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 implemented to a small number of large units. The synthetic control methods face two major challenges: (a) estimating weights for each donor 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 Bayesian inference, which is from the duality between a penalised least squares approach and a Bayesian maximum a posteriori (MAP) approach. Our approach differs from the recent Bayesian approach, which allow violating the convex hull restriction and face the potential extrapolation bias. Simulation results indicate that the proposed method leads to smaller biases compared to alternatives. We revisit Abadie and Gardeazabal (2003) by applying our proposed method.