Why Synthetic Control estimators are biased and what to do about it: Introducing Relaxed and Penalized Synthetic Controls

This paper extends the literature on the theoretical properties of synthetic controls to the case of non-linear generative models, showing that the synthetic control estimator is generally biased in such settings. I derive a lower bound for the bias, showing that the only component of it that is affected by the choice of synthetic control is the weighted sum of pairwise differences between the treated unit and the untreated units in the synthetic control. To address this bias, I propose a novel synthetic control estimator that allows for a constant difference of the synthetic control to the treated unit in the pre-treatment period, and that penalizes the pairwise discrepancies. Allowing for a constant offset makes the model more flexible, thus creating a larger set of potential synthetic controls, and the penalization term allows for the selection of the potential solution that will minimize bias. I study the properties of this estimator and propose a data-driven process for parameterizing the penalization term.