Bootstrap with Clustering in Two or More Dimensions

We propose a bootstrap procedure for data that may exhibit cluster dependence in two or more dimensions. We use insights from the theory of generalized U-statistics to analyze the large-sample properties of statistics that are sample averages from the observations pooled across clusters. The asymptotic distribution of these statistics may be non-standard if observations are dependent but uncorrelated within clusters. We show that there exists no procedure for estimating the limiting distribution of the sample mean under two-way clustering that achieves uniform consistency. However, we propose (a) one bootstrap procedure that is adaptive and point-wise consistent for any fixed data-generating process (DGP), (b) an alternative procedure that is uniformly consistent if we exclude the case of dependence with no correlation. The two procedures can be combined for uniformly valid, but conservative inference. For pivotal statistics, either procedure also provides pointwise asymptotic refinements over the Gaussian approximation when the limiting distribution is normal. We discuss several special cases and extensions, including V-statistics, subgraph densitities for network data, and non-exhaustive samples of matched data. JEL Classification: C1, C12, C23, C33