  Thousands of papers have reported two-way cluster-robust (TWCR) standard
errors. However, the recent econometrics literature points out the potential
non-gaussianity of two-way cluster sample means, and thus invalidity of the
inference based on the TWCR standard errors. Fortunately, simulation studies
nonetheless show that the gaussianity is rather common than exceptional. This
paper provides theoretical support for this encouraging observation.
Specifically, we derive a novel central limit theorem for two-way clustered
triangular arrays that justifies the use of the TWCR under very mild and
interpretable conditions. We, therefore, hope that this paper will provide a
theoretical justification for the legitimacy of most, if not all, of the
thousands of those empirical papers that have used the TWCR standard errors. We
provide a guide in practice as to when a researcher can employ the TWCR
standard errors.
