Dynamical systems theory for causal inference with application to synthetic control methods

To estimate treatment effects in panel data, suitable control units need to be selected to generate counterfactual outcomes. To guard against cherry-picking of potential controls, which is an important concern in practice, we leverage results from dynamical systems theory. Specifically, key results on delay embeddings in dynamical systems~\citep{Takens1981} show that under fairly general assumptions a dynamical system can be reconstructed up to a one-to-one mapping from scalar observations of the system. This suggests a quantified measure of strength of the dynamical relationship between any two time series variables. The key idea in this paper is to use this measure to ensure that selected control units are dynamically related to treated units, and thus guard against cherry-picking of controls. We illustrate our approach on the synthetic control methodology of~\citet{Abadie2003}, which generates counterfactuals using a model of treated unit outcomes fitted on outcomes from control units. In this setting, we propose to screen out control units that have a weak dynamical relationship to the single treated unit before the model is fit. In simulated studies, we show that the standard synthetic control methodology can be biased towards any desirable direction by adversarially creating artificial control units, but the bias is largely mitigated if we apply the aforementioned screening. In real-world applications, the proposed approach contributes to more reliable control selection, and thus more robust estimation of treatment effects.