Theory of Weak Identification in Semiparametric Models

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We provide general formulation of weak identification in semiparametric models and a novel efficiency concept. Weak identification occurs when a parameter is weakly regular, i.e., when it depends on the score asymptotically. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that behind every weakly regular parameter there exists an underlying parameter that is regular and fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down the desirable choice. If the estimation of minimal sufficient underlying parameters is inefficient, it introduces noise in the corresponding estimation of weakly regular parameters, whence we can improve the estimators by local asymptotic Rao-Blackwellization. We call an estimator weakly efficient if it attains an asymptotic distribution that does not admit such improvement. We demonstrate in heteroskedastic linear IV models that popular estimators can be improved under some conditions.