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We consider the asymptotic properties of the Synthetic Control (SC) estimatorwhen both the number of pre-treatment periods and control units are large. Ifpotential outcomes follow a linear factor model, we provide conditions underwhich the factor loadings of the SC unit converge in probability to the factorloadings of the treated unit. This happens when there are weights diluted amongan increasing number of control units such that a weighted average of thefactor loadings of the control units asymptotically reconstructs the factorloadings of the treated unit. In this case, the SC estimator is asymptoticallyunbiased even when treatment assignment is correlated with time-varyingunobservables. This result can be valid even when the number of control unitsis larger than the number of pre-treatment periods.

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