Alastair Hall (University of Manchester)

"GMM and Indirect Inference in models that are second order identified"  (with Prosper Dovonon)

Abstract:  Standard first order asymptotic theory of GMM and Indirect inference (II) estimators is premised on the assumption that the parameters are first order identified in the sense that the relevant Jacobian matrix is full rank. The validity of this assumption has been questioned in a number of econometric models. In a seminal article, Dufour (1997) demonstrated that the possibility of identification failure renders the standard Wald-based confidence intervals invalid. This result helped to stimulate the development of alternative frameworks for GMM based inference that are robust to identification failure. To date, the focus has mostly been on cases in which identification fails in the limit as, for example, in the weak identification framework in which Jacobian converges to a rank deficient limit at rate equal to the inverse of the square root of the sample size.  However, first order identification failure does not necessitate identification failure. In some models, the parameters can be shown to be second order identified even though first order identification fails; examples include factor models of variance and linear dynamic panel data models. In this paper, we present a limiting distribution theory for both GMM and II estimators when first order identification fails but the parameters are second order identified. We show how these results can be used to construct valid confidence intervals for the parameters. Our results are illustrated using the dynamic linear panel data model considered by Madsen (2009) in which the parameters are estimated using second moments. We compare the finite sample properties of GMM and II estimators. We also discuss the relevance of our results to recent concerns about the identification of DSGE models, for example see Canova & Sala (2009).