  We consider the asymptotic properties of the Synthetic Control (SC) estimator
when both the number of pre-treatment periods and control units are large. If
potential outcomes follow a linear factor model, we provide conditions under
which the factor loadings of the SC unit converge in probability to the factor
loadings of the treated unit. This happens when there are weights diluted among
an increasing number of control units such that a weighted average of the
factor loadings of the control units asymptotically reconstructs the factor
loadings of the treated unit. In this case, the SC estimator is asymptotically
unbiased even when treatment assignment is correlated with time-varying
unobservables. This result can be valid even when the number of control units
is larger than the number of pre-treatment periods.
