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A Design-Based Perspective on Synthetic Control Methods

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Document pages: 37 pages

Abstract: Since their introduction in Abadie and Gardeazabal (2003), Synthetic Control(SC) methods have quickly become one of the leading methods for estimatingcausal effects in observational studies with panel data. Formal discussionsoften motivate SC methods by the assumption that the potential outcomes weregenerated by a factor model. Here we study SC methods from a design-basedperspective, assuming a model for the selection of the treated unit(s), e.g.,random selection as guaranteed in a randomized experiment. We show that SCmethods offer benefits even in settings with randomized assignment, and thatthe design perspective offers new insights into SC methods for observationaldata. A first insight is that the standard SC estimator is not unbiased underrandom assignment. We propose a simple modification of the SC estimator thatguarantees unbiasedness in this setting and derive its exact,randomization-based, finite sample variance. We also propose an unbiasedestimator for this variance. We show in settings with real data that underrandom assignment this Modified Unbiased Synthetic Control (MUSC) estimator canhave a root mean-squared error (RMSE) that is substantially lower than that ofthe difference-in-means estimator. We show that such an improvement is weaklyguaranteed if the treated period is similar to the other periods, for example,if the treated period was randomly selected. The improvement is most likely tobe substantial if the number of pre-treatment periods is large relative to thenumber of control units.

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