A limit theorem for a smooth class of semiparametric estimators

We consider an econometric model based on a set of moment conditions which are indexed by both a finite-dimensional vector of interest, θ, and an infinite-dimensional parameter, h, which in turn depends upon both θ and another infinite-dimensional parameter, τ. The population moment conditions equal zero at θ = θ0. Estimators of θ0 are obtained by forming nonparametric estimates of h and τ, substituting them into the sample analog of the moment conditions, and choosing that value of θ that makes the sample moments as ‘close as possible’ to zero. Using independence and smoothness assumptions the paper provides consistency, √n consistency, and asymptotic normality proofs for the resultant estimator. As an example, we consider Olley and Pakes' (1991) use of semiparametric techniques to control for both simultaneity and selection biases in estimating production functions. The example illustrates how semiparametric techniques an be used to overcome both computational problems and the need for strong functional form restrictions in obtaining estimates from structural models. It also illustrates the impacts of (i) alternative estimators for the nonparametric components of the problem and (ii) alternative estimators for the standard errors of the estimated θ.