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Likelihood Ratio Specification Tests

Andrew Chesher
Richard J. Smith

Abstract

Misspecification tests for parametric models, $f(\mathbf{y}, \theta)$, that examine data for failure of moment conditions implied by the maintained parametric distribution are interpreted as score tests of $H_0: \lambda = \mathbf{0}$ in the context of a parametric family of distributions $r(\mathbf{y}; \theta, \lambda)$. This family contains the maintained distribution as a special case $(\lambda = \mathbf{0})$ and has the property that only in that special case do the chosen moment conditions hold. A likelihood ratio test of $H_0: \lambda = \mathbf{0}$ therefore constitutes an alternative test of the validity of the moment conditions. This test admits a Bartlett correction, unlike conventional moment tests for which adjustments based on second order asymptotic theory may behave badly. The dependence of the Bartlett correction and of the $O(n^{-1/2})$ local power of the test on the way in which $r(\mathbf{y}; \theta, \lambda)$ is constructed is studied. In many cases the correction can be made to vanish leading to a specification test whose distribution is chi-square to order $O_p(n^{-2})$.