# Why doesn't Wilks 1938 proof work for misspecified models?

In a famous 1938 paper ("The large-sample distribution of the likelihood ratio for testing composite hypotheses", Annals of Mathematical Statistics, 9:60-62), Samuel Wilks derived the asymptotic distribution of $2 \times LLR$ (log likelihood ratio) for nested hypotheses, under the assumption that the larger hypothesis is correctly specified. The limiting distribution is $\chi^2$ (chi-squared) with $h-m$ degrees of freedom, where $h$ is the number of parameters in the larger hypothesis and $m$ is the number of free parameters in the nested hypothesis. However, it is supposedly well-known that this result does not hold when the hypotheses are misspecified (i.e., when the larger hypothesis is not the true distribution for the sampled data).

Can anyone explain why? It seems to me that the proof should still work with minor modifications. It relies on the asymptotic normality of the maximum likelihood estimate (MLE), which still holds with misspecified models (assuming an invertible covariance matrix). The only difference is the nature of the covariance matrix: for correctly specified models, we can approximate it with the inverse Fisher information matrix $J^{-1}$, with misspecification, we can use the sandwich estimate of the covariance matrix ($J^{-1} K J^{-1}$). The latter reduces to the inverse of the Fisher information matrix when the model is correctly specified (since $J = K$). AFAICT, Wilks proof doesn't care where the estimate of the covariance matrix comes from, as long as we have an invertible asymptotic covariance matrix of the multivariate normal for the MLEs.

Update 20140619: See also [Vuong 1989], cor. 7.3 et seq. and cor. 7.5 et seq. where Wilks's result holds if the larger model is correctly specified. (If it is not, its limiting distribution is not necessarily $\chi^2$ with the correct number of degrees of freedom.) The easiest failure is getting the wrong number of d.o.f. by having redundancy. Other failure modes are possible. This result doesn't seem to notice whether the smaller model is or is not misspecified. (It seems to be acceptable for the larger model to incorporate any misspecification of the smaller model.)
• Vuong's paper is actually one of the motivations for my question. Has Vuong actually shown that the limiting distribution is not $\chi^2$ under misspecification? It is not clear to me. But my question is about Wilks' proof, which appears to still work under misspecification. (Aside: AIUI, the smaller model inherits specification/misspecification from the larger model). – ratsalad Jun 19 '14 at 16:28
• No. Vuong has shown that under misspecification the limiting distribution may not be $\chi^2$, or may be $\chi^2$ with the wrong number of d.o.f. It is still possible for a happy coincidence to give the right distribution, so no such proof can exist (without additional hypotheses capturing that such a coincidence does not occur). Wilks's theorem has no additional hypotheses capturing that such a coincidence has occurred, so Wilks's proof cannot work under uncontrolled misspecification. (I may have the inheritance sentence backwards; it's been a while since I thought about this in detail.) – Eric Towers Jun 19 '14 at 17:03