  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 the analysis from "large units"
(e.g., states) to "small units" (e.g., individuals in states). Under this
re-formulation, we derive sufficient conditions for the non-parametric causal
identification of the causal effect. 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, and (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.
