Non-normal variation and regression to the mean
05 May 1997
Journal Article, Econometrica,
Vol. 65, No. 3, pp. 627-646
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 Op(n-2).