On the Role of Covariates in the Synthetic Control Method

This note revisits the role of time-invariant observed covariates in the Synthetic Control (SC) method. We first derive conditions under which the original result of Abadie et al (2010) regarding the bias of the SC estimator remains valid when we relax the assumption of a perfect match on observed covariates and assume only a perfect match on pre-treatment outcomes. We then show that, even when the conditions for the first result are valid, a perfect match on pre-treatment outcomes does not generally imply an approximate match for all covariates. This will only be true for those that are both relevant and whose effects (over time) are not collinear with the effects of other observed and unobserved covariates. Taken together, our results show that a perfect match on covariates should not be required for the SC method, as long as there is a perfect match on a long set of pre-treatment outcomes.