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On Estimating Multiple Treatment Effects with Regression


  • Paul Goldsmith-Pinkham

    (Yale University)

  • Peter Hull

    (University of Chicago)

  • Michal Kolesár

    (Princeton University)


We study the causal interpretation of regressions on multiple dependent treatments and flexible controls. Such regressions are often used to analyze randomized control trials with multiple intervention arms, and to estimate institutional quality (e.g. teacher value-added) with observational data. We show that, unlike with a single binary treatment, these regressions do not generally estimate convex averages of causal effects—even when the treatments are conditionally randomly assigned and the controls fully address omitted variables bias. We discuss different solutions to this issue, and propose as a solution a new class of efficient estimators of weighted average treatment effects.

Suggested Citation

  • Paul Goldsmith-Pinkham & Peter Hull & Michal Kolesár, 2021. "On Estimating Multiple Treatment Effects with Regression," Working Papers 2021-41, Princeton University. Economics Department..
  • Handle: RePEc:pri:econom:2021-41

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    References listed on IDEAS

    1. Richard K. Crump & V. Joseph Hotz & Guido W. Imbens & Oscar A. Mitnik, 2009. "Dealing with limited overlap in estimation of average treatment effects," Biometrika, Biometrika Trust, vol. 96(1), pages 187-199.
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    3. Alan B. Krueger, 1999. "Experimental Estimates of Education Production Functions," The Quarterly Journal of Economics, Oxford University Press, vol. 114(2), pages 497-532.
    4. David Arnold & Will Dobbie & Peter Hull, 2022. "Measuring Racial Discrimination in Bail Decisions," American Economic Review, American Economic Association, vol. 112(9), pages 2992-3038, September.
    5. Clément de Chaisemartin & Xavier D'Haultfœuille, 2020. "Two-Way Fixed Effects Estimators with Heterogeneous Treatment Effects," American Economic Review, American Economic Association, vol. 110(9), pages 2964-2996, September.
    6. Angrist, Joshua D. & Krueger, Alan B., 1999. "Empirical strategies in labor economics," Handbook of Labor Economics, in: O. Ashenfelter & D. Card (ed.), Handbook of Labor Economics, edition 1, volume 3, chapter 23, pages 1277-1366, Elsevier.
    7. Thomas J. Kane & Douglas O. Staiger, 2008. "Estimating Teacher Impacts on Student Achievement: An Experimental Evaluation," NBER Working Papers 14607, National Bureau of Economic Research, Inc.
    8. Cattaneo, Matias D., 2010. "Efficient semiparametric estimation of multi-valued treatment effects under ignorability," Journal of Econometrics, Elsevier, vol. 155(2), pages 138-154, April.
    9. Chetty, Nadarajan & Friedman, John & Rockoff, Jonah E., 2014. "Measuring the Impacts of Teachers I: Evaluating Bias in Teacher Value-Added Estimates," Scholarly Articles 30749073, Harvard University Department of Economics.
    10. Raj Chetty & John N. Friedman & Jonah E. Rockoff, 2014. "Measuring the Impacts of Teachers I: Evaluating Bias in Teacher Value-Added Estimates," American Economic Review, American Economic Association, vol. 104(9), pages 2593-2632, September.
    11. Peter Hull, 2018. "Estimating Treatment Effects in Mover Designs," Papers 1804.06721,
    12. Joshua D. Angrist & Jörn-Steffen Pischke, 2009. "Mostly Harmless Econometrics: An Empiricist's Companion," Economics Books, Princeton University Press, edition 1, number 8769.
    13. Imbens, Guido W & Angrist, Joshua D, 1994. "Identification and Estimation of Local Average Treatment Effects," Econometrica, Econometric Society, vol. 62(2), pages 467-475, March.
    14. Nicole Maestas & Kathleen J. Mullen & Alexander Strand, 2013. "Does Disability Insurance Receipt Discourage Work? Using Examiner Assignment to Estimate Causal Effects of SSDI Receipt," American Economic Review, American Economic Association, vol. 103(5), pages 1797-1829, August.
    15. Jason Abaluck & Mauricio Caceres Bravo & Peter Hull: & Amanda Starc, 2021. "Mortality Effects and Choice Across Private Health Insurance Plans," The Quarterly Journal of Economics, Oxford University Press, vol. 136(3), pages 1557-1610.
    16. Matias D. Cattaneo, 2010. "multi-valued treatment effects," The New Palgrave Dictionary of Economics,, Palgrave Macmillan.
    17. Patrick Kline & Christopher R. Walters, 2019. "On Heckits, LATE, and Numerical Equivalence," Econometrica, Econometric Society, vol. 87(2), pages 677-696, March.
    18. Jeffrey R. Kling, 2006. "Incarceration Length, Employment, and Earnings," American Economic Review, American Economic Association, vol. 96(3), pages 863-876, June.
    19. Joshua D. Angrist & Peter D. Hull & Parag A. Pathak & Christopher R. Walters, 2017. "Leveraging Lotteries for School Value-Added: Testing and Estimation," The Quarterly Journal of Economics, Oxford University Press, vol. 132(2), pages 871-919.
    20. Joshua D. Angrist, 1998. "Estimating the Labor Market Impact of Voluntary Military Service Using Social Security Data on Military Applicants," Econometrica, Econometric Society, vol. 66(2), pages 249-288, March.
    21. Callaway, Brantly & Sant’Anna, Pedro H.C., 2021. "Difference-in-Differences with multiple time periods," Journal of Econometrics, Elsevier, vol. 225(2), pages 200-230.
    22. Victor Chernozhukov & Whitney K. Newey & Rahul Singh, 2022. "Automatic Debiased Machine Learning of Causal and Structural Effects," Econometrica, Econometric Society, vol. 90(3), pages 967-1027, May.
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    Cited by:

    1. Goller, Daniel & Diem, Andrea & Wolter, Stefan C., 2023. "Sitting next to a dropout: Academic success of students with more educated peers," Economics of Education Review, Elsevier, vol. 93(C).
    2. de Chaisemartin, Clément & D’Haultfœuille, Xavier, 2023. "Two-way fixed effects and differences-in-differences estimators with several treatments," Journal of Econometrics, Elsevier, vol. 236(2).
    3. Clément de Chaisemartin & Xavier D’haultfœuille, 2023. "Two-way fixed effects and differences-in-differences estimators with several treatments," Post-Print hal-04187969, HAL.
    4. Callaway, Brantly & Li, Tong, 2023. "Policy evaluation during a pandemic," Journal of Econometrics, Elsevier, vol. 236(1).
    5. Manudeep Bhuller & Henrik Sigstad, 2022. "2SLS with Multiple Treatments," Papers 2205.07836,, revised Oct 2023.

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    More about this item


    regressions; treatment effect;

    JEL classification:

    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General

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