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The Wild Bootstrap with a “Small” Number of “Large” Clusters

Author

Listed:
  • Ivan A. Canay

    (Northwestern University)

  • Andres Santos

    (UCLA)

  • Azeem M. Shaikh

    (University of Chicago)

Abstract

This paper studies the wild bootstrap–based test proposed in Cameron, Gelbach, and Miller (2008). Existing analyses of its properties require that number of clusters is “large.” In an asymptotic framework in which the number of clusters is “small,” we provide conditions under which an unstudentized version of the test is valid. These conditions include homogeneity-like restrictions on the distribution of covariates. We further establish that a studentized version of the test may only overreject the null hypothesis by a “small” amount that decreases exponentially with the number of clusters. We obtain a qualitatively similar result for “score” bootstrap-based tests, which permit testing in nonlinear models.

Suggested Citation

  • Ivan A. Canay & Andres Santos & Azeem M. Shaikh, 2021. "The Wild Bootstrap with a “Small” Number of “Large” Clusters," The Review of Economics and Statistics, MIT Press, vol. 103(2), pages 346-363, May.
    • Handle: RePEc:tpr:restat:v:103:y:2021:i:2:p:346-363
    DOI: 10.1162/rest_a_00887
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    Cited by:

    1. Iyer, Sriya & Larcom, Shaun & Shah, Jaimin & She, Po-Wen, 2025. "Do religious people cope better in a crisis? Evidence from the UK pandemic lockdowns," Journal of Economic Behavior & Organization, Elsevier, vol. 237(C).
    2. Heckman, James & Pinto, Rodrigo & Shaikh, Azeem M., 2024. "Dealing with imperfect randomization: Inference for the highscope perry preschool program," Journal of Econometrics, Elsevier, vol. 243(1).
    3. MacKinnon, James G., 2023. "Fast cluster bootstrap methods for linear regression models," Econometrics and Statistics, Elsevier, vol. 26(C), pages 52-71.
    4. Ampofo, Akwasi & Cheng, Terence C. & Doko Tchatoka, Firmin, 2022. "Oil extraction and spillover effects into local labour market: Evidence from Ghana," Energy Economics, Elsevier, vol. 106(C).
    5. MacKinnon, James G. & Nielsen, Morten Ørregaard & Webb, Matthew D., 2023. "Cluster-robust inference: A guide to empirical practice," Journal of Econometrics, Elsevier, vol. 232(2), pages 272-299.
    6. Weiss, Amanda, 2024. "How Much Should We Trust Modern Difference-in-Differences Estimates?," OSF Preprints bqmws, Center for Open Science.
    7. James G. MacKinnon, 2025. "When can we trust cluster-robust inference?," Canadian Stata Users' Group Meetings 2025 11, Stata Users Group.
    8. Òscar Jordà & Alan M. Taylor, 2024. "Local Projections," NBER Working Papers 32822, National Bureau of Economic Research, Inc.
    9. Balmford, Ben & Collins, Joseph & Day, Brett & Lindsay, Luke & Peacock, James, 2023. "Pricing rules for PES auctions: Evidence from a natural experiment," Journal of Environmental Economics and Management, Elsevier, vol. 122(C).
    10. Kojevnikov, Denis & Song, Kyungchul, 2023. "Some impossibility results for inference with cluster dependence with large clusters," Journal of Econometrics, Elsevier, vol. 237(2).
    11. Atsushi Inoue & Òscar Jordà & Guido M Kuersteiner, 2026. "Inference for local projections," The Econometrics Journal, Royal Economic Society, vol. 29(1), pages 2-26.
    12. Hwang, Jungbin, 2021. "Simple and trustworthy cluster-robust GMM inference," Journal of Econometrics, Elsevier, vol. 222(2), pages 993-1023.
    13. François Bareille & Raja Chakir, 2024. "Structural identification of weather impacts on crop yields: Disentangling agronomic from adaptation effects," American Journal of Agricultural Economics, John Wiley & Sons, vol. 106(3), pages 989-1019, May.
    14. Felipe González & Luis R Martínez & Pablo Muñoz & Mounu Prem, 2024. "Higher Education and Mortality: Legacies of an Authoritarian College Contraction," Journal of the European Economic Association, European Economic Association, vol. 22(4), pages 1762-1797.
    15. James G. MacKinnon & Matthew D. Webb, 2020. "When and How to Deal with Clustered Errors in Regression Models," Working Paper 1421, Economics Department, Queen's University.
    16. Yong Cai, 2021. "A Modified Randomization Test for the Level of Clustering," Papers 2105.01008, arXiv.org, revised Jan 2022.
    17. repec:osf:osfxxx:bqmws_v1 is not listed on IDEAS
    18. Laurent Davezies & Xavier D'Haultfoeuille & Yannick Guyonvarch, 2018. "Asymptotic results under multiway clustering," Papers 1807.07925, arXiv.org, revised Aug 2018.
    19. Yong Cai, 2021. "Some Finite Sample Properties of the Sign Test," Papers 2103.01412, arXiv.org, revised Feb 2024.
    20. Li, Wenchao & Lian, Fangzhou & Zhong, Ninghua, 2025. "Highways to prosperity: Unlocking their potential through anti-poverty interventions," Journal of Economic Behavior & Organization, Elsevier, vol. 239(C).
    21. Chiara Berneri & Shaun Larcom & Congmin Peng & Po-Wen She, 2024. "The impact of law on moral and social norms: evidence from facemask fines in the UK," European Journal of Law and Economics, Springer, vol. 57(3), pages 311-346, June.
    22. Cai, Yong & Rafi, Ahnaf, 2024. "On the performance of the Neyman Allocation with small pilots," Journal of Econometrics, Elsevier, vol. 242(1).
    23. Yu Zheng & Honggang Fan, 2025. "Fast Cluster Bootstrap Methods for Spatial Error Models," Mathematics, MDPI, vol. 13(18), pages 1-16, September.
    24. James G. MacKinnon & Morten Ørregaard Nielsen & Matthew D. Webb, 2023. "Fast and reliable jackknife and bootstrap methods for cluster‐robust inference," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 38(5), pages 671-694, August.
    25. David M. Ritzwoller & Joseph P. Romano & Azeem M. Shaikh, 2024. "Randomization Inference: Theory and Applications," Papers 2406.09521, arXiv.org, revised Feb 2025.

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