Revisiting the Synthetic Control Estimator

• Grant number: 16/20877-2
• Support Opportunities: Regular Research Grants
• Start date: March 01, 2017
• End date: February 28, 2019
• Field of knowledge: Applied Social Sciences - Economics - Quantitative Methods Applied to Economics
• Principal Investigator: Bruno Ferman
• Grantee: Bruno Ferman
• Host Institution: Escola de Economia de São Paulo (EESP). Fundação Getúlio Vargas (FGV). São Paulo , SP, Brazil
• Associated researchers: Cristine Campos de Xavier Pinto

Abstract

The synthetic control (SC) method has been recently proposed as an alternative to estimate treatment effects in comparative case studies. The SC relies on the assumption that there is a weighted average of the control units that reconstructs the potential outcome of the treated unit in the absence of treatment. If these weights were known, then one could estimate the counterfactual for the treated unit using this weighted average. With these weights, the SC would provide an unbiased estimator for the treatment effect even if selection into treatment is correlated with the unobserved heterogeneity. In this paper, we revisit the SC method in a linear factor model where the SC weights are considered nuisance parameters that are estimated to construct the SC estimator. We show that, when the number of control units is fixed, the estimated SC weights will generally not converge to the weights that reconstruct the factor loadings of the treated unit, even when the number of pre-intervention periods goes to infinity. As a consequence, the SC estimator will be asymptotically biased if treatment assignment is correlated with the unobserved heterogeneity. We suggest a slight modification in the SC method that guarantees that the SC estimator is asymptotically unbiased and has a lower asymptotic variance than the difference-in-differences (DID) estimator when the DID identification assumption is satisfied. We will also suggest an alternative way to estimate the SC weights based on instrumental variables that guarantee an unbiased SC estimator if we impose additional assumptions on the structure of the errors. Finally, we will consider the implications of our results to the permutation test proposed in Abadie et al. (2010). (AU)