Formal interpretation of likelihood ratio tests?

Question

The following is a hypothetical question and I don't have a specific model in mind...

Consider two models $M_0, M_1$ such that $M_0$ is nested within $M_1$.

I perform the likelihood ratio test with $M_0$ being considered the "null" model and obtain a test statistic $\chi$. I understand that if the null model $M_0$ is true, then asymptotically $\chi$ will follow a chi-square distribution with $k_1 - k_0$ degrees of freedom, the difference in parameters between the two models.

What I'm having trouble understanding is how to actually interpret a rejection.

So suppose $\chi$ is such that we obtain a very small p-value and reject the null hypothesis.

What does this actually mean with respect to the model $M_1$? Does it actually tell us anything about whether it is a good fit?

Aren't we simply rejecting that the null model is true, so that it only tells us the model $M_0$ is unlikely to hold, and does not tell us anything about the validity of $M_1$?

The reason I am confused is that many examples use the likelihood ratio test in order to compare the two models. But how does this test tell us about the comparitive fit, rather than simply "$M_0$ does not hold"?

Hopefully I've made my confusion clear.

Answer 1

You are assessing whether or not M1 provides a statistically better fit than M0, that’s all. It does not indicate whether or not M0 (or M1 for that matter) is a good fit. You can imagine a situation where your models are misspecified such that neither would be considered ‘good’, but because M1 slightly leads to lower error, it is significantly better.

Answer 2

Consider the following special case. Assume model $M_1$ is sufficiently flexible and Contains the distribution which generated the data. In addition, assume $M_0$ is fully nested within $M_1$, then rejecting the null hypothesis that $M_1$ and $M_0$ provide equally good fits implies That $M_0$ is not a good fit for the class of statistical environments specified by $M_1$. So in this very special case where $M_1$ is very flexible The likelihood ratio test is detecting lack of fit In model $M_0$.