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The Wild Bootstrap with a Small Number of Large Clusters

Author

Listed:
  • Ivan A. Canay

    (Institute for Fiscal Studies and Northwestern University)

  • Andres Santos

    (Institute for Fiscal Studies and UC San Diego)

  • Azeem M. Shaikh

    (Institute for Fiscal Studies and University of Chicago)

Abstract
This paper studies the properties of the wild bootstrap-based test proposed in Cameron et al. (2008) for testing hypotheses about the coefficients in a linear regression model with clustered data. Cameron et al. (2008) provide simulations that suggest this test works well even in settings with as few as fi ve clusters, but existing theoretical analyses of its properties all rely on an asymptotic framework in which the number of clusters is "large." In contrast to these analyses, we employ an asymptotic framework in which the number of clusters is "small," but the number of observations per cluster is "large." In this framework, we provide conditions under which an unstudentized version of the test is valid in the sense that it has limiting rejection probability under the null hypothesis that does not exceed the nominal level. Importantly, these conditions require, among other things, certain homogeneity restrictions on the distribution of covariates. In contrast, we establish that a studentized version of the test may only over-reject the null hypothesis by a "small" amount in the sense that it has limiting rejection probability under the null hypothesis that does not exceed the nominal level by more than an amount that decreases exponentially with the number of clusters. We obtain results qualitatively similar to those for the studentized version of the test for closely related "score" bootstrap-based tests, which permit testing hypotheses about parameters in nonlinear models. We illustrate the relevance of our theoretical or applied work via a simulation study and empirical application.

Suggested Citation

  • Ivan A. Canay & Andres Santos & Azeem M. Shaikh, 2019. "The Wild Bootstrap with a Small Number of Large Clusters," CeMMAP working papers CWP40/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:40/19
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    References listed on IDEAS

    as
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    Cited by:

    1. 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).
    2. Hwang, Jungbin, 2021. "Simple and trustworthy cluster-robust GMM inference," Journal of Econometrics, Elsevier, vol. 222(2), pages 993-1023.
    3. 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.
    4. Yong Cai, 2021. "A Modified Randomization Test for the Level of Clustering," Papers 2105.01008, arXiv.org, revised Jan 2022.
    5. Laurent Davezies & Xavier D'Haultfoeuille & Yannick Guyonvarch, 2018. "Asymptotic results under multiway clustering," Papers 1807.07925, arXiv.org, revised Aug 2018.
    6. Yong Cai, 2021. "Some Finite Sample Properties of the Sign Test," Papers 2103.01412, arXiv.org, revised Feb 2024.

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