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

2025-11-07

AI Summary

This paper addresses statistical inference challenges in difference-in-differences (DID) designs with a single treated cluster and a fixed number of control clusters. Under weak assumptions permitting arbitrary unknown intra-cluster dependence, we propose a variance-free t-test that avoids estimating the asymptotic variance. The method requires only a user-specified bound on the relative heteroskedasticity between treated and control clusters; it then constructs customized critical values—either analytically or via numerical optimization—to achieve valid inference at any desired significance level. Unlike conventional approaches, it does not rely on asymptotic normality or large numbers of clusters, thereby substantially improving inference reliability in small-sample and limited-control-group settings. Extensive simulations and empirical applications demonstrate the method’s robustness and high statistical power. A table of commonly used critical values is provided for immediate implementation by applied researchers.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

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.

Problem

• Addresses inference with single treated cluster and fixed controls
• Develops t-test critical values without variance estimation
• Handles unknown within-cluster dependence in difference-in-differences designs

Innovation

• Uses t-test for single treated cluster inference
• Develops critical values without variance estimation
• Requires only relative heterogeneity specification