A Short Note on Event-Study Synthetic Difference-in-Differences Estimators

Abstract: I propose an event study extension of Synthetic Difference-in-Differences (SDID) estimators. I show that, in simple and staggered adoption designs, estimators from Arkhangelsky et al. (2021) can be disaggregated into dynamic treatment effect estimators, comparing the lagged outcome differentials of treated and synthetic controls to their pre-treatment average. Estimators presented in this note can be computed using the sdid_event Stata package.

• The paper introduces a novel extension of SDID estimators for event-study analysis, accommodating staggered treatment adoption.
• It decomposes the SDID estimator into dynamic event-study components using optimal weights for accurate post-treatment effect evaluation.
• It provides a practical implementation via the sdid_event Stata package to enhance causal inference in longitudinal studies.

An Analysis of Event-Study Synthetic Difference-in-Differences Estimators

The paper explores an extension of the Synthetic Difference-in-Differences (SDID) estimators, originally introduced by Arkhangelsky et al. (2021), to accommodate event-study analysis. This adaptation is relevant in contexts where staggered adoption of treatments is observed, allowing for a dynamic treatment effect estimation across cohorts with differing treatment initiation periods. The author disaggregates SDID estimators into a set of dynamic estimators, which compare the outcome differentials between treated and synthetic control units, aligning these deviations with the pre-treatment averages.

Methodological Insights

In settings with balanced panel structures, the paper describes a procedure where the cohort-specific SDID estimator, $\hat{\tau}^{sdid}_a$, is decomposed into event-study estimators $\hat{\tau}^{sdid}_{a, \ell}$. These estimators assess the impact of treatment across different time periods, allowing researchers to observe the dynamic effects post-treatment. The fundamental precept is to derive post-treatment effects by leveraging optimal weights, $\lambda_t$ and $\omega_i$, that best approximate the pre-treatment outcome trajectories.

The formulation of these estimators specifies treatment effects as aggregated measures across all cohorts participating in each period post-adoption. As such, $\hat{\tau}^{sdid}_a$ represents the sample average of dynamic estimators for each cohort over its treatment duration. Similarly, the paper outlines the aggregation of these cohort-specific effects into overall event-study estimates, denoted as $\hat{\tau}^{sdid}_\ell$, capturing treatment impact at equivalent temporal follow-ups across different cohorts.

Implementation and Computational Utility

The proposed estimators can be computed using the sdid_event Stata package, available on GitHub, which facilitates estimation of both the Average Treatment Effect on the Treated (ATT) and time-specific treatment effects. This tool provides a streamlined approach to accessing disaggregated cohort-specific treatment effects alongside aggregate estimates, offering a practical resource for empirical researchers employing event-study designs.

Theoretical and Practical Implications

The extension of SDID to event-study contexts enhances the flexibility of causal inference methods in empirical research by addressing the complexities associated with staggered treatment adoption. The incorporation of dynamic treatment effects allows for more granular analyses, which are critical in settings where treatment impacts evolve over time.

Practically, this methodological advancement expands the toolkit for researchers conducting longitudinal studies in fields such as economics, political science, and public policy, where treatment effects may not manifest uniformly across units or time. The capacity to evaluate cohort-specific and aggregated treatment impacts can facilitate more nuanced policy evaluations and strengthen causal attributions in observational study designs.

Looking ahead, these methodological enhancements encourage further exploration into optimizing synthetic controls for dynamic contexts, potentially spurring developments in both software efficiency and theoretical foundations of SDID methods. This line of inquiry may yield robust procedures that further refine the credibility of causal estimates drawn from complex observational data structures.