[Paper Review] Asymptotic results under multiway clustering

Laurent Davezies, Xavier D’Haultfœuille+1 more|arXiv (Cornell University)|Jul 20, 2018

Monetary Policy and Economic Impact4 citations

TL;DR

This paper establishes asymptotic theory for multiway clustering in econometrics, proving weak convergence of empirical processes and consistency of cluster-robust variance estimators, including a new positive-definite alternative to Cameron et al. (2011). It further validates the pigeonhole bootstrap for multiway clustering, showing improved inference accuracy even with few clusters, thus providing theoretical justification for widely used empirical practices.

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).

Motivation & Objective

• To provide theoretical justification for the widespread use of multiway cluster-robust standard errors in applied econometrics.
• To establish weak convergence of empirical processes under multiway clustering, enabling central limit theorems and asymptotic normality for nonlinear estimators.
• To prove consistency of cluster-robust variance estimators, including Cameron et al. (2011) and a new positive-definite alternative.
• To validate the pigeonhole bootstrap as an asymptotically valid resampling method under multiway clustering.
• To demonstrate through Monte Carlo simulations that the proposed methods improve inference accuracy, especially with few clusters.

Proposed method

• Proves weak convergence of empirical processes indexed by a class of functions under multiway clustering, under conditions nearly equivalent to i.i.d. data.
• Establishes consistency of three asymptotic variance estimators, including Cameron et al. (2011) and a new estimator that is guaranteed to be positive-definite.
• Introduces and proves the asymptotic validity of the pigeonhole bootstrap, which resamples clusters independently across multiple dimensions to preserve multiway dependence structure.
• Uses a general framework allowing for random, unbounded, and endogenous cell sizes, accommodating cluster heterogeneity.
• Applies the results to derive asymptotic normality for linear and nonlinear estimators, such as GMM estimators, under multiway clustering.
• Employs moment restrictions adapted to multiway clustering, ensuring theoretical validity under minimal assumptions.

Experimental results

Research questions

• RQ1 Under what conditions does the empirical process converge weakly under multiway clustering, enabling central limit theorems?
• RQ2 Are the cluster-robust variance estimators proposed by Cameron et al. (2011) consistent under multiway clustering, and can they be improved?
• RQ3 Can a new variance estimator be constructed that is consistent and always positive-definite?
• RQ4 Is the pigeonhole bootstrap asymptotically valid for inference under multiway clustering?
• RQ5 How does inference based on the proposed methods compare to existing methods in finite samples, especially with few clusters?

Key findings

• The empirical process under multiway clustering converges weakly under conditions nearly identical to those for i.i.d. data, enabling standard asymptotic theory for estimators.
• The variance estimator of Cameron et al. (2011) is consistent under the proposed framework, but may be negative in finite samples.
• A new variance estimator is proposed that is consistent and guaranteed to be positive-definite, improving practical reliability.
• The pigeonhole bootstrap is proven to be asymptotically valid for multiway clustering, providing a robust resampling alternative.
• Monte Carlo simulations show that inference based on the new variance estimator and the pigeonhole bootstrap is accurate even with very few clusters, outperforming Cameron et al. (2011).
• The theoretical results extend existing asymptotic normality results for GMM and other nonlinear estimators to the multiway clustering setting.