Orthogonalized Synthetic Controls

Abstract

When conducting inference for the average treatment effect on the treated with a Synthetic Control Estimator, the vector of control weights is a nuisance parameter that is often constrained, high-dimensional, and may be only partially identified even when the average treatment effect on the treated is point-identified. All three of these features of a nuisance parameter can lead to failure of asymptotic normality for the estimate of the parameter of interest when using standard methods. I provide a new method that yields asymptotic normality for an estimate of average treatment effects, even when all three complications are present. This is accomplished by first estimating the control weights and any other nuisance parameters using a regularization penalty to achieve identification, and then estimating average treatment effects using moment conditions that are orthogonalized with respect to the nuisance parameters. Additionally, I extend results from the fixed-smoothing literature to provide tests that control size without requiring consistent standard errors. I present high-level sufficient conditions applicable to the traditional Synthetic Control Estimator as well as other weighting-based panel data methods, and verify them in an example involving instrumental variables.