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.  
 A: Misspecification via redundant variables will produce proportional rows (or columns, if you perfer) in the covariance matrix.  This causes the defect you observe -- the inverse covariance does not exist.  (Other defects may do this, depending on details not available to a general analysis.)
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.)
A: Vuong actually showed that if you don't assume the information matrix equality holds then the wilks 1938 statistics has a weighted chi square distribution. In the special case of correct specification
(Which implies the information matrix equality holds)
 then the weights are all equal to one and this reduces to the 1938 result. You might find my paper on this to be helpful 
"Discrepancy risk model selection test theory for comparing possibly misspecified or non nested models" psychometrika 2003 vol 68 issue 2 
229-249. Richard Golden
