Choosing Right Bayesian Tools: A Comparative Study of Modern Bayesian Methods in Spatial Econometric Models

We compare three modern Bayesian approaches, Hamiltonian Monte Carlo (HMC), Variational Bayes (VB), and Integrated Nested Laplace Approximation (INLA), for two classic spatial econometric specifications: the spatial lag model and spatial error model. Our Monte Carlo experiments span a range of sample sizes and spatial neighborhood structures to assess accuracy and computational efficiency. Overall, posterior means exhibit minimal bias for most parameters, with precision improving as sample size grows. VB and INLA deliver substantial computational gains over HMC, with VB typically fastest at small and moderate samples and INLA showing excellent scalability at larger samples. However, INLA can be sensitive to dense spatial weight matrices, showing elevated bias and error dispersion for variance and some regression parameters. Two empirical illustrations underscore these findings: a municipal expenditure reaction function for Île-de-France and a hedonic price for housing in Ames, Iowa. Our results yield actionable guidance. HMC remains a gold standard for accuracy when computation permits; VB is a strong, scalable default; and INLA is attractive for large samples provided the weight matrix is not overly dense. These insights help practitioners select Bayesian tools aligned with data size, spatial neighborhood structure, and time constraints.