Revisiting the synthetic control estimator

Abstract

VERSÃO ATUALIZADA DE ABRIL DE 2018 DISPONÍVEL.

The synthetic control (SC) method has been recently proposed as an alternative to estimate treatment effects in comparative case studies. The idea of the SC method is to use the pre-treatment periods to estimate weights such that a weighted average of the control units reconstructs the pre-treatment outcomes of the treated unit, and then use these weights to construct a counterfactual for the treated unit. \cite{Abadie2010} show that, if the pre-treatment match is close to perfect, then the bias of the SC estimator is bounded by a term that goes to zero with the number of pre-treatment periods ($T_0$). In this paper, we revisit the SC method in a linear factor model setting and consider the asymptotic properties of the SC estimator when $T_0$ goes to infinity. Differently from \cite{Abadie2010}, we do not condition the analysis on a close-to-perfect pre-treatment match, as the probability that this happens goes to zero when $T_0$ is large. We show that, in our setting, the SC estimator is asymptotically biased if treatment assignment is correlated with the unobserved heterogeneity. If errors are stationary, then the asymptotic bias of the SC estimator goes to zero when the transitory shocks are small, which is also the case in which it is more likely that the pre-treatment match will be good for a given $T_0$. Still, we show that the SC method can substantially improve over the difference-in-differences (DID) estimator even when a close-to-perfect fit is not achieved. However, in this case the method would rely on stronger identification assumptions. If a subset of the common factors is non-stationary, then we show that the SC weights might not reconstruct the factor loadings related to stationary common factors, even conditional on a close-to-perfect fit. While this is a scenario where the SC method significantly improves relative to DID, an important qualification is that the identification assumption in this case relies on orthogonality between treatment assignment and the stationary common factors. Finally, we suggest a modification in the permutation test proposed by \cite{Abadie2010} that has good asymptotic properties if the SC estimator is unbiased.