Distributionally Robust Synthetic Control: Ensuring Robustness Against Highly Correlated Controls and Weight Shifts

Abstract: The synthetic control method estimates the causal effect by comparing the outcomes of a treated unit to a weighted average of control units that closely match the pre-treatment outcomes of the treated unit. This method presumes that the relationship between the potential outcomes of the treated and control units remains consistent before and after treatment. However, the estimator may become unreliable when these relationships shift or when control units are highly correlated. To address these challenges, we introduce the Distributionally Robust Synthetic Control (DRoSC) method by accommodating potential shifts in relationships and addressing high correlations among control units. The DRoSC method targets a new causal estimand defined as the optimizer of a worst-case optimization problem that checks through all possible synthetic weights that comply with the pre-treatment period. When the identification conditions for the classical synthetic control method hold, the DRoSC method targets the same causal effect as the synthetic control. When these conditions are violated, we show that this new causal estimand is a conservative proxy of the non-identifiable causal effect. We further show that the limiting distribution of the DRoSC estimator is non-normal and propose a novel inferential approach to characterize this non-normal limiting distribution. We demonstrate its finite-sample performance through numerical studies and an analysis of the economic impact of terrorism in the Basque Country.