On the Informativeness of Specification Tests for Estimator Validity

Empirical researchers often use model specification tests, such as Hausman tests and overidentifying restrictions tests, to assess the validity of estimators rather than the correctness of models. This paper examines the extent to which such tests are informative about the presence of asymptotic bias in estimators. Under a local misspecification framework, we show that the directions of local deviation from a benchmark distribution to which locally unbiased specification tests have nontrivial power are orthogonal to those that induce asymptotic bias in asymptotically efficient estimators. Consequently, specification tests generally provide limited information about estimator validity unless researchers impose additional untestable assumptions on the form of misspecification. We further demonstrate that, when used as estimator selection rules, Hausman tests can lead to the choice of inefficient estimators that exhibit asymptotic bias, even when efficient estimators remain asymptotically unbiased. These findings highlight fundamental limitations in using specification tests to guide estimator choice.