Inference with few heterogeneous clusters
Abstract
Suppose estimating a model on each of a small number of potentially heterogeneous clusters yields approximately independent, unbiased, and Gaussian parameter estimators. We make two contributions in this setup. First, we show how to compare a scalar parameter of interest between treatment and control units using a two-sample t-statistic, extending previous results for the one-sample t-statistic. Second, we develop a test for the appropriate level of clustering; it tests the null hypothesis that clustered standard errors from a much finer partition are correct. We illustrate the approach by revisiting empirical studies involving clustered, time series, and spatially correlated data.
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Canay, Romano and Shaikh (2017), Ibragimov and Müller (2016), Hagemann (2021), and Chernozhukov, Wüthrich and Zhu (2021) propose procedures that relax distributional assumptions about common shocks and accommodate richer heterogeneity.
“Canay, Romano and Shaikh (2017), Ibragimov and Müller (2016), Hagemann (2021), and Chernozhukov, Wüthrich and Zhu (2021) respectively propose permutation-based, t -test based, adjusted permutation-based, and conformal inference-based procedures that allow one to relax distributional assumptions about common shocks and accommodate richer forms of heterogeneity.”
From What's trending in difference-in-differences? A synthesis of the recent econometrics literature (2023) · cites this paper as Ibragimov and Müller (2016)
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“For instance, Canay et al. (2017), Ibragimov and Müller (2016), Hagemann (2021), and Chernozhukov et al. (2021) respectively propose permutation-based, t -test based, adjusted permutation-based, and conformal inference-based procedures that allow one to relax distributional assumptions about common…”
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Huang, Z., & Ibragimov, R. (2022) · Dependence Modeling · doi:10.1515/demo-2022-0109 · Cited by 2 · see citing works
Self-citationSign tests for dependent observations
Brown, D. J., & Ibragimov, R. (2018) · Econometrics and Statistics · doi:10.1016/j.ecosta.2018.11.001 · Cited by 2 · see citing works
Self-citationAsymptotic Theory for Clustered SamplesPreprint
Hansen, B. E., & Lee, S. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3099187 · Cited by 2 · see citing works
Андреевич, С. А., Pedersen, R. S., & Ibragimov, R. (2020) · SSRN Electronic Journal · doi:10.2139/ssrn.3580916 · Cited by 2 · see citing works
Self-citationUsing Multiple Outcomes to Adjust Standard Errors for Spatial Correlation
DellaVigna, S., Imbens, G. W., Kim, W., & Ritzwoller, D. M. (2025) · National Bureau of Economic Research · doi:10.3386/w33716 · Cited by 2 · see citing works
Some impossibility results for inference with cluster dependence with large clusters
Kojevnikov, D., & Song, K. (2023) · Journal of Econometrics · doi:10.1016/j.jeconom.2023.105524 · Cited by 1 · see citing works
Jackknife Standard Errors for Clustered Regression
Hansen, B. E. (2026) · The Review of Economic Studies · doi:10.1093/restud/rdag050 · Cited by 1 · see citing works
Midões, C., & de Crombrugghe, D. (2023) · The Journal of Economic Inequality · doi:10.1007/s10888-023-09574-w · Cited by 1 · see citing works
Panel Data Inference with Dependent Clusters
Powell, D. (2017) · Cited by 1 · see citing works
Inference and Empirical Examples
Ibragimov, M., Ibragimov, R., & Waldén, J. (2015) · Lecture notes in statistics · doi:10.1007/978-3-319-16877-7_3
Self-citationHwang, J. (2016) · eScholarship (California Digital Library)
Dependence-Robust Inference Using Randomized SubsamplingPreprint
Leung, M. P. (2017) · SSRN Electronic Journal · doi:10.2139/ssrn.3036626
Robust Inference on Income Inequality: \textit{t-}Statistic Based ApproachesPreprint
Ibragimov, R., Kattuman, P. A., & Андреевич, С. А. (2021) · arXiv (Cornell University)
Self-citationPanel Data with Unknown ClustersPreprint
Cai, Y. (2021) · RePEc: Research Papers in Economics · doi:10.48550/arxiv.2106.05503
Inference with Correlated ClustersPreprint
Powell, D. (2016) · SSRN Electronic Journal · doi:10.2139/ssrn.2786125
Unbiased estimation of the OLS covariance matrix when the errors are clustered
Boot, T., Niccodemi, G., & Wansbeek, T. J. (2023) · Empirical Economics · doi:10.1007/s00181-023-02379-w
A Market Crash or Tail Risk? Heavy Tails and Asymmetry of Returns in the Chinese Stock Market
Xing, Z., & Ibragimov, R. (2023) · Emerald Group Publishing Limited eBooks · doi:10.1108/s0731-90532023000045b009
Self-citationEling, M., Ibragimov, R., & Ning, D. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4827872
Self-citationUnbiased estimation of the OLS covariance matrix when the errors are clustered
Boot, T., Niccodemi, G., & Wansbeek, T. J. (2023) · Advanced studies in theoretical and applied econometrics · doi:10.1007/978-3-031-48385-1_3
Using Multiple Outcomes to Adjust Standard Errors for Spatial CorrelationPreprint
DellaVigna, S., Imbens, G. W., Kim, W., & Ritzwoller, D. M. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5233074
Gradient Wild Bootstrap for Instrumental Variable Quantile Regressions with Weak and Few Clusters
Wang, W., & Zhang, Y. (2026) · Journal of Business and Economic Statistics · doi:10.1080/07350015.2026.2618311
All that legislates is not gold: Representation and redistribution in Chile
Contreras, C. M., & Corvalán, A. (2026) · European Journal of Political Economy · doi:10.1016/j.ejpoleco.2026.102818
Statistical Comparison of Forecasts Made at Different Frequencies
Zhang, R., Zu, Y., & Li, W. (2026) · Journal of Business and Economic Statistics · doi:10.1080/07350015.2026.2634825
Drachal, K., & Tabash, M. I. (2026) · International May Conference on Strategic Management (IMCSM) · doi:10.5937/imcsm26407d
Tail risk, large fluctuations and downfalls in renewable energy markets
Hasanov, A. S., Ibragimov, R., Yıldırım, R., Chong, S. K., & Abrorov, S. (2026) · PLoS ONE · doi:10.1371/journal.pone.0351106
Self-citation
- Authors
- Ibragimov, R., & Müller, U. K.
- Affiliations
- Imperial College London (GB)
- Princeton University (US)
- ORCID
- Ibragimov, Rustam
- Venue
- Review of Economics and Statistics
- Publisher
- MIT Press - Journals
- Published
- 2016-03
- Volume
- 98
- Issue
- 1
- Pages
- 83-96
- ISSN
- 0034-6535 (ISSN-L)
- 1530-9142
- OpenAlex
- W1942265867
- Cited by
- 99
- References
- 47
- Citations by year
- 2026: 6 · 2025: 7 · 2024: 5 · 2023: 15 · 2022: 9 · 2021: 9 · 2020: 13 · 2019: 10 · 2018: 9 · 2017: 8 · 2016: 5 · 2015: 2 · 2014: 1
- FWCI
- 7.14
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- The Review of Economics and Statistics (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Spiral (Imperial College London) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Keywords
- heterogeneous clusters, clustered standard errors, treatment-control comparison, small number of clusters, time series, spatially correlated data
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