High dimensional semiparametric moment restriction models

Moment restriction semiparametric models, where both the dimension of parameter and the number of restrictions are divergent and an unknown function is involved, are studied using the generalized method of moments (GMM) and sieve method dealing with the nonparametric parameter. The consistency and normality for the GMM estimators are established. Meanwhile, a new test statistic is proposed for over-identification issue, which also is workable for the traditional moment restriction models. In addition, the potential sparsity under our setting is investigated via the combination of GMM methodology and penalty function approach. Numerical examples are used to verify the established theory.