Large-Sample Properties of the Synthetic Control Method under Selection on Unobservables

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We analyze the properties of the synthetic control (SC) method in settings with a large number of units. We assume that the selection into treatment is based on unobserved permanent heterogeneity and pretreatment information, thus allowing for both strictly and sequentially exogenous assignment processes. Exploiting duality, we interpret the solution of the SC optimization problem as an estimator for the underlying treatment probabilities. We use this to derive the asymptotic representation for the SC method and characterize sufficient conditions for its asymptotic normality. We show that the critical property that determines the behavior of the SC method is the ability of input features to approximate the unobserved heterogeneity. Our results imply that the SC method delivers asymptotically normal estimators for a large class of linear panel data models as long as the number of pretreatment periods is large, making it a natural alternative to conventional methods built on the Difference-in-Differences.