Asymptotic Size of Kleibergen's LM and Conditional LR Tests for Moment Condition Models

120 Pages Posted: 24 Dec 2014 Last revised: 5 Jan 2015

See all articles by Donald W. K. Andrews

Donald W. K. Andrews

Yale University - Cowles Foundation

Patrik Guggenberger

Pennsylvania State University, College of the Liberal Arts - Department of Economic

Date Written: December 22, 2014

Abstract

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 theta 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.

Keywords: Asymptotics, Conditional likelihood ratio test, Confidence set, Identification, Inference, Lagrange multiplier test, Moment conditions, Robust, Test, Weak identification, Weak instruments

JEL Classification: C10, C12

Suggested Citation

Andrews, Donald W. K. and Guggenberger, Patrik, Asymptotic Size of Kleibergen's LM and Conditional LR Tests for Moment Condition Models (December 22, 2014). Cowles Foundation Discussion Paper No. 1977, Available at SSRN: https://ssrn.com/abstract=2541746 or http://dx.doi.org/10.2139/ssrn.2541746

Donald W. K. Andrews (Contact Author)

Yale University - Cowles Foundation ( email )

Box 208281
New Haven, CT 06520-8281
United States
203-432-3698 (Phone)
203-432-6167 (Fax)

Patrik Guggenberger

Pennsylvania State University, College of the Liberal Arts - Department of Economic ( email )

524 Kern Graduate Building
University Park, PA 16802-3306
United States

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