Degrees of Freedom and Information Criteria for the Synthetic Control Method

We provide an analytical characterization of the model flexibility of the synthetic control method (SCM) in the familiar form of degrees of freedom. We obtain estimable information criteria. These may be used to circumvent cross-validation when selecting either the weighting matrix in the SCM with covariates, or the tuning parameter in model averaging or penalized variants of SCM. We assess the impact of car license rationing in Tianjin and make a novel use of SCM; while a natural match is available, it and other donors are noisy, inviting the use of SCM to average over approximately matching donors. The very large number of candidate donors calls for model averaging or penalized variants of SCM and, with short pre-treatment series, model selection per information criteria outperforms that per cross-validation.