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Difference-in-Differences with Multiple Time Periods

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  • Brantly Callaway
  • Pedro H. C. Sant'Anna

Abstract

In this article, we consider identification, estimation, and inference procedures for treatment effect parameters using Difference-in-Differences (DiD) with (i) multiple time periods, (ii) variation in treatment timing, and (iii) when the "parallel trends assumption" holds potentially only after conditioning on observed covariates. We show that a family of causal effect parameters are identified in staggered DiD setups, even if differences in observed characteristics create non-parallel outcome dynamics between groups. Our identification results allow one to use outcome regression, inverse probability weighting, or doubly-robust estimands. We also propose different aggregation schemes that can be used to highlight treatment effect heterogeneity across different dimensions as well as to summarize the overall effect of participating in the treatment. We establish the asymptotic properties of the proposed estimators and prove the validity of a computationally convenient bootstrap procedure to conduct asymptotically valid simultaneous (instead of pointwise) inference. Finally, we illustrate the relevance of our proposed tools by analyzing the effect of the minimum wage on teen employment from 2001--2007. Open-source software is available for implementing the proposed methods.

Suggested Citation

  • Brantly Callaway & Pedro H. C. Sant'Anna, 2018. "Difference-in-Differences with Multiple Time Periods," Papers 1803.09015, arXiv.org, revised Dec 2020.
  • Handle: RePEc:arx:papers:1803.09015
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    References listed on IDEAS

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    3. Irene Botosaru & Federico H. Gutierrez, 2018. "Difference‐in‐differences when the treatment status is observed in only one period," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 33(1), pages 73-90, January.
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    9. Ekaterina Jardim & Mark C. Long & Robert Plotnick & Emma van Inwegen & Jacob Vigdor & Hilary Wething, 2017. "Minimum Wage Increases, Wages, and Low-Wage Employment: Evidence from Seattle," NBER Working Papers 23532, National Bureau of Economic Research, Inc.
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    11. Jonathan Meer & Jeremy West, 2016. "Effects of the Minimum Wage on Employment Dynamics," Journal of Human Resources, University of Wisconsin Press, vol. 51(2), pages 500-522.
    12. Brantly Callaway & Tong Li, 2019. "Quantile treatment effects in difference in differences models with panel data," Quantitative Economics, Econometric Society, vol. 10(4), pages 1579-1618, November.
    13. Dale Belman & Paul J. Wolfson, 2014. "What Does the Minimum Wage Do?," Books from Upjohn Press, W.E. Upjohn Institute for Employment Research, number wdmwd, December.
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    15. David H. Autor & William R. Kerr & Adriana D. Kugler, 2007. "Does Employment Protection Reduce Productivity? Evidence From US States," Economic Journal, Royal Economic Society, vol. 117(521), pages 189-217, June.
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