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As the question says, how are the two concepts related? As far as I understand, both approaches correct for heteroscedasticity and autocorrelation. Yet, they are different. Would applying one of the both methods in a case where the other one would be more "appropriate" yield biased SE´s?

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Both estimators belong to the broad class of "sandwich covariances" so they are related. Both are robust to certain kinds of autocorrelation (and heteroscedasticity) in the data. However, clustered covariances are for clustered data while Newey-West are for time-series data. Thus, the former guards against within-cluster correlation of independent clusters while the latter guards against serial correlation. For an overview of these and other related methods see:

Berger S, Graham N, Zeileis A (2020). "Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R." Journal of Statistical Software, 95(1), 1-36. Forthcoming, preprint at http://EconPapers.RePEc.org/RePEc:inn:wpaper:2017-12.

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