Design-based Analysis in Difference-In-Differences Settings with Staggered Adoption

Abstract: In this paper we study estimation of and inference for average treatment effects in a setting with panel data. We focus on the setting where units, e.g., individuals, firms, or states, adopt the policy or treatment of interest at a particular point in time, and then remain exposed to this treatment at all times afterwards. We take a design perspective where we investigate the properties of estimators and procedures given assumptions on the assignment process. We show that under random assignment of the adoption date the standard Difference-In-Differences estimator is is an unbiased estimator of a particular weighted average causal effect. We characterize the proeperties of this estimand, and show that the standard variance estimator is conservative.

• The paper presents a design-based approach that uses random assignment of adoption dates to yield an unbiased DID estimator for average treatment effects.
• It derives an exact variance of the DID estimator that improves upon traditional methods, offering less conservative inference.
• The methodology is pivotal for policy evaluation, especially in settings with staggered interventions and heterogeneous treatment effects.

Design-Based Analysis in Difference-In-Differences Settings with Staggered Adoption

This study addresses the challenge of estimating and inferring average treatment effects in scenarios characterized by staggered adoption using panel data. A staggered adoption design (SAD) is analyzed where units, such as individuals, firms, or states, adopt a policy at various points in time but do not revert to a non-treatment status.

Key Contributions

1. Framework and Assumptions: The authors approach the problem with a focus on design-based inference, contrasting the traditional sampling-based perspective. The core assumptions involve the random assignment of adoption dates and exclusion restrictions that facilitate the simplification of potential outcomes into binary treatment scenarios.
2. Estimation Methodology: They establish that the standard Difference-In-Differences (DID) estimator is unbiased for a particular weighted average causal effect under the assumption of random assignment of adoption dates. The study rigorously characterizes the properties of this estimand.
3. Variance Estimation: A significant finding is that the standard variance estimator commonly employed in DID settings is conservative. The paper derives the exact variance of the DID estimator within the context of the assumed design, highlighting improvements over traditional methods like the Liang-Zeger variance estimator and the clustered bootstrap.

Numerical Results

• The study conducts simulations incorporating various adoption distributions and designs for potential outcome distributions. Findings indicate that the proposed estimator holds promise in providing more accurate and less conservative variance estimates than existing methodologies, such as the Liang-Zeger method.

Implications and Future Directions

• Policy Evaluation: This research introduces robust tools for evaluating staggered policy interventions with enhanced statistical properties. These results are particularly relevant in policy scenarios where staggered implementation is unavoidable, such as education reforms or healthcare interventions.
• Inference Improvements: Future advancements could explore extensions to non-linear models or settings with interactive fixed effects, expanding the utility of this robust framework.
• Broader Application: The methodology could be adapted for more complex settings, such as those involving heterogeneous treatment effects or spillover effects across units, pushing the boundaries of causal inference in observational studies.

This paper contributes to the econometrics literature by refining the approach to staggered adoption settings, offering improved interpretative and inferential techniques that bolster the credibility of causal claims in empirical research.