I've only seen the Mundlak-Chamberlain (aka Mundlak aka Chamberlain* aka "correlated random effects": henceforth MC) specification applied in the random effects (RE) context. But is there any reason it can't be applied in a generalized estimating equations (GEE) framework?
I would think that, in both linear and nonlinear modeling settings, applying the MC specification in a GEE framework would yield the same coefficients on the time-varying (or cluster-varying, depending on the context, but for concision I will just talk about time-varying going forward) regressors as would be obtained from applying the MC specification in an RE framework -- or at least, that they would have the same interpretation: that is, a "within" (using econometric jargon) or "subject-specific" (using biometric jargon) effect. Is this correct? I know that in general GEE only estimates "population-averaged" effects, but this seems to me like a way to separate the subject-specific effects from the population-averaged effects for the time-varying regressors.
The main difference between applying MC in a GEE framework compared to an RE framework, I would think, would be in terms of the assumptions and interpretation of the coefficients on the variables that don't vary over time. Of course, for these to even be estimable** at all, we must work with a slightly more restrictive version of the MC specification, namely that the unobserved heterogeneity is uncorrelated with the time-invariant regressors. But let's work with this assumption.
Then, when the MC specification is applied in an RE framework, the mean of the distribution of the unobserved heterogeneity is allowed to depend on the time-varying regressors, but the key restrictive assumptions are: (1) that the unobserved heterogeneity follows a normal distribution, with (2) homoskedastic variance. These assumptions enable the estimated coefficients on the time-invariant regressors to have a "subject-specific" interpretation (even though they are only identified on the basis of cross-subject variation combined with these assumptions.. but that is an argument for another post).
But what I'm thinking is, if the MC specification is applied in a GEE framework, could that allow us to do away with distributional assumptions (1) and (2) needed in the RE setting? Of course, as always, the interpretation of the coefficients on the time-invariant regressors must be population-averaged effects, but for me that is a worthwhile price to pay for the benefit of being able to relax the aforementioned distributional assumptions.
Is this approach valid? Are there any errors in my thought process?
*I am aware that the Chamberlain specification is actually slightly more general than the Mundlak specification but the difference is not important for my question.
**Both Merriam-Webster and dictionary.com list "capable of being estimated" as a secondary definition of "estimable".