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In order to use clustered standard errors, I want to make sure that the residuals in each cluster are coming from the same distribution (and thus, we expect correlation), thus clustering is needed. Is there a way to ensure that the residuals in one cluster are from the same distribution? Is it possible to do some inspection by visualization?

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If you have enough data in a cluster (say 20 or more obs), you can plot the residuals against other variables and look for patterns of heteroscedasticity and/or nonlinearity. If you don't have that many observations you may have to resort to pooling the resdiuals from different clusters and performing similar analyses.

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