Permutation Inference under Multi-way Clustering and Missing Data

TL;DR

This paper tackles finite-sample inference under multi-way clustering by developing a permutation-based test that relies on conditional double exchangeability of regression errors given covariates. The authors formulate a three-step procedure—partialling-out via orthogonal projections, minorization, and a two-way permutation randomization—to produce finite-sample valid p-values and inversion-based confidence regions for the coefficient of interest in dyadic regression, with extensions to missing data and higher-way clustering. They establish power results under heavy-tailed errors, demonstrate finite-sample validity in simulations, and apply the method to bilateral trade flows, showing more conservative but robust inference compared to standard asymptotic approaches. The framework generalizes to random effects, panel models, irregular designs, and multi-way layouts, with a graph-theoretic approach to construct fully observed blocks in the presence of missing data. Overall, the work provides a practical, robust alternative for inference in complex dependence structures often encountered in economics and related fields.

Abstract

Econometric applications with multi-way clustering often feature a small number of effective clusters or heavy-tailed data, making standard cluster-robust and bootstrap inference unreliable in finite samples. In this paper, we develop a framework for finite-sample valid permutation inference in linear regression with multi-way clustering under an assumption of conditional exchangeability of the errors. Our assumption is closely related to the notion of separate exchangeability studied in earlier work, but can be more realistic in many economic settings as it imposes minimal restrictions on the covariate distribution. We construct permutation tests of significance that are valid in finite samples and establish theoretical power guarantees, in contrast to existing methods that are justified only asymptotically. We also extend our methodology to settings with missing data and derive power results that reveal phase transitions in detectability. Through simulation studies, we demonstrate that the proposed tests maintain correct size and competitive power, while standard cluster-robust and bootstrap procedures can exhibit substantial size distortions.

Figures (6)

• Figure 1: Left: The missing mask matrix $M$ induces outcomes with missing values (marked "$\mathrm{NA}$"). Right: Permutation 1 swaps columns 1 and 3 and rows 1 and 3, thus permuting observed values with missing values as indicated by the shaded cells. In contrast, permutation 2 swaps columns 2 and 3 and rows 2 and 3. This only permutes data from fully observed entries. Procedure \ref{['proc1']} is infeasible under permutation 1, but feasible under permutation 2.
• Figure 2: Test power based on 10,000 simulations at the 5% significance level.
• Figure 3: Confidence intervals for different variables in the regression model. We refer readers to silva2006log for detailed variable descriptions.
• Figure 4: Type I errors based on 10,000 simulations at the 5% significance level.
• Figure 5: Widths of IPT confidence intervals over the inverse of effective variances.

Theorems & Definitions (30)

• Theorem 1
• Remark 1
• Theorem 2
• Theorem 3
• Definition 1
• Theorem 4
• Remark 2: Construction of $\mathcal{F}_M$
• Proposition 1
• Theorem 5
• Remark 3