Asymptotic Normality of Single-Equation Estimators for the Case with a Large Number of Weak Instruments∗

This paper analyzes conditions under which various single-equation estimators are asymptotically normal in a simultaneous equations framework with many weak instruments. In particular, our paper adds to the many instruments asymptotic normality literature, including papers by Morimune (1983), Bekker (1994), Angrist and Krueger (1995), Donald and Newey (2001), Hahn, Hausman, and Kuersteiner (2001), and Stock and Yogo (2003). We consider the case where instrument weakness is such that rn, the rate of growth of the concentration parameter, is slower than Kn, the growth rate of the number of instruments, but such that √ Kn rn → 0 as n →∞. In this case, the rate of convergence is shown to be rn √ Kn . We also show that formulae for the asymptotic variances of various single-equation estimators are different from those obtained under assumptions of stronger instruments, i.e., cases where rn is assumed to grow at the same rate or at a faster rate than Kn. An interesting finding of this paper is that, for the case we study here, both the LIML and the Fuller estimators can be shown to be asymptotically more efficient than the B2SLS estimator not just for the case where the error distributions are assumed to be Gaussian but for all error distributions that lie within the elliptical family. JEL classification: C13, C31.