Identification and Inference for Synthetic Controls with Confounding

Abstract: This paper studies inference on treatment effects in panel data settings with unobserved confounding. We model outcome variables through a factor model with random factors and loadings. Such factors and loadings may act as unobserved confounders: when the treatment is implemented depends on time-varying factors, and who receives the treatment depends on unit-level confounders. We study the identification of treatment effects and illustrate the presence of a trade-off between time and unit-level confounding. We provide asymptotic results for inference for several Synthetic Control estimators and show that different sources of randomness should be considered for inference, depending on the nature of confounding. We conclude with a comparison of Synthetic Control estimators with alternatives for factor models.