Inference based on cluster-robust standard errors is known to fail when the number of clusters is small, and the wild cluster bootstrap fails dramatically when the number of treated clusters is very small. We propose a family of new procedures called the sub- cluster wild bootstrap. In the case of pure treatment models, where all the observations in each cluster are either treated or not, the new procedures can work astonishingly well. The key requirement is that the sizes of the treated and untreated clusters should be very similar. Unfortunately, the analog of this requirement is not likely to hold for difference-in-differences regressions. Our theoretical results are supported by extensive simulations and an empirical example.
CRVE grouped data clustered data wild bootstrap wild cluster bootstrap subclustering treatment model difference-in-differences robust inference