Wild Bootstrap Inference for Wildly Different Cluster Sizes

The cluster robust variance estimator (CRVE) relies on the number of clusters being large. The precise meaning of `large' is ambiguous, but a shorthand `rule of 42' has emerged in the literature. We show that this rule depends crucially on the assumption of equal-sized clusters. Monte Carlo evidence suggests that rejection frequencies can be much higher when a dataset has 50 clusters proportional to the populations of the US states than when it has 50 equal-sized clusters. In contrast, using a cluster wild bootstrap procedure generally works well in both cases. We also show that, when the test regressor is a dummy variable, as in a difference-in-differences framework, both conventional and bootstrap tests perform badly when the proportion of clusters treated is very small or very large. However, bootstrap tests perform very well when that is not the case. A third set of simulations studies placebo laws and finds that bootstrap tests usually perform very much better than conventional ones.