Synthetic Interventions

Consider a panel data setting with observations of $N$ units over $T$ time periods. Each unit undergoes one of $D$ interventions at time period $T_0$, with $1 \le T_0 < T$, prior to which all units are under control. We present synthetic interventions (SI), a framework to estimate counterfactual outcomes of each unit under each of the $D$ interventions, averaged over the post-intervention time periods. We prove identification of this causal parameter under a latent factor model across time, units, and interventions. We furnish an estimator for this causal parameter and establish its consistency and asymptotic normality. In doing so, we establish novel identification and inference results for the synthetic controls (SC) literature. Further, we introduce a hypothesis test to validate when to use SI (and thereby SC). Through simulations and an empirical case-study, we demonstrate efficacy of the SI framework. Lastly, we discuss connections between SI and tensor estimation.