Difference-in-Differences with a Continuous Treatment

TL;DR

This paper extends difference-in-differences methodology to continuous treatments, clarifying identification, addressing selection bias, and proposing new estimation methods with empirical validation.

Contribution

It introduces identification conditions for continuous treatments, discusses limitations of existing estimands, and proposes alternative estimation procedures.

Findings

Treatment-on-the-treated parameters are identifiable under parallel trends.

Multiple interpretations of two-way fixed effects estimands exist, with limitations.

New estimation procedures are demonstrated empirically.

Abstract

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.