  Since their introduction in Abadie and Gardeazabal (2003), Synthetic Control
(SC) methods have quickly become one of the leading methods for estimating
causal effects in observational studies in settings with panel data. Formal
discussions often motivate SC methods by the assumption that the potential
outcomes were generated by a factor model. Here we study SC methods from a
design-based perspective, assuming a model for the selection of the treated
unit(s) and period(s). We show that the standard SC estimator is generally
biased under random assignment. We propose a Modified Unbiased Synthetic
Control (MUSC) estimator that guarantees unbiasedness under random assignment
and derive its exact, randomization-based, finite-sample variance. We also
propose an unbiased estimator for this variance. We document in settings with
real data that under random assignment, SC-type estimators can have root
mean-squared errors that are substantially lower than that of other common
estimators. We show that such an improvement is weakly guaranteed if the
treated period is similar to the other periods, for example, if the treated
period was randomly selected. While our results only directly apply in settings
where treatment is assigned randomly, we believe that they can complement
model-based approaches even for observational studies.
