Combining Clusters for the Approximate Randomization Test

TL;DR

This paper introduces efficient methods for combining clusters in the approximate randomization test to address identification issues in small-sample inference, demonstrating improved performance through simulations and empirical analysis.

Contribution

It develops novel procedures for combining clusters when the target parameter isn't identified within individual clusters, enhancing the test's applicability and power.

Findings

Procedures perform well in simulations.

Methods improve inference in small samples.

Empirical application confirms effectiveness.

Abstract

This paper develops procedures to combine clusters for the approximate randomization test proposed by Canay, Romano, and Shaikh (2017). Their test can be used to conduct inference with a small number of clusters and imposes weak requirements on the correlation structure. However, their test requires the target parameter to be identified within each cluster. A leading example where this requirement fails to hold is when a variable has no variation within clusters. For instance, this happens in difference-in-differences designs because the treatment variable equals zero in the control clusters. Under this scenario, combining control and treated clusters can solve the identification problem, and the test remains valid. However, there is an arbitrariness in how the clusters are combined. In this paper, I develop computationally efficient procedures to combine clusters when this identification requirement does not hold. Clusters are combined to maximize local asymptotic power. The simulation study and empirical application show that the procedures to combine clusters perform well in various settings.