Overidentification in Regular Models

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.