Synthetic control method with convex hull restrictions: a Bayesian maximum a posteriori approach

Synthetic control methods have gained popularity among causal studies with observational data, particularly when estimating the impacts of the interventions that are implemented to a small number of large units. Implementing the synthetic control methods faces two major challenges: a) estimating weights for each control unit to create a synthetic control and b) providing statistical inferences. To overcome these challenges, we propose a Bayesian framework that implements the synthetic control method with the parallelly shiftable convex hull and provides a useful Bayesian inference, which is drawn from the duality between a penalized least squares method and a Bayesian Maximum A Posteriori (MAP) approach. Simulation results indicate that the proposed method leads to smaller biases compared to alternatives. We apply our Bayesian method to the real data example of Abadie and Gardeazabal (2003) and find that the treatment effects are statistically significant during the subset of the post-treatment period.