arXiv 7 Nov 2025 · Econometrics
arXiv:2511.05710 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Hagemann, Andres (2024) Inference with a single treated cluster | 1.000 | 28 | 6 | 100% |
| 2 | Conley, Timothy G. and Taber, Christopher R (2011) Inference with “Difference in Differences” with a Small Number of Policy Changes | 0.811 | 4 | 2 | 100% |
| 3 | Bakirov, N. K. and Székely, G. J (2006) Student's t-test for Gaussian scale mixtures | 0.794 | 8 | 3 | 50% |
| 4 | Ibragimov, Rustam and Müller, Ulrich K (2010) t-Statistic Based Correlation and Heterogeneity Robust Inference | 0.737 | 3 | 2 | 100% |
| 5 | Ibragimov, Rustam and Müller, Ulrich K (2016) Inference with Few Heterogeneous Clusters | 0.737 | 3 | 2 | 100% |
| 6 | Depew, Briggs and Swensen, Isaac (2022) The Effect of Concealed-Carry and Handgun Restrictions on Gun-Related Deaths: Evidence from the Sullivan Act of 1911 | 0.644 | 4 | 1 | 100% |
| 7 | Ivan A. Canay and Joseph P. Romano and Azeem M. Shaikh (2017) Randomization Tests under an Approximate Symmetry Assumption | 0.644 | 2 | 2 | 100% |
| 8 | Ferman, Bruno and Pinto, Cristine (2019) Inference in Differences-in-Differences with Few Treated Groups and Heteroskedasticity | 0.644 | 2 | 2 | 100% |
| 9 | Andreas Hagemann (2022) Permutation inference with a finite number of heterogeneous clusters | 0.644 | 2 | 2 | 100% |
| 10 | Lau, Chun Pong (2025) Combining Clusters for the Approximate Randomization Test self | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 38 scored citations.