Asymptotic results under multiway clustering

Abstract

If multiway cluster-robust standard errors are used routinely in applied economics, surprisingly few theoretical results justify this practice. This paper aims to fill this gap. We first prove, under nearly the same conditions as with i.i.d. data, the weak convergence of empirical processes under multiway clustering. This result implies central limit theorems for sample averages but is also key for showing the asymptotic normality of nonlinear estimators such as GMM estimators. We then establish consistency of various asymptotic variance estimators, including that of Cameron et al. (2011) but also a new estimator that is positive by construction. Next, we show the general consistency, for linear and nonlinear estimators, of the pigeonhole bootstrap, a resampling scheme adapted to multiway clustering. Monte Carlo simulations suggest that inference based on our two preferred methods may be accurate even with very few clusters, and significantly improve upon inference based on Cameron et al. (2011).

Algorithmic Subsampling Under Multiway Clustering

“…Lemmas D.3 and D.4 follow closely from Theorem 3.4(i) in Davezies et al (2021) and Lemma D.12 in Davezies et al (2018), respectively. Lemma D.3 (Glivenko-Cantelli for two-way clustered random variables).…”

“…The first step is to get the asymptotic normality for A NM . Our setup satisfies the first part of Assumption 3 in Davezies, D'Haultfoeuille, and Guyonvarch (2018), since class F is finite and E[F 2 ] < ∞. Under our Assumptions 1, 2(i), (iii), and 3, applying Theorem 3.1 of Davezies et al (2018) yields…”

Inference in Difference-in-Differences: How Much Should We Trust in Independent Clusters?

“…If we had a setting with many periods and consider a CRVE at the time level, then the reverse result would hold. A possible alternative in this case, if both N and T are large, could be the use of two-way cluster at the group and time dimensions (see Cameron et al (2011), Thompson (2011, Davezies et al (2018), Menzel (2017), and MacKinnon et al ( 2019)). While some of these methods report good performance in simulations with few clusters in one dimension, if common factors are serially correlated, then this solution would not take into account the correlation between η jt and η j t , for j = j and t = t , which would lead to over-rejection.…”

Dyadic double/debiased machine learning for analyzing determinants of free trade agreements

“…Also related to, but different from, the literature on dyadic data is the set of studies which investigate multiway clustered data (e.g., Cameron, Gelbach, and Miller, 2011;Thompson, 2011;Cameron and Miller, 2015;Menzel, 2021;Davezies, D'Haultfoeuille, and Guyonvarch, 2018;MacKinnon, Nielsen, and Webb, 2021;Chiang, Kato, Ma, and Sasaki, 2021a). The robust variance formulas exposited in this literature are related to those relevant for dyadic cases, but are sufficiently different that they cannot be applied in settings with dyadic dependence.…”