Continuous difference-in-differences with double/debiased machine learning

Abstract

This paper extends difference-in-differences to settings with continuous treatments. Specifically, the average treatment effect on the treated (ATT) at any level of treatment intensity is identified under a conditional parallel trends assumption. Estimating the ATT in this framework requires first estimating infinite-dimensional nuisance parameters, particularly the conditional density of the continuous treatment, which can introduce substantial bias. To address this challenge, we propose estimators for the causal parameters under the double/debiased machine learning framework and establish their asymptotic normality. Additionally, we provide consistent variance estimators and construct uniform confidence bands based on a multiplier bootstrap procedure. To demonstrate the effectiveness of our approach, we revisit a previous study on the 1983 Medicare Prospective Payment System reform, reframing it as a DiD with continuous treatment and non-parametrically estimating its effects.