# Permutation inference with a finite number of heterogeneous clusters

## TL;DR

This paper proposes a simple, robust permutation testing method for evaluating treatment effects across finite, heterogeneous clusters, effectively controlling size and accommodating variability among clusters.

## Contribution

It introduces an easy-to-implement permutation procedure with level adjustments that reliably tests hypotheses in clustered experimental designs with heterogeneity.

## Key findings

- Controls size asymptotically with level-adjusted permutation test
- Performs well with at least four treated and control clusters
- Robust to high variability among clusters

## Abstract

I introduce a simple permutation procedure to test conventional (non-sharp) hypotheses about the effect of a binary treatment in the presence of a finite number of large, heterogeneous clusters when the treatment effect is identified by comparisons across clusters. The procedure asymptotically controls size by applying a level-adjusted permutation test to a suitable statistic. The adjustments needed for most empirically relevant situations are tabulated in the paper. The adjusted permutation test is easy to implement in practice and performs well at conventional levels of significance with at least four treated clusters and a similar number of control clusters. It is particularly robust to situations where some clusters are much more variable than others. Examples and an empirical application are provided.