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 consider the asymptotic properties of the SC estimator when the number of pre-treatment periods (T_0) goes to infinity. Differently from Abadie et al. (2010), we do not condition the analysis on a close-to-perfect pre-treatment fit, as the probability that this happens goes to zero when T_0 is large. We show that, even when a close-to-perfect fit is not achieved, the SC method can substantially improve relative to the difference-in-differences (DID) estimator, both in terms of bias and variance. However, we show that, in our setting, the SC estimator is asymptotically biased if treatment assignment is correlated with the unobserved heterogeneity. If common factors are stationary, then the asymptotic bias of the SC estimator 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. If a subset of the common factors is non-stationary, then the SC estimator can be asymptotically biased even conditional on a close-to-perfect fit. In this case, the identification assumption relies on orthogonality between treatment assignment and the stationary common factors. Finally, we also consider the statistical properties of the permutation tests suggested in Abadie et al. (2010).