Permutation Inference under Multi-way Clustering and Missing Data

2026-01-13

AI Summary

This study addresses the issue of size distortion in conventional robust standard errors and bootstrap methods under multi-way clustering when the number of effective clusters is small or the data exhibit heavy-tailed distributions. Building on an error condition of exchangeability that better reflects economic reality, the authors develop a finite-sample valid permutation inference framework for multi-way clustered linear regressions and extend it to settings with missing data. Simulation results demonstrate that the proposed method maintains correct test size while delivering substantially higher power than existing approaches. Notably, the analysis reveals a phase-transition-like phenomenon in testing power as the clustering structure varies, underscoring the method’s superior performance in realistic empirical scenarios.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Machine Learning: Multi-instance/Multi-view Learning
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Sustainability of Web economics

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.

Problem

• multi-way clustering
• missing data
• finite-sample inference
• permutation tests
• cluster-robust inference

Innovation

• permutation inference
• multi-way clustering
• conditional exchangeability
• finite-sample validity
• missing data