[Paper Review] Difference-in-Differences with a Continuous Treatment

Brantly Callaway, Andrew Goodman-Bacon+1 more|arXiv (Cornell University)|Jul 6, 2021

Optimal Experimental Design Methods235 citations

TL;DR

The paper develops identification, estimation, and interpretation tools for difference-in-differences with continuous or multi-valued treatments, showing TWFE can mislead and offering nonparametric estimators and meaningful summary measures, with an empirical illustration on Medicare reform.

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.

Motivation & Objective

• Identify how average level treatment effects and average causal responses can be identified under parallel trends with continuous treatments.
• Show limitations of standard TWFE specifications in continuous-treatment DiD settings.
• Develop nonparametric DiD estimators with favorable convergence and inference properties.
• Provide interpretable summary measures and event-study tools for continuous DiD analyses.
• Illustrate the methods with an application to Medicare reform and discuss interpretation challenges.

Proposed method

• Define level treatment effects (ATT(d|d)) and average treatment effects (ATE(d)) for continuous treatments.
• Define average causal responses (ACR(d)) and their derivatives (ACRT) to capture marginal dose effects.
• Introduce parallel-trends-based identification for ATT(d|d) and ATT^o under a two-period, continuous-treatment DiD framework.
• Demonstrate that TWFE regressions can yield biased or hard-to-interpret parameters due to heterogeneity and weighting issues.
• Propose nonparametric DiD estimators adapted from data-driven sup-norm estimation to recover ATT(d|d) and ACR(d) with strong parallel trends.
• Provide summary measures (ATT^o, ATE^o, ACR^o, ACRT^o) using dose-density weights and discuss event-study construction.

Experimental results

Research questions

• RQ1 How can ATT(d|d) and ATE(d) be identified when treatment is continuous or multi-valued discrete under parallel trends?
• RQ2 What are the limitations and interpretation issues of TWFE estimators in continuous-treatment DiD designs?
• RQ3 How can we nonparametrically estimate and conduct inference for the dose–response (ATT and ACR) functions?
• RQ4 What summary measures best capture treatment effects across the dose distribution, and how can they be constructed?

Key findings

• TWFE coefficients in continuous-treatment DiD can be decomposed into various building blocks and may assign negative weights, hindering clear causal interpretation.
• Under a generalized parallel trends assumption, ATT(d|d) and ATT^o are identified, but ATE(d) may not be without stronger assumptions.
• Comparisons across dose groups can be contaminated by selection bias unless strong parallel trends hold and treatment effect heterogeneity is limited.
• Nonparametric DiD estimators yield uniformly convergent estimates with narrow confidence bands and do not rely on rigid functional forms.
• Estimated effects in the Medicare reform application show larger TWFE-based estimates than some decompositions, highlighting interpretation differences.
• The paper provides practical summary measures (ATT^o, ATE^o, ACR^o, ACRT^o) and discusses event-study representations to assess parallel trends.