Bayesian causal inference with a single treated unit

Abstract

Causal inference with observational data has been of great interest in various areas of study. One of the important research questions in observational studies is how to estimate the treatment effect when the treatment is assigned to a single unit. To address this problem, the Synthetic Control Method (SCM) was proposed to construct a weighted combination of control units that approximates the treated unit. However, SCM has several limitations, including difficulties in conducting formal statistical inference. The Generalized Synthetic Control Method (GSCM) was proposed to address the issues of SCM. GSCM is based on a factor modeling framework that requires rank selection and creates some difficulties with making statistical inferences. We address these issues by developing a Bayesian GSCM. The proposed approach enables Bayesian inference through Markov chain Monte Carlo (MCMC). We further extend the model to allow for flexible, time-varying treatment effects using Bspline representations. To improve computational efficiency, we incorporate parameter expansion (PX) within the MCMC framework to enhance mixing and convergence. In addition, rank selection is performed using posterior model probabilities. To demonstrate the validity of the proposed method, we conduct simulation studies and compare the proposed method with existing SCM and GSCM. Finally, the proposed method is applied to two different real datasets to study treatment effects in distinct empirical settings. This dissertation consists of five chapters. In Chapter 1, we discuss the background and motivation for our study. In Chapter 2, we develop a fully Bayesian approach for estimating the treatment effect when the treatment is assigned to a single unit. Chapter 3 extends the Bayesian framework to model time-varying treatment effects and introduces a parameter expansion technique to improve the efficiency of the Gibbs sampler. Chapter 4 introduces a Bayesian procedure for rank selection in the generalized synthetic control framework. Chapter 5 discusses some remarks and future directions.