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.

翻译：在多向聚类的计量经济学应用中，有效聚类数量较少或数据重尾分布的情况时常出现，这使得标准聚类稳健推断与自助法推断在有限样本下不可靠。本文基于误差条件可交换性假设，为多向聚类线性回归建立了有限样本有效的置换推断框架。我们的假设与早期研究中提出的分离可交换性概念密切相关，但由于其对协变量分布施加了最小限制，在许多经济设定中可能更为现实。我们构建了有限样本有效的显著性置换检验，并建立了理论功效保证，这与现有仅具有渐近合理性的方法形成对比。我们还将该方法拓展至存在缺失数据的设定，并推导出揭示可检测性相变现象的功效结果。通过模拟研究，我们证明所提出的检验能保持正确的检验水平与具有竞争力的功效，而标准聚类稳健方法与自助法程序则可能出现显著的检验水平失真。