Coordinated Testing for Identification Failure and Correct Model Specification

In the context of GMM or Minimum Distance (MD) inference, practitioners are often interested in testing both for correct specification (e.g. with an overidentification test) and for weak identification. In this paper, our focus is twofold. On the one hand, we develop an overidentification test that is robust to weak identification. And we show that this requires properly adjusting the critical values to avoid over-rejection. Conversely, we also develop a test of the null of weak identification that is robust to misspecification and powerful through a conditional approach. More generally, we emphasize that these two testing goals require some coordination to ensure there is enough power to test for weak identification, which is critical for valid and powerful specification testing. We illustrate the performance of our different testing strategies through three applications: Discrete Choice models with simultaneity and the New Keynesian Philips curve (in a GMM framework), as well as Asset Pricing models with stochastic volatility and leverage (in a MD framework).