On the Assumptions of Synthetic Control Methods

Synthetic control (SC) methods have been widely applied to estimate the causal effect of large-scale interventions, e.g., the state-wide effect of a change in policy.The idea of synthetic controls is to approximate one unit's counterfactual outcomes using a weighted combination of some other units' observed outcomes.The motivating question of this paper is: how does the SC strategy lead to valid causal inferences?We address this question by re-formulating the causal inference problem targeted by SC with a more fine-grained model, where we change the unit of analysis from large units" (e.g., states) tosmall units" (e.g., individuals in states).Under the re-formulation, we derive sufficient conditions for the non-parametric causal identification of the causal effect.We show that, in some settings, existing linear SC estimators are valid even when the data generating process is non-linear.We highlight two implications of the reformulation: 1) it clarifies where ``linearity" comes from, and how it falls naturally out of the more fine-grained and flexible model; 2) it suggests new ways of using available data with SC methods for valid causal inference, in particular, new ways of selecting observations from which to estimate the counterfactual.