Speci cation Tests for Strong Identi cation

We consider a general GMM framework where weaker patterns of identi cation may arise: typically, the data generating process is allowed to depend on the sample size. We are interested in providing inference about the identi cation of the structural parameter. First, we propose a consistent Hausman-type test to test whether additional moment conditions enhance its identi cation. Failing to reject the null means that we consider that the candidate additional moment restrictions are useless to improve the strength of identi cation of the structural parameter. Second, for a given set of moment conditions, we propose a J-type test of the null that the structural parameter is not strongly identi ed. In other words, it is only as long as that the null cannot be rejected that it remains necessary to resort to alternative (nearly) weak identi cation asymptotics. The above testing strategy is developed for both unconditional and conditional moment restrictions and illustrated with the Epstein and Zin asset pricing model. JEL Classi cation: C32; C12; C13; C51.