Exact Density Functions and Approximate Critical Regions for Likelihood Ratio Identifiability Test Statistics

Abstract

The exact finite-sample density function of the likelihood ratio (LR) test of overidentifying restrictions in linear equations with any number of included endogenous and exogenous variables is presented, along with its derivation. Properties of the density function and the LR test are presented and discussed. A new approximation for the LR test critical region is proposed and studied. 1. INTRODUCTION AND SUMMARY THE CRITICAL NATURE of identification in simultaneous equation models is well recognized. The identification status of equations within a simultaneous equation system helps determine which estimator to use and dictates some properties of the alternative estimators. It represents a statistical hypothesis that is logically prior to many other standard estimation and testing procedures carried out for model construction and evaluation. The major purposes of this paper are (i) analytical exploration of the likelihood ratio identifiability test statistic for equations with any number of endogenous and exogenous variables and (ii) proposal of a new approximate critical region for testing overidentifying restrictions in such equations. Anderson and Rubin [2] proposed a likelihood ratio test of overidentifying restrictions imposed on the structural parameter space along with their derivation of LIML estimates. Basmann [5] proposed a similar statistic based on the GCL estimation method. McDonald [20] derived the exact finite sample density function for the LR test statistic for an equation containing exactly two endogenous variables. Richardson [26] derived the finite sample density function for the two endogenous variable GCL version of the identifiability test statistic. Both McDonald and Richardson showed that the moments of the exact functions converge to moments of the F distribution as the concentration parameter increases without bound. Each gave necessary conditions for existence of exact moments. McDonald showed that the GCL test statistic will not possess more moments than the LIML statistic in the two-equation case. In further analysis, Kadane [13, 14] showed that both tests are consistent and that both the LR and GCL identifiability test statistics converge to the F distribution as the structural disturbances grow small. Some parallel developments have occurred in multivariate statistical analysis. These are not reviewed here because they are mainly tangential or else are cited at appropriate places in this paper.