Is it useful to implement clustered SE in the probit-type models? For my research, I am implementing a two-stage Heckman procedure. I am working with panel data, so I was wondering if it is common and actually needed to use clustered standard errors for the first stage (probit)? There are many discussions and a lot of literature about clustered SE for linear models, however very limited information regarding non-linear ones. Would be thankful for any suggestions.
 A: TL;DR: In clustered data - and if you can assume that you have specified the marginal model correctly - it is a valid strategy to use a standard probit model (assuming independence) but account for cluster correlation by clustered standard errors.
Details: While in linear models clustered standard errors are useful for accounting for both (a) heteroscedasticity and (b) cluster correlations, only the latter is true for clustered standard errors in logit and probit models.
In general, for "robust" standard errors to work the main estimating equation must be correctly specified. In binary regression models this means that the mean equation (yielding the "success probability") is correctly specified (correct link function, correct linear predictor).
The robust standard errors can then account for the rest of the (joint) likelihood being misspecified. However:

*

*In independent binary data there is no room for further misspecification (as already pointed out by @Repmat in the comments) because you have already fully specified the entire likelihood when you have specified the mean aka success probability. Thus, either your entire model is misspecified (and then the parameter estimates are inconsistent) or your entire model is correctly specified (and then there is no need for robust standard errors).


*In clustered data correct specification of the mean aka success probability means that you have correctly specified the marginal distributions. There may still be correlation in the joint distribution that you haven't accounted for if you used a basic probit model that assumed independent observations.
Illustration: Simulations illustrating that the above strategy works in clustered binary responses can be found in Experiment II in Section 6 of

Achim Zeileis, Susanne Köll, Nathaniel Graham (2020).
"Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R.”
Journal of Statistical Software, 95(1), 1-36.
doi:10.18637/jss.v095.i01

