The wild bootstrap with a “small” number of “large” clusters

This paper studies the wild bootstrap–based test proposed in Cameron, Gelbach, and Miller (2008). Existing analyses of its properties require that number of clusters is “large.” In an asymptotic framework in which the number of clusters is “small,” we provide conditions under which an unstudentized version of the test is valid. These conditions include homogeneity-like restrictions on the distribution of covariates. We further establish that a studentized version of the test may only overreject the null hypothesis by a “small” amount that decreases exponentially with the number of clusters. We obtain a qualitatively similar result for “score” bootstrap-based tests, which permit testing in nonlinear models.