Wild Bootstrap and Asymptotic Inference With Multiway Clustering

Abstract

We study two cluster-robust variance estimators (CRVEs) for regression models with clustering in two dimensions and give conditions under which t-statistics based on each of them yield asymptotically valid inferences. In particular, one of the CRVEs requires stronger assumptions about the nature of the intra-cluster correlations. We then propose several wild bootstrap procedures and state conditions under which they are asymptotically valid for each type of t-statistic. Extensive simulations suggest that using certain bootstrap procedures with one of the t-statistics generally performs very well. An empirical example confirms that bootstrap inferences can differ substantially from conventional ones.Comparing the results in Theorems 1 and 2, it is immediate that only the three-term CRVE,V 3 , is consistent in all three of the cases considered. In contrast,V 2 is consistent only in the first case. In the other cases, it is not even valid asymptotically. Instead, it would yield standard errors that are too large, asymptotically, by a factor of √ 2. The asymptotic distributions of the t-statistics in (10), i.e. t a,2 and t a,3 , follow immediately from Theorems 1 and 2, and we state these as a corollary. 9 Corollary 1. Suppose Assumptions 1-6 are satisfied and the null hypothesis, H 0 , is true. (16) is true, then it holds that t a,2 d −→ N(0, 1) and t a,3 d −→ N(0, 1).(ii) If either (17) or (18) is true, then it holds that t a,2 d −→ N(0, 1/2) and t a,3 d −→ N(0, 1).The result in Corollary 1 shows the fundamental problem with the two-term CRVE, namely, that it is not robust to cases where the assumed cluster structure is in fact not present in the DGP. In such situations, shown in case (ii) of Corollary 1, a t-test based on t a,2 and critical values from the N(0, 1) distribution is not valid, even asymptotically. On the other hand, a t-test based on t a,3 is valid in all cases considered. This may or may not outweigh the numerical advantages of the two-term CRVE discussed earlier, depending on how close the (unknown) DGP is to the special cases in (17) and (18).More generally, the results in Corollary 1 justify the use of critical values and P values from a normal approximation to perform t-tests and construct confidence intervals based on the three-term CRVE,V 3 . However, results in Bester, Conley, and Hansen (2011) suggest that it will often be more accurate to use the t(G − 1) distribution in the one-way case; see Cameron and Miller (2015) for a discussion of this issue. In the two-way case, CGM suggests using the t(R − 1) distribution (recall from Assumption 5 that R = min{G, H}), and we do this in Sections 5 and 6 below.

Fast and wild: Bootstrap inference in Stata using boottest

“…As discussed in section 4.2, there are three natural choices for the level of bootstrap clustering in this case: by ethno-linguistic group, by country, or by the intersection of the two. Simulation results in MacKinnon, Nielsen, and Webb (2017) favor clustering in the dimension with the fewest clusters, which, in this case, is the 49 countries. For illustration, we perform tests for all three levels.…”

The Wild Bootstrap with a 'Small' Number of 'Large' Clusters

“…The use of the wild bootstrap in applications with such a small number of clusters contrasts sharply with existing analyses of its theoretical properties, which, to the best of our knowledge, all employ an asymptotic framework where the number of clusters tends to infinity. See, for example, Carter et al (2017), Djogbenou et al (2017), and MacKinnon et al (2017). In this paper, we address this discrepancy by studying its properties in an asymptotic framework in which the number of clusters is fixed, but the number of observations per cluster tends to infinity.…”

How cluster‐robust inference is changing applied econometrics

“…MacKinnon et al. () show that the WCR bootstrap often works well in this case, and boottest makes it easy to do this.…”

“…() and Thompson () to propose CRVEs that allow for clustering in two or more dimensions; see MacKinnon et al. ().…”