In this note, we characterize the semiparametric eﬀiciency bound for a class of semiparametric models in which the unknown nuisance functions are identiﬁed via nonparametric conditional moment restrictions with possibly non-nested or over-lapping conditioning sets, and the ﬁnite dimensional parameters are potentially over-identiﬁed via unconditional moment restrictions involving the nuisance functions. We discover a surprising result that semiparametric two-step optimally weighted GMM estimators achieve the eﬀiciency bound, where the nuisance functions could be estimated via any consistent nonparametric procedures in the ﬁrst step. Regardless of whether the eﬀiciency bound has a closed form expression or not, we provide easy-to-compute sieve based optimal weight matrices that lead to asymptotically eﬀicient two-step GMM estimators.
Chen, Xiaohong; Hahn, Jinyong; and Liao, Zhipeng, "Asymptotic Efficiency of Semiparametric Two-step GMM" (2012). Cowles Foundation Discussion Papers. 2248.