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Irene Botosaru, Bruno Ferman, On the role of covariates in the synthetic control method, The Econometrics Journal, Volume 22, Issue 2, May 2019, Pages 117–130, https://doi.org/10.1093/ectj/utz001
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Summary
Abadie et al. (2010) derive bounds on the bias of the synthetic control estimator under a perfect balance assumption on both observed covariates and pre-treatment outcomes. In the absence of a perfect balance on covariates, we show that it is still possible to derive such bounds, albeit at the expense of relying on stronger assumptions about the effects of observed and unobserved covariates and of generating looser bounds. We also show that a perfect balance on pre-treatment outcomes does not generally imply an approximate balance for all covariates, even when they are all relevant. Our results have important implications for the implementation of the method.