Consider a panel data setting with observations of N units over T time steps. Each of the N units undergoes exactly one of D interventions at time step T_0, with 1 \le T_0 < T, prior to which all units experience no intervention, i.e., control. We present a causal framework, synthetic interventions (SI), to estimate the counterfactual outcome of each unit under each of the D interventions, averaged over the post-intervention time period . We prove identification of the causal parameter of interest under a latent factor model across time, units, and interventions. We furnish an algorithm to estimate the causal parameter , which utilizes principal component regression (PCR) as a key subroutine. We argue that PCR implicitly de-noises the observations, which are corrupted by idiosyncratic measurement error, and thus advocate for its usage in panel data settings. Formally, we establish consistency and asymptotic normality of the estimated causal parameter. We then compare our assumptions and results with those in the synthetic control (SC) literature. In doing so, we establish  identification and inference results for SC as well. We further introduce a novel hypothesis test , with provable guarantees, to validate when to use SI (and thereby SC). Empirically, we showcase the efficacy of the SI framework on synthetic and real-world data. Finally , we discuss connections between the SI causal framework and tensor estimation.
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.
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