Andrew Goodman-Bacon, Federal Reserve Bank of Minneapolis

Difference-in-Differences with a Continuous Treatment

Abstract

This paper analyzes difference-in-differences setups 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 two-way fixed-effect specifications can be hard to interpret, even when there are only two time periods. We introduce alternative estimation strategies that do not suffer from these drawbacks. Our results also cover cases where (i) there is no available untreated comparison group and (ii) there are multiple periods and variation in treatment timing, which are both common in empirical work.

Andrew Goodman-Bacon is an economist at the Opportunity and Inclusive Growth Institute at the Federal Reserve Bank of Minneapolis. His research is in economic history, health economics, public economics, and applied econometrics.

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CEBI contact: Miriam Wüst (Joint with Danmarks Nationalbank)