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 at the five percent level can be more than twice the desired size when a dataset has 50 clusters proportional to the populations of the US states. In contrast, using a cluster wild bootstrap procedure for the same dataset usually results in very accurate rejection frequencies. We also show that, when the test regressor is a dummy variable, both conventional and bootstrap tests perform badly when the proportion of clusters treated is very small or very large. A third set of simulations uses placebo laws to see whether similar results hold in a difference-in-differences framework.