Identification of Treatment Effects from Synthetic Controls

Abstract

This paper considers the Generalized Synthetic Control method of Xu (2015). It presents a characterization of the identified set for this model. In addition it presents a version in which the matrix factorization is unique without strong restrictions on the matrix factors. The approach assumes a panel data set represented by a linear factor model with no structural breaks other than the treatment for the treated unit(s). The method calculates the unit specific treatment effect using matrix factorization of the pre-treatment panel data. The paper numerically determines the identified set of the average treatment on the treated. Statistical uncertainty in the size of the treatment effect is calculated by boot-strapping the errors using the observed residuals from linear factor model estimated in the pre-treatment period. Like Xu (2015) the approach assumes that the researcher has a large number of pre-treatment periods. The estimator and identification results for the matrix factorization is presented in Adams (2016b). The approach is illustrated using synthetic data from Xu (2015), the treatment effect of electoral law changes and country level data from OECD countries before and after the reunification of Germany. The analysis presents bounds on the average treatment effect that do not include zero but do include the actual average treatment effect in the synthetic data. It estimates bounds on the average treatment on the treated for same day voter registration that includes zero and shows that reunification systematically decreased the output of the former West Germany.

Keywords: Treatment Effects, Synthetic Controls, Linear Factor Model, Identification, Panel Data

JEL Classification: C14, C23, C33