In models deﬁned by unconditional moment restrictions, speciﬁcation tests are possible and estimators can be ranked in terms of eﬀiciency whenever the number of moment restrictions exceeds the number of parameters. We show that a similar relationship between potential refutability of a model and semiparametric eﬀiciency is present in a much broader class of settings. Formally, we show a condition we name local overidentiﬁcation is required for both speciﬁcation tests to have power against local alternatives and for the existence of both eﬀicient and ineﬀicient estimators of regular parameters. Our results immediately imply semiparametric conditional moment restriction models are typically locally overidentiﬁed, and hence their proper speciﬁcation is locally testable. We further study nonparametric conditional moment restriction models and obtain a simple characterization of local overidentiﬁcation in that context. As a result, we are able to determine when nonparametric conditional moment restriction models are locally testable, and when plug-in and two stage estimators of regular parameters are semiparametrically eﬀicient.
Chen, Xiaohong and Santos, Andres, "Overidentification in Regular Models" (2015). Cowles Foundation Discussion Papers. 2433.