  To infer the treatment effect for a single treated unit using panel data,
synthetic control methods construct a linear combination of control units'
outcomes that mimics the treated unit's pre-treatment outcome trajectory. This
linear combination is subsequently used to impute the counterfactual outcomes
of the treated unit had it not been treated in the post-treatment period, and
used to estimate the treatment effect. Existing synthetic control methods rely
on correctly modeling certain aspects of the counterfactual outcome generating
mechanism and may require near-perfect matching of the pre-treatment
trajectory. Inspired by proximal causal inference, we obtain two novel
nonparametric identifying formulas for the average treatment effect for the
treated unit: one is based on weighting, and the other combines models for the
counterfactual outcome and the weighting function. We introduce the concept of
covariate shift to synthetic controls to obtain these identification results
conditional on the treatment assignment. We also develop two treatment effect
estimators based on these two formulas and the generalized method of moments.
One new estimator is doubly robust: it is consistent and asymptotically normal
if at least one of the outcome and weighting models is correctly specified. We
demonstrate the performance of the methods via simulations and apply them to
evaluate the effectiveness of a Pneumococcal conjugate vaccine on the risk of
all-cause pneumonia in Brazil.
