Should we always use clustered standard errors in panel data with only two time points, in a multilevel logistic regression? This is the situation:

*

*I have a binary outcome at two timepoints (T1 and T2);


*I'm using a random-effect logistic regression (I mean: just the random intercept, no random slopes) to estimate subject-specific effects.
My question is: should I use cluster-robust standard errors? Since the time points are just two, here there is no risk of wrong specification of the autocorrelation structure (I mean: there's no difference between "unstrctured", "AR1" and "exchangeble"), as long as we are confident that there is residual correlation between the observations of the same person (otherwise, the random intercept should be removed). So, the only reason I see to include cluster-robust standard errors is: to take into account the error in estimating the autocorrelation parameter. Am I right? In case: is it something advisable even when the number of clusters is relatively large (hundreds of people)?
I can add some other features of my situation, in case they were relevant: my panel is strongly unbalanced (T2 observations are available only for $43.5\%$ of observations); I'm controlling for outcome values at baseline (T0) and including "time" both as a predictor and in interaction terms, so that the effect of each covariate is allowed to differ between timepoints; finally, all covariates are constant across time.

If you've controlled for autocorrelation using AR(1), there's no need to account for it a second time using cluster-robust errors. Make sure you actually have controlled for the autocorrelation and are getting the right SE, though; assuming the within-subject observations have already been adjusted away will get the wrong answer, and that's a pretty easy mistake to make if you're not careful with your code.
If you have a large sample, bootstrapping the clusters themselves is a pretty easy way to get a good standard error without having to triple-check that your code is actually doing what you think it is, or you can check the bootstrapped errors closely match what your software is putting out.
