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Finite-Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness

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  • Timothy B. Armstrong
  • Michal Koles'ar

Abstract

We consider estimation and inference on average treatment effects under unconfoundedness conditional on the realizations of the treatment variable and covariates. Given nonparametric smoothness and/or shape restrictions on the conditional mean of the outcome variable, we derive estimators and confidence intervals (CIs) that are optimal in finite samples when the regression errors are normal with known variance. In contrast to conventional CIs, our CIs use a larger critical value that explicitly takes into account the potential bias of the estimator. When the error distribution is unknown, feasible versions of our CIs are valid asymptotically, even when $\sqrt{n}$-inference is not possible due to lack of overlap, or low smoothness of the conditional mean. We also derive the minimum smoothness conditions on the conditional mean that are necessary for $\sqrt{n}$-inference. When the conditional mean is restricted to be Lipschitz with a large enough bound on the Lipschitz constant, the optimal estimator reduces to a matching estimator with the number of matches set to one. We illustrate our methods in an application to the National Supported Work Demonstration.

Suggested Citation

  • Timothy B. Armstrong & Michal Koles'ar, 2017. "Finite-Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness," Papers 1712.04594, arXiv.org, revised Jan 2021.
  • Handle: RePEc:arx:papers:1712.04594
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    Cited by:

    1. Zichen Deng & Maarten Lindeboom, 2022. "Early‐life famine exposure, hunger recall, and later‐life health," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(4), pages 771-787, June.
    2. Zhexiao Lin & Peng Ding & Fang Han, 2023. "Estimation Based on Nearest Neighbor Matching: From Density Ratio to Average Treatment Effect," Econometrica, Econometric Society, vol. 91(6), pages 2187-2217, November.
    3. Cl'ement de Chaisemartin, 2021. "Trading-off Bias and Variance in Stratified Experiments and in Matching Studies, Under a Boundedness Condition on the Magnitude of the Treatment Effect," Papers 2105.08766, arXiv.org, revised Jan 2024.
    4. Dmitry Arkhangelsky & David Hirshberg, 2023. "Large-Sample Properties of the Synthetic Control Method under Selection on Unobservables," Papers 2311.13575, arXiv.org, revised Dec 2023.
    5. Timothy B. Armstrong & Michal Koles'ar & Soonwoo Kwon, 2020. "Bias-Aware Inference in Regularized Regression Models," Papers 2012.14823, arXiv.org, revised Aug 2023.
    6. Susan Athey & Stefan Wager, 2021. "Policy Learning With Observational Data," Econometrica, Econometric Society, vol. 89(1), pages 133-161, January.
    7. Pengzhou Wu & Kenji Fukumizu, 2021. "$\beta$-Intact-VAE: Identifying and Estimating Causal Effects under Limited Overlap," Papers 2110.05225, arXiv.org.
    8. D’Amour, Alexander & Ding, Peng & Feller, Avi & Lei, Lihua & Sekhon, Jasjeet, 2021. "Overlap in observational studies with high-dimensional covariates," Journal of Econometrics, Elsevier, vol. 221(2), pages 644-654.
    9. Timothy B. Armstrong & Michal Kolesár, 2021. "Sensitivity analysis using approximate moment condition models," Quantitative Economics, Econometric Society, vol. 12(1), pages 77-108, January.
    10. Deng, Zichen & Lindeboom, Maarten, 2021. "Early-Life Famine Exposure, Hunger Recall and Later-Life Health," IZA Discussion Papers 14487, Institute of Labor Economics (IZA).
    11. Ferman, Bruno, 2021. "Matching estimators with few treated and many control observations," Journal of Econometrics, Elsevier, vol. 225(2), pages 295-307.
    12. Sokbae Lee & Martin Weidner, 2021. "Bounding Treatment Effects by Pooling Limited Information across Observations," Papers 2111.05243, arXiv.org, revised Dec 2023.
    13. Laurent Davezies & Xavier D'Haultfoeuille & Louise Laage, 2021. "Identification and Estimation of Average Marginal Effects in Fixed Effects Logit Models," Papers 2105.00879, arXiv.org, revised Oct 2022.
    14. Huiming Zhang & Haoyu Wei & Guang Cheng, 2023. "Tight Non-asymptotic Inference via Sub-Gaussian Intrinsic Moment Norm," Papers 2303.07287, arXiv.org, revised Jan 2024.
    15. Kohei Yata, 2021. "Optimal Decision Rules Under Partial Identification," Papers 2111.04926, arXiv.org, revised Aug 2023.
    16. Ganesh Karapakula, 2023. "Stable Probability Weighting: Large-Sample and Finite-Sample Estimation and Inference Methods for Heterogeneous Causal Effects of Multivalued Treatments Under Limited Overlap," Papers 2301.05703, arXiv.org, revised Jan 2023.
    17. Zichen Deng & Maarten Lindeboom, 2021. "Early-life Famine Exposure, Hunger Recall and Later-life Health," Tinbergen Institute Discussion Papers 21-054/V, Tinbergen Institute.
    18. Zichen Deng & Maarten Lindeboom, 2021. "Early-life Famine Exposure, Hunger Recall and Later-life Health," Papers 2021-04, Centre for Health Economics, Monash University.
    19. Christoph Breunig & Ruixuan Liu & Zhengfei Yu, 2022. "Double Robust Bayesian Inference on Average Treatment Effects," Papers 2211.16298, arXiv.org, revised Feb 2024.
    20. Max Cytrynbaum, 2021. "Optimal Stratification of Survey Experiments," Papers 2111.08157, arXiv.org, revised Aug 2023.

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    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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