  This paper proves a new central limit theorem for a sample that exhibits
two-way dependence and heterogeneity across clusters. Statistical inference for
situations with both two-way dependence and cluster heterogeneity has thus far
been an open issue. The existing theory for two-way clustering inference
requires identical distributions across clusters (implied by the so-called
separate exchangeability assumption). Yet no such homogeneity requirement is
needed in the existing theory for one-way clustering. The new result therefore
theoretically justifies the view that two-way clustering is a more robust
version of one-way clustering, consistent with applied practice. In an
application to linear regression, I show that a standard plug-in variance
estimator is valid for inference.
