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In the unconditional moment restriction model of Hansen (1982), speciﬁcation tests and more eﬀicient estimators are both available whenever the number of moment restrictions exceeds the number of parameters of interest. We show a similar relationship between potential refutability of a model and existence of more eﬀicient estimators is present in much broader settings. Speciﬁcally, a condition we name local overidentiﬁcation is shown to be equivalent to both the existence of speciﬁcation tests with nontrivial local power and the existence of more eﬀicient estimators of some “smooth” parameters in general semi/nonparametric models. Under our notion of local overidentiﬁcation, various locally nontrivial speciﬁcation tests such as Hausman tests, incremental Sargan tests (or optimally weighted quasi-likelihood ratio tests) naturally extend to general semi/nonparametric settings. We further obtain simple characterizations of local overidentiﬁcation for general models of nonparametric conditional moment restrictions with possibly diﬀerent conditioning sets. The results are applied to determining when semi/nonparametric models with endogeneity are locally testable, and when nonparametric plug-in and semiparametric two-step GMM estimators are semiparametrically eﬀicient. Examples of empirically relevant semi/nonparametric structural models are presented.
Chen, Xiaohong and Santos, Andres, "Overidentification in Regular Models" (2015). Cowles Foundation Discussion Papers. 2434.