Bayesian Donor Set Selection in Synthetic Controls

2026-07-09

AI Summary

This study addresses the sensitivity of synthetic control methods (SCM) to donor pool composition, as existing approaches typically assume a fixed donor set and struggle to exclude irrelevant or weakly related units. To overcome this limitation, the authors propose a Bayesian hierarchical model that jointly infers donor inclusion indicators and synthetic weights via a Gamma-Bernoulli prior structure. This framework achieves automatic sparsity while preserving the simplex constraint on weights: excluded donors receive exact zero weights, and posterior mass concentrates on low-dimensional faces of the simplex. Theoretical analysis establishes posterior consistency, and numerical experiments demonstrate that the method substantially improves donor selection accuracy and weight estimation in noisy donor pools, remains competitive when all donors are relevant, and successfully replicates the canonical synthetic control analysis of West Germany’s GDP trajectory.

Abstract

The Synthetic Control Method (SCM) is a widely used approach for assessing the effects of interventions by constructing a synthetic counterfactual using a donor set of untreated units. However, the effectiveness of SCM heavily relies on the careful selection of an appropriate donor set. In this paper, we propose a Bayesian hierarchical model that performs donor set selection while preserving the standard SCM simplex constraint on donor weights. Unlike approaches that assume a fixed donor set, our model allows for the simultaneous estimation of the synthetic control weights and the active donor set. By using a hierarchical Gamma-Bernoulli construction for the donor weights, the proposed model assigns posterior mass to simplex faces and allows exact zero weights for excluded donors. We establish a posterior donor-set consistency result under a simplified pre-intervention model. Through numerical simulations, we show that our model improves donor recovery and weight estimation when the donor pool contains irrelevant or weakly related units, while remaining competitive in full-donor settings. Finally, we apply our model to the GDP trajectory of West Germany, illustrating its practical applicability. Our findings suggest that incorporating donor set selection offers a more parsimonious and flexible extension of existing Bayesian synthetic control methods.

Problem

• Synthetic Control Method
• Donor Set Selection
• Bayesian Hierarchical Model
• Counterfactual Estimation
• Intervention Effects

Innovation

• Bayesian synthetic control
• donor set selection
• Gamma-Bernoulli prior
• simplex constraint
• posterior consistency