[Paper Review] Difference-in-Differences Estimators for Treatments Continuously Distributed at Every Period

Clément de Chaisemartin, Xavier D’Haultfœuille+3 more|arXiv (Cornell University)|Jan 18, 2022

Pharmaceutical Economics and Policy4 citations

TL;DR

This paper introduces novel difference-in-differences estimators for continuous treatments in panel data, distinguishing between switchers (units whose treatment changes) and stayers (whose treatment is constant). It proposes two estimands—Average of Switchers' Slopes (AOSS) and Weighted Average of Switchers' Slopes (WAOSS)—under a parallel trends assumption, showing that WAOSS offers superior estimation efficiency and robustness, including doubly-robust estimation, despite AOSS being more intuitive for policy inference.

ABSTRACT

When one studies the effects of taxes, tariffs, or prices using panel data, the treatment is often continuously distributed in every period. We propose difference-in-differences (DID) estimators for such cases. We assume that between consecutive periods, the treatment of some units, the switchers, changes, while the treatment of other units, the stayers, remains constant. We show that under a parallel-trends assumption, the slopes of switchers' potential outcomes are nonparametrically identified by difference-in-differences estimands comparing the outcome evolutions of switchers and stayers with the same baseline treatment. Controlling for the baseline treatment ensures that our estimands remain valid if the treatment's effect changes over time. We consider two weighted averages of switchers' slopes, and discuss their respective advantages. For each weighted average, we propose a doubly-robust, nonparametric, and $\sqrt{n}$-consistent estimator. We generalize our results to the instrumental-variable case. We apply our method to estimate the price-elasticity of gasoline consumption.

Motivation & Objective

• To address the limitations of two-way fixed effects (TWFE) regressions in estimating treatment effects under heterogeneous effects, especially with continuous treatments.
• To develop a difference-in-differences framework that accommodates continuously distributed treatments across time periods, not just binary or discrete changes.
• To define and estimate two interpretable causal parameters: the Average of Switchers' Slopes (AOSS) and the Weighted Average of Switchers' Slopes (WAOSS), under a parallel trends assumption.
• To demonstrate the superiority of WAOSS over AOSS in terms of estimation efficiency, asymptotic variance, and compatibility with doubly-robust estimation.
• To apply the proposed estimators to estimate the price-elasticity of gasoline consumption using US state-level panel data.

Proposed method

• Propose a parallel trends assumption comparing outcome trends of switchers and stayers with the same initial treatment level, under a counterfactual where switchers' treatment does not change.
• Define the AOSS as the unweighted average slope of switchers’ potential outcome functions from period one to two treatment levels.
• Define the WAOSS as a weighted average of switchers’ slopes, with weights proportional to the absolute value of their treatment change.
• Show that under shape restrictions (e.g., convexity or concavity), AOSS can inform the effect of unobserved treatment changes, while WAOSS supports cost-benefit analysis.
• Establish that WAOSS can be estimated at the parametric rate even when treatment changes are arbitrarily small, unlike AOSS.
• Extend the framework to instrumental variables (IV) by proposing an IV-WAOSS estimator, which is doubly robust and achieves lower asymptotic variance than AOSS.

Experimental results

Research questions

• RQ1 Can difference-in-differences estimators be generalized to continuous treatments while maintaining causal interpretability under heterogeneous effects?
• RQ2 How do the AOSS and WAOSS estimands differ in economic interpretation, and when should each be preferred?
• RQ3 Does the WAOSS estimator achieve better estimation efficiency and robustness than the AOSS estimator, particularly in settings with small or continuous treatment changes?
• RQ4 Can the WAOSS estimator be extended to instrumental variable settings, and does it maintain favorable properties like double robustness?
• RQ5 What is the empirical performance of the WAOSS estimator in estimating the price-elasticity of gasoline consumption using real-world panel data?

Key findings

• The WAOSS estimator can be estimated at the parametric rate even when treatment changes are arbitrarily small, a property not shared by the AOSS estimator.
• Under certain conditions, the asymptotic variance of the WAOSS estimator is strictly smaller than that of the AOSS estimator, implying greater precision.
• The WAOSS estimator is amenable to doubly-robust estimation, while the AOSS estimator is not, enhancing its robustness to model misspecification.
• In an empirical application to gasoline consumption, the standard error of the WAOSS estimator was nearly three times smaller than that of the AOSS estimator.
• The IV-WAOSS estimate of the price-elasticity of gasoline consumption was -0.726 (95% CI: [-1.349, -0.328]), while the 2SLS-TWFE estimate was -1.084, with a 27% wider confidence interval, indicating that robustness does not always come at a precision cost.
• The confidence intervals for the IV-WAOSS and 2SLS-TWFE estimators overlapped, and a bootstrap-based equality test found no significant difference between them, despite the WAOSS estimator being more efficient.