Asymptotic Properties of the Distributional Synthetic Controls

This paper enhances our comprehension of the Distributional Synthetic Control (DSC) proposed by Gunsilius (2023), focusing on its asymptotic properties. We first establish the DSC estimator's asymptotic optimality. The essence of this optimality lies in the treatment effect estimator given by DSC achieves the lowest possible squared prediction error among all potential treatment effect estimators that depend on an average of quantiles of control units. We also establish the convergence of the DSC weights when some requirements are met, as well as the convergence rate. A significant aspect of our research is that we find DSC synthesis forms an optimal weighted average, particularly in situations where it is impractical to perfectly fit the treated unit's quantiles through the weighted average of the control units' quantiles. To corroborate our theoretical insights, we provide empirical evidence derived from simulations.