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My question relates to recent work on correlated random effects probit models (see these slides from Wooldridge) and comparing them to GEE population averaged probit models: Is one approach better as accounting for within-subject correlations than the other?

CRE (more mixed effect?) models account for within-subject variation by including a within-subject variable ($\bar{x}_{i}$) capturing the mean of each time-variant explanatory variable ($x_{it}$). The statistical significance of this parameter is then a test of traditional RE assumption that the individual effects are independent of the explanatory variables. It also serves as a correction for violations of this assumption, but requires (I think) assuming that the individual effect is normally distributed around the mean of the individual's values.

In GEE models, from what I understand, within subject correlations are accounted for via the correlation structure of the error term. What I don't quite understand is if/why the betas are not affected by within-subject correlations and the extent to which we need to worry about that.

I understand that GEEs can't recover the individual effect, but I don't follow why much of the action is in the errors and there is little discussion of how within-subject variation might bias the population estimate of the betas.

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It looks like I was mistaking what sort of problems correlated data introduce to the estimation process. I had it in my head that there was a bias problem with correlated data since FE probit/logit are biased. But this bias, it seems, comes from the errors (incidental parameters problem) and thus the focus on the errors in GEE population-averaged estimates makes sense.

CRE methods deal with incidental parameters problem by using a modified RE model that doesn't have same incidental parameters problem that a FE probit has (see this paper by Bell and Jones). Then it deals with assumptions about the correlation between explanatory variables and errors by modeling some of this dependence with subject-specific parameters a la Mundlak 1978.

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