Difference-in-Differences with Variation in Treatment Timing
The canonical difference-in-differences (DD) model contains two time periods, “pre” and “post”, and two groups, “treatment” and “control”. Most DD applications, however, exploit variation across groups of units that receive treatment at different times. This paper derives an expression for this general DD estimator, and shows that it is a weighted average of all possible two-group/two-period DD estimators in the data. This result provides detailed guidance about how to use regression DD in practice. I define the DD estimand and show how it averages treatment effect heterogeneity and that it is biased when effects change over time. I propose a new balance test derived from a unified definition of common trends. I show how to decompose the difference between two specifications, and I apply it to models that drop untreated units, weight, disaggregate time fixed effects, control for unit-specific time trends, or exploit a third difference.
I thank Michael Anderson, Martha Bailey, Marianne Bitler, Brantly Callaway, Kitt Carpenter, Eric Chyn, Bill Collins, John DiNardo, Andrew Dustan, Federico Gutierrez, Brian Kovak, Emily Lawler, Doug Miller, Sayeh Nikpay, Pedro Sant’Anna, Jesse Shapiro, Gary Solon, Isaac Sorkin, Sarah West, and seminar participants at the Southern Economics Association, ASHEcon 2018, the University of California, Davis, University of Kentucky, University of Memphis, University of North Carolina Charlotte, the University of Pennsylvania, and Vanderbilt University. All errors are my own. The views expressed herein are those of the author and do not necessarily reflect the views of the National Bureau of Economic Research.
Andrew Goodman-Bacon, 2021. "Difference-in-differences with variation in treatment timing," Journal of Econometrics, vol 225(2), pages 254-277.