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Placebo inference on treatment effects when the number of clusters is small

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  • Hagemann, Andreas
Abstract
I introduce a general, Fisher-style randomization testing framework to conduct nearly exact inference about the lack of effect of a binary treatment in the presence of very few, large clusters when the treatment effect is identified across clusters. The proposed randomization test formalizes and extends the intuitive notion of generating null distributions by assigning placebo treatments to untreated clusters. I show that under simple and easily verifiable conditions, the placebo test leads to asymptotically valid inference in a very large class of empirically relevant models. Examples discussed explicitly are (i) least squares regression with cluster-level treatment, (ii) difference-in-differences estimation, and (iii) binary choice models with cluster-level treatment. A simulation study and an empirical example are provided. The proposed inference procedure is easy to implement and performs well with as few as three treated and three untreated clusters.

Suggested Citation

  • Hagemann, Andreas, 2019. "Placebo inference on treatment effects when the number of clusters is small," Journal of Econometrics, Elsevier, vol. 213(1), pages 190-209.
    • Handle: RePEc:eee:econom:v:213:y:2019:i:1:p:190-209
    DOI: 10.1016/j.jeconom.2019.04.011
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    References listed on IDEAS

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    Cited by:

    1. MacKinnon, James G. & Webb, Matthew D., 2020. "Randomization inference for difference-in-differences with few treated clusters," Journal of Econometrics, Elsevier, vol. 218(2), pages 435-450.
    2. Hao Wang & Tao Zhang & Xi Wang, 2024. "High-speed railways reduces carbon emissions: mediating effects of green innovation and the resilience of environmental investment," Palgrave Communications, Palgrave Macmillan, vol. 11(1), pages 1-24, December.
    3. Kojevnikov, Denis & Song, Kyungchul, 2023. "Some impossibility results for inference with cluster dependence with large clusters," Other publications TiSEM 80b8e4ed-54bc-4a34-883f-f, Tilburg University, School of Economics and Management.
    4. Kevin Lang & Kaiwen Leong & Huailu Li & Haibo Xu, 2019. "Lending to the Unbanked: Relational Contracting with Loan Sharks," NBER Working Papers 26400, National Bureau of Economic Research, Inc.
    5. Rik Chakraborti & Gavin Roberts, 2023. "How price-gouging regulation undermined COVID-19 mitigation: county-level evidence of unintended consequences," Public Choice, Springer, vol. 196(1), pages 51-83, July.
    6. Wang, Wenjie & Zhang, Yichong, 2024. "Wild bootstrap inference for instrumental variables regressions with weak and few clusters," Journal of Econometrics, Elsevier, vol. 241(1).
    7. 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.
    8. Kojevnikov, Denis & Song, Kyungchul, 2023. "Some impossibility results for inference with cluster dependence with large clusters," Journal of Econometrics, Elsevier, vol. 237(2).
    9. Hwang, Jungbin, 2021. "Simple and trustworthy cluster-robust GMM inference," Journal of Econometrics, Elsevier, vol. 222(2), pages 993-1023.
    10. Kevin Lang, 2020. "Effort and wages: Evidence from the payroll tax," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 53(1), pages 108-139, February.
    11. 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.
    12. Felipe Lozano‐Rojas & Patrick Carlin, 2022. "The effect of soda taxes beyond beverages in Philadelphia," Health Economics, John Wiley & Sons, Ltd., vol. 31(11), pages 2381-2410, November.
    13. Bruno Ferman, 2023. "Inference in difference‐in‐differences: How much should we trust in independent clusters?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 38(3), pages 358-369, April.
    14. Yingfei Mu & Edward A. Rubin & Eric Zou, 2021. "What’s Missing in Environmental (Self-)Monitoring: Evidence from Strategic Shutdowns of Pollution Monitors," NBER Working Papers 28735, National Bureau of Economic Research, Inc.
    15. Andreas Hagemann, 2019. "Permutation inference with a finite number of heterogeneous clusters," Papers 1907.01049, arXiv.org, revised Feb 2023.
    16. Brantly Callaway, 2022. "Difference-in-Differences for Policy Evaluation," Papers 2203.15646, arXiv.org.
    17. Dao, Chi Danh & Fenig, Guidon & Sator, Georg & Yoon, Jin Young, 2024. "Assessing Robustness to Varying Clustering Methods and Samples in Ambuehl, Bernheim, and Lusardi (2022): Replication and Sensitivity Analysis," I4R Discussion Paper Series 110, The Institute for Replication (I4R).

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    More about this item

    Keywords

    Cluster-robust inference; Randomization; Permutation;
    All these keywords.

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models

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