On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls

We consider the asymptotic properties of the synthetic control (SC) estimator when both the number of pretreatment periods and control units are large. If potential outcomes follow a linear factor model, we provide conditions under which the SC unit asymptotically recovers the factor structure of the treated unit, even when the pretreatment fit is imperfect. This happens when there are weights diluted among an increasing number of control units such that a weighted average of the factor structure of the control units asymptotically reconstructs the factor structure of the treated unit. In this case, the SC estimator is asymptotically unbiased even when treatment assignment is correlated with time-varying unobservables. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.