We have revised 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. We also establish that a studentized version of the test may only over-reject the null hypothesis by a “small” amount in the sense that it has limiting rejection probability under the null hypothesis that does not exceed the nominal level by more than an amount that decreases exponentially with the number of clusters. We obtain results qualitatively similar to those for the studentized version of the test for closely related “score” bootstrap-based tests, which permit testing hypotheses about parameters in nonlinear models. We illustrate the relevance of our theoretical results for applied work via a simulation study and empirical application. An important lesson our results is that when these “homogeneity” 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).