Before revision:  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;
After revision:  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 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;