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Difference-in-Differences with a Continuous Treatment
(Forthcoming Article) - This paper analyzes difference-in-differences designs with a continuous treatment. We show that
treatment-on-the-treated-type parameters are identified under a parallel trends assumption analogous
to the binary treatment case. However, comparing these parameters across treatments is
challenging because parallel trends does not rule out selection bias. We discuss alternative, typically
stronger, assumptions that eliminate selection bias. We further show that popular two-way
fixed effects estimands admit multiple interpretations, depending on the underlying causal building
block, all having important limitations as meaningful summaries of treatment effects. Finally,
we introduce alternative estimation procedures that avoid these drawbacks and demonstrate them
in an empirical application.
