Cluster-robust inference with a single treated cluster using the t-test

TL;DR

This paper develops a t-test-based inference method for settings with a single treated cluster and multiple control clusters, allowing for unknown dependence and heterogeneity, without variance estimation.

Contribution

It introduces a novel t-test procedure with suitable critical values for valid inference in clustered data with one treated cluster, applicable under weak assumptions.

Findings

The test provides valid inference with at least two control clusters.

Critical values can be computed easily for common significance levels.

Simulations and empirical applications demonstrate the method's effectiveness.

Abstract

This paper considers inference when there is a single treated cluster and a fixed number of control clusters, a setting that is common in empirical work, especially in difference-in-differences designs. We use the t-statistic and develop suitable critical values to conduct valid inference under weak assumptions allowing for unknown dependence within clusters. In particular, our inference procedure does not involve variance estimation. It only requires specifying the relative heterogeneity between the variances from the treated cluster and some, but not necessarily all, control clusters. Our proposed test works for any significance level when there are at least two control clusters. When the variance of the treated cluster is bounded by those of all control clusters up to some prespecified scaling factor, the critical values for our t-statistic can be easily computed without any optimization for many conventional significance levels and numbers of clusters. In other cases, one-dimensional numerical optimization is needed and is often computationally efficient. We have also tabulated common critical values in the paper so researchers can use our test readily. We illustrate our method in simulations and empirical applications.