High dimensional semiparametric moment restriction models

We consider nonlinear moment restriction semiparametric models where both the dimension of the parameter vector and the number of restrictions are divergent with sample size and an unknown smooth function is involved. We propose an estimation method based on the sieve generalized method of moments (sieve GMM). We establish consistency and asymptotic normality for the estimated quantities when the number of parameters increases modestly with sample size. We also consider the case where the number of potential parameters/covariates is very large, i.e., increases rapidly with sample size, but the true model exhibits sparsity. We use a penalized sieve GMM approach to select the relevant variables, and establish the oracle property of our method in this case. We also provide new results for inference. We propose several new test statistics for the over-identi fication and establish their large sample properties. We provide a simulation study that shows the performance of our methodology. We also provide an application to modelling the effect of schooling on wages using data from the NLSY79 used by Carneiro et al.