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Unformatted text preview: Specification Tests for Strong Identification Bertille Antoine ∗ and Eric Renault † January 20, 2010 PRELIMINARY Abstract: We consider a general GMM framework where weaker patterns of identification may arise: typically, the data generating process is allowed to depend on the sample size. We are interested in providing inference about the identification of the structural parameter. First, for a given set of moment conditions, we propose a J-type test of the null that the structural parameter is not strongly identified. 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 identification asymptotics. Second, we propose a consistent Hausman-type test to test whether additional moment conditions enhance its identification. Failing to reject the null means that we consider that the candidate additional moment restrictions are useless to improve the strength of identification of the structural parameter. Finally, we propose a consistent test to test whether additional moment conditions are valid. JEL Classification: C32; C12; C13; C51. Keywords: Specification test; GMM; Weak identification. ∗ Simon Fraser University. Email: [email protected] † University of North Carolina at Chapel Hill, CIRANO and CIREQ. Email: [email protected] 1 1 Introduction The practical relevance of the asymptotic theory of the Generalized Method of Moments (GMM) as developed by Hansen (1982) is known to be limited by the weak instruments issue. Since the seminal work of Stock and Wright (2000), it is common to capture the impact of instruments weakness by a drifting data generating process (hereafter DGP) such that the informational content of estimating equations ρ T ( θ ) = 0 about structural parameters of interest θ ∈ Θ ⊂ R p is impaired by the fact that ρ T ( θ ) becomes zero when the sample size T goes to infinity. While genuine weak identification as defined by Stock and Wright (2000) assumes that some components of ρ T ( θ ) may go to zero as fast as the convergence rate (root- T ) of the sample counterparts, others including Hahn and Kuersteiner (2002), Antoine and Renault (2008, 2009), and Caner (2009) have considered the intermediate case, dubbed nearly-weak identification, where it may go to zero at a slower rate T α with < α < 1 / 2 . Generally speaking, we are interested in this paper in any set of estimating equations such that for some sequence A T of deterministic matrices, A T ρ T ( θ ) converges to some asymptotic estimating equations suﬃcient to identify the true unknown value θ of θ : lim T [ A T ρ T ( θ )] = c ( θ ) with c ( θ ) = 0 ⇔ θ = θ The rate of convergence of coeﬃcients of the matrix A T towards infinity characterizes the degree of global identification weakness. We show in this paper that the degree of local identification weakness is similarly characterized through a sequence of matrices ˜ A T such that: lim T [ ∂ρ...
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