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.
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1$\begingroup$ In general no, its not usefull. For a probit model heteroscedasticity is not different from model miss specification. That is, if you need to use clustered SE for probit it's akin to saying "my model is miss specified". $\endgroup$– RepmatCommented Apr 22, 2021 at 19:09
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$\begingroup$ @Repmat , thank you for your reply. This is true but if I have panel data (e.g. I am studying airline delay propagation, so I have data for many airlines), I need to implement probit to see whether some factors lead to the propagation or not. So then I suppose I need to do clustering per airline? $\endgroup$– AnnaCommented Apr 23, 2021 at 8:56
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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
answered Apr 30, 2021 at 8:41
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$\begingroup$ I agree that under clustering the marginal distribution is correctly specified, but wouldn't the likelihood function be misspecified? $\prod_{i=1}^n f(y|x,\beta)$ assumes independence, which clustering violates. Aren't there concerns of inconsistency? $\endgroup$ Commented May 26, 2022 at 19:30
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$\begingroup$ No, that's the whole motivation for this approach. For consistency, you don't need to specify the entire likelihood correctly - it is sufficient if the model "scores" are correctly specified. In this case, this means: how the probabilities depend on the regressors needs to be correctly specified. But the default standard errors will be inconsistent which is why "robust" or "clustered" standard erorrs are plugged in. These are consistent under certain departures from the assumptions (but only if the coefficient estimates are in fact consistent). $\endgroup$ Commented May 27, 2022 at 2:27