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Difference-in-Differences with a Continuous Treatment

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  • Brantly Callaway
  • Andrew Goodman-Bacon
  • Pedro H. C. Sant'Anna

Abstract

This paper analyzes difference-in-differences designs with a continuous treatment. We show that treatment effect on the treated-type parameters can be identified under a generalized parallel trends assumption that is similar to the binary treatment setup. However, interpreting differences in these parameters across different values of the treatment can be particularly challenging due to treatment effect heterogeneity. We discuss alternative, typically stronger, assumptions that alleviate these challenges. We also provide a variety of treatment effect decomposition results, highlighting that parameters associated with popular linear two-way fixed-effect (TWFE) specifications can be hard to interpret, \emph{even} when there are only two time periods. We introduce alternative estimation procedures that do not suffer from these TWFE drawbacks, and show in an application that they can lead to different conclusions.

Suggested Citation

  • Brantly Callaway & Andrew Goodman-Bacon & Pedro H. C. Sant'Anna, 2021. "Difference-in-Differences with a Continuous Treatment," Papers 2107.02637, arXiv.org, revised Jan 2024.
    • Handle: RePEc:arx:papers:2107.02637
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    References listed on IDEAS

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    8. Michelle Marcus & Pedro H. C. Sant’Anna, 2021. "The Role of Parallel Trends in Event Study Settings: An Application to Environmental Economics," Journal of the Association of Environmental and Resource Economists, University of Chicago Press, vol. 8(2), pages 235-275.
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    More about this item

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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