Moment based tests for mispecification of parametric models (e.g., of mean equals variance in a Poisson model) are studied. The moment restrictions under test are embedded in an extension of the model so that the moment test is a score test of the hypothesis that a vector of added parameters is zero. Second-order asymptotic properties of the likelihood ratio version of this test are studied. Unlike the conventional test, the likelihood ratio version is Bartlett correctable. The correction depends on the curvature at the origin of the function used to incorporate the moment restriction in the extended model.
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Chesher et al. (1997) studied this question.
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