Revisiting the Synthetic Control Estimator

The synthetic control (SC) method has been recently proposed as an alternative to estimate treatment effects in comparative case studies. In this paper, we revisit the SC method in a linear factor model setting and derive conditions under which the SC estimator is asymptotically unbiased when the number of pre-treatment periods goes to infinity. If the pre-treatment averages of the first and second moments of the common factors converge, then we show that the SC estimator is asymptotically biased if there is selection on unobservables. In this case, the bias goes to zero when the variance of the transitory shocks is small, which is also the case in which it is more likely that the pre-treatment fit will be good. In models with non-stationary common factors, however, we show that the asymptotic bias may not go to zero even when the pre-treatment fit is almost perfect. Finally, we show that a demeaned version of the SC estimator can substantially improve relative to the difference-in-differences (DID) estimator, both in terms of bias and variance. Overall, our results show that the SC method can substantially improve relative to the DID estimator. However, researchers should be more careful in interpreting the identification assumptions required for this method.