Non-normal variation and regression to the mean

Misspecification tests for parametric models, f(y, θ), that examine data for failure of moment conditions implied by the maintained parametric distribution are interpreted as score tests of H0: λ = 0 in the context of a parametric family of distributions r(y;θ, λ). This family contains the maintained distribution as a special case (λ = 0) and has the property that only in that special case do the chosen moment conditions hold. A likelihood ratio test of H0: λ = 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(y; θ, λ) 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).