Asymptotic Size of Kleibergen's LM and Conditional LR Tests for Moment Condition Models
An influential paper by Kleibergen (2005) introduces Lagrange multiplier (LM) and conditional likelihood ratio-like (CLR) tests for nonlinear moment condition models. These procedures aim to have good size performance even when the parameters are unidentiﬁed or poorly identiﬁed. However, the asymptotic size and similarity (in a uniform sense) of these procedures has not been determined in the literature. This paper does so. This paper shows that the LM test has correct asymptotic size and is asymptotically similar for a suitably chosen parameter space of null distributions. It shows that the CLR tests also have these properties when the dimension p of the unknown parameter θ equals 1: When p ≥ 2; however, the asymptotic size properties are found to depend on how the conditioning statistic, upon which the CLR tests depend, is weighted. Two weighting methods have been suggested in the literature. The paper shows that the CLR tests are guaranteed to have correct asymptotic size when p ≥ 2 with one weighting method, combined with the Robin and Smith (2000) rank statistic. The paper also determines a formula for the asymptotic size of the CLR test with the other weighting method. However, the results of the paper do not guarantee correct asymptotic size when p ≥ 2 with the other weighting method, because two key sample quantities are not necessarily asymptotically independent under some identiﬁcation scenarios. Analogous results for conﬁdence sets are provided. Even for the special case of a linear instrumental variable regression model with two or more right-hand side endogenous variables, the results of the paper are new to the literature.
Andrews, Donald W.K. and Guggenberger, Patrik, "Asymptotic Size of Kleibergen's LM and Conditional LR Tests for Moment Condition Models" (2014). Cowles Foundation Discussion Papers. 2392.