We develop a method to help quantify the impact different levels of mobility restrictions could have had on COVID-19 related deaths across nations. Synthetic control (SC) has emerged as a standard tool in such scenarios to produce counterfactual estimates if a particular interventionhad not occurred, using just observational data.However, it remains an important open problem of how to extend SC to obtain counterfactual estimates if a particular intervention had occurred - this is exactly the question of the impact of mobility restrictions stated above. As our main contribution, we introduce\le  synthetic interventions (SI), which helps resolve this open problem by allowing one to produce counterfactual estimates if there are multiple interventions of interest . We prove SI produces consistent counterfactual estimates under a tensor factor model . Our finite sample analysis shows the test error decays as 1/T_0, where T_0 is the amount of observed pre-intervention data. As a special case, this improves upon the 1/\sqrt{T_0 , we establish the parameter estimation and test error for Principal Component Regression (a key subroutine in SI and several SCvariants)under the setting of error-in-variable regression decays as 1/T_0, where T_0 is the number of samples observed; this improves the best prior test error bound of 1/\sqrt{T_0 .
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 .
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We develop a method to help quantify the impact different levels of mobility restrictions could have had on COVID-19 related deaths across nations. Synthetic control (SC) has emerged as a standard tool in such scenarios to produce counterfactual estimates if a particular interventionhad not occurred, using just observational data.However, it remains an important open problem of how to extend SC to obtain counterfactual estimates if a particular intervention had occurred - this is exactly the question of the impact of mobility restrictions stated above. As our main contribution, we introduce
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
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T_0 < T, prior to which all units experience no intervention, i.e., control. We present a causal framework,
R
which helps resolve this open problem by allowing one to produce counterfactual estimates if there are multiple interventions of interest
to estimate the counterfactual outcome of each unit under each of the D interventions, averaged over the post-intervention time period
R
SI produces consistent counterfactual estimates under a tensor factor model . Our finite sample analysis shows the test error decays as 1/T_0, where T_0 is the amount of observed pre-intervention data. As a special case, this improves upon the 1/\sqrt{T_0
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
R
the parameter estimation and test error for Principal Component Regression (a key subroutine in SI and several SCvariants)under the setting of error-in-variable regression decays as 1/T_0, where T_0 is the number of samples observed; this improves the best prior test error bound of 1/\sqrt{T_0
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