally weighted minimum distance estimator of θ attains the semiparametric efficiency bound. We illustrate our results with two examples: a partially linear regression with an endogenous nonparametric part, and a partially additive IV regression with a link function.

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