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Trading-off Bias and Variance in Stratified Experiments and in Staggered Adoption Designs, Under a Boundedness Condition on the Magnitude of the Treatment Effect

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  • Clément de Chaisemartin

    (ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique)

Abstract

I consider estimation of the average treatment effect (ATE), in a population composed of G groups, when one has unbiased and uncorrelated estimators of each group's conditional average treatment effect (CATE). These conditions are met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by B standard deviations of the outcome, for some known B. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This minimax-linear estimator assigns a weight equal to group g's share in the population to the most precisely estimated CATEs, and a weight proportional to one over the estimator's variance to the least precisely estimated CATEs. I also derive the minimax-linear estimator when the CATEs' estimators are positively correlated, a condition that may be met by differences-indifferences estimators in staggered adoption designs.

Suggested Citation

  • Clément de Chaisemartin, 2022. "Trading-off Bias and Variance in Stratified Experiments and in Staggered Adoption Designs, Under a Boundedness Condition on the Magnitude of the Treatment Effect," Working Papers hal-03873919, HAL.
    • Handle: RePEc:hal:wpaper:hal-03873919
    Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-03873919
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    References listed on IDEAS

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    1. Clément de Chaisemartin & Luc Behaghel, 2020. "Estimating the Effect of Treatments Allocated by Randomized Waiting Lists," Econometrica, Econometric Society, vol. 88(4), pages 1453-1477, July.
    2. Xinran Li & Peng Ding, 2017. "General Forms of Finite Population Central Limit Theorems with Applications to Causal Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1759-1769, October.
    3. Luc Behaghel & Clément de Chaisemartin & Marc Gurgand, 2017. "Ready for Boarding? The Effects of a Boarding School for Disadvantaged Students," American Economic Journal: Applied Economics, American Economic Association, vol. 9(1), pages 140-164, January.
    4. Sun, Liyang & Abraham, Sarah, 2021. "Estimating dynamic treatment effects in event studies with heterogeneous treatment effects," Journal of Econometrics, Elsevier, vol. 225(2), pages 175-199.
    5. Will Dobbie & Roland G. Fryer, 2011. "Are High-Quality Schools Enough to Increase Achievement among the Poor? Evidence from the Harlem Children's Zone," American Economic Journal: Applied Economics, American Economic Association, vol. 3(3), pages 158-187, July.
    6. Atila Abdulkadiroğlu & Joshua D. Angrist & Susan M. Dynarski & Thomas J. Kane & Parag A. Pathak, 2011. "Accountability and Flexibility in Public Schools: Evidence from Boston's Charters And Pilots," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 126(2), pages 699-748.
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    More about this item

    Keywords

    Bias-variance trade-off; Average treatment effect; Mean-squared error; Minimaxlinear estimator; Bounded normal mean model; Stratified randomized experiments; Differencesin-differences; Staggered adoption designs; Shrinkage;
    All these keywords.

    JEL classification:

    • 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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