I understand that if I have two models A and B and A is nested in B then, given some data, I can fit the parameters of A and B using MLE and apply the generalized log likelihood ratio test. In particular, the distribution of the test should be $\chi^2$ with $n$ degrees of freedom where $n$ is the difference in the number of parameters that $A$ and $B$ have.
However, what happens if $A$ and $B$ have the same number of parameters but the models are not nested? That is they are simply different models. Is there any way to apply a likelihood ratio test or can one do something else?
Best Answer
The paper Vuong, Q. H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 307-333. has the full theoretical treatment and test procedures. It distinguishes between three situations, "Strictly Non-nested Models", "Overlapping Models", "Nested Models", and also examines cases of misspecification. It is therefore no-accident that it finds that for some cases, the test statistic is distributed as a linear combination of chi-squares.
The paper is not light, neither it proposes an "off-the-shelf" testing procedure. But, for once, its (close to) 3,000 citations speak of its merits, being an inspired combination of classical testing framework and the information-theoretic approach.