Objective Bayesian analysis of SAR models

The simultaneously autoregressive model (abbreviated as SAR) has been extensively applied for lattice (regional summary) data. A Bayesian approach has been studied by De Oliveira and Song (2008), but they only considered two versions of Jeffreys priors, Jeffreys-rule and independence Jeffreys priors. They recommended the independence Jeffreys prior for a default prior. This prior is known to have the potential problem of posterior impropriety. In this paper, we consider the reference priors including the commonly used reference and “exact” reference priors for the SAR model. We show that common reference priors typically result in improper posterior distributions. Next, two “exact” reference priors are developed and are shown to yield proper posterior distributions. Frequentist properties of inferences based on two “exact” reference and Jeffreys-rule priors are studied by means of simulation. For illustrative purposes, we apply the method to SAT verbal scores across the US.