We just finished a paper, joint with Azeem Shaikh and Andres Santos, on the formal properties of the Wild Cluster Bootstrap when the data contains few, but large, clusters [See paper here].

Cameron et al. (2008) provide simulations that suggest the wild bootstrap test works well even in settings with as few as five clusters, but existing theoretical analyses of its properties all rely on an asymptotic framework in which the number of clusters is “large.”

In contrast to these analyses, we employ an asymptotic framework in which the number of clusters is “small,” but the number of observations per cluster is “large.” In this framework, we provide conditions under which the limiting rejection probability of an un-Studentized version of the test does not exceed the nominal level. Importantly, these conditions require, among other things, certain homogeneity restrictions on the distribution of covariates. The practical relevance of these conditions in finite samples is confirmed via a small simulation study. In addition, our results can help explain the remarkable behavior of these tests in the simulations of Cameron et al. (2008). It follows from our results that when these conditions are implausible and there are few clusters, researchers may wish to consider methods that do not impose such conditions, such as Ibragimov and Muller (2010) and Canay, Romano, and Shaikh (2017).