Non-Gaussian Simultaneous Autoregressive Models with Missing Data

Abstract: Standard simultaneous autoregressive (SAR) models are usually assumed to have normally distributed errors, an assumption that is often violated in real-world datasets, which are frequently found to exhibit non-normal, skewed, and heavy-tailed characteristics. New SAR models are proposed to capture these non-Gaussian features. In this project, the spatial error model (SEM), a widely used SAR-type model, is considered. Three novel SEMs are introduced that extend the standard Gaussian SEM by incorporating Student's $t$-distributed errors after a one-to-one transformation is applied to the response variable. Variational Bayes (VB) estimation methods are developed for these models, and the framework is further extended to handle missing response data. Standard variational Bayes (VB) methods perform well with complete datasets; however, handling missing data requires a Hybrid VB (HVB) approach, which integrates a Markov chain Monte Carlo (MCMC) sampler to generate missing values. The proposed VB methods are evaluated using both simulated and real-world datasets, demonstrating their robustness and effectiveness in dealing with non-normal data and missing data in spatial models. Although the method is demonstrated using SAR models, the proposed model specifications and estimation approaches are widely applicable to various types of models for handling non-Gaussian data with missing values.