  This paper provides new insights into the asymptotic properties of the
synthetic control method (SCM). We show that the synthetic control (SC) weight
converges to a limiting weight that minimizes the mean squared prediction risk
of the treatment-effect estimator when the number of pretreatment periods goes
to infinity, and we also quantify the rate of convergence. Observing the link
between the SCM and model averaging, we further establish the asymptotic
optimality of the SC estimator under imperfect pretreatment fit, in the sense
that it achieves the lowest possible squared prediction error among all
possible treatment effect estimators that are based on an average of control
units, such as matching, inverse probability weighting and
difference-in-differences. The asymptotic optimality holds regardless of
whether the number of control units is fixed or divergent. Thus, our results
provide justifications for the SCM in a wide range of applications. The
theoretical results are verified via simulations.
