Multiply robust difference-in-differences estimation of causal effect curves for continuous exposures

Researchers commonly use difference-in-differences (DiD) designs to evaluate public policy interventions. While methods exist for estimating effects in the context of binary interventions, policies often result in varied exposures across regions implementing the policy. Yet, existing approaches for incorporating continuous exposures face substantial limitations in addressing confounding variables associated with intervention status, exposure levels, and outcome trends. These limitations significantly constrain policymakers' ability to fully comprehend policy impacts and design future interventions. In this work, we propose new estimators for causal effect curves within the DiD framework, accounting for multiple sources of confounding. Our approach accommodates misspecification of a subset of intervention, exposure, and outcome models while avoiding any parametric assumptions on the effect curve. We present the statistical properties of the proposed methods and illustrate their application through simulations and a study investigating the heterogeneous effects of a nutritional excise tax under different levels of accessibility to cross-border shopping.