Serial correlation and clustered correlation in FE and FD estimators

Question

I'm running a regression on panel data, one time with Fixed Effects and one time with First Differences. The estimators are really different (the FE estimator is statistically significant and the FD isn't). While I'm running the regressions with the Clustered Standard Errors, it affects a lot the S.D of the FD estimator but not the S.D of the FE estimator. When I'm running Wooldridge test for serial correlation I get that there is no serial correlation. My questions:

Is it possible to be within clustered correlation for the FD model but not for the FE?
If there is no serial correlation, is it possible to be a within clustered correlation?

chan1142 · Answer

Yes. If the idiosyncratic error is iid, xtreg (FE) with no options (ordinary se) is valid, but the ordinary se for FD, reg d.(y x), is invalid because the differenced error is serially correlated.
Yes, it is. It's cluster-robust, which means OK  regardless of the presence of within-group correlation.

Serial correlation and clustered correlation in FE and FD estimators

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