Bayesian Synthetic Control with a Soft Simplex Constraint

Abstract: Whether the synthetic control method should be implemented with the simplex constraint and how to implement it in a high-dimensional setting have been widely discussed. To address both issues simultaneously, we propose a novel Bayesian synthetic control method that integrates a soft simplex constraint with spike-and-slab variable selection. Our model is featured by a hierarchical prior capturing how well the data aligns with the simplex assumption, which enables our method to efficiently adapt to the structure and information contained in the data by utilizing the constraint in a more flexible and data-driven manner. A unique computational challenge posed by our model is that conventional Markov chain Monte Carlo sampling algorithms for Bayesian variable selection are no longer applicable, since the soft simplex constraint results in an intractable marginal likelihood. To tackle this challenge, we propose to update the regression coefficients of two predictors simultaneously from their full conditional posterior distribution, which has an explicit but highly complicated characterization. This novel Gibbs updating scheme leads to an efficient Metropolis-within-Gibbs sampler that enables effective posterior sampling from our model and accurate estimation of the average treatment effect. Simulation studies demonstrate that our method performs well across a wide range of settings, in terms of both variable selection and treatment effect estimation, even when the true data-generating process does not adhere to the simplex constraint. Finally, application of our method to two empirical examples in the economic literature yields interesting insights into the impact of economic policies.