  The spatial error model (SEM) is a type of simultaneous autoregressive (SAR)
model for analysing spatially correlated data. Markov chain Monte Carlo (MCMC)
is one of the most widely used Bayesian methods for estimating SEM, but it has
significant limitations when it comes to handling missing data in the response
variable due to its high computational cost. Variational Bayes (VB)
approximation offers an alternative solution to this problem. Two VB-based
algorithms employing Gaussian variational approximation with factor covariance
structure are presented, joint VB (JVB) and hybrid VB (HVB), suitable for both
missing at random and not at random inference. When dealing with many missing
values, the JVB is inaccurate, and the standard HVB algorithm struggles to
achieve accurate inferences. Our modified versions of HVB enable accurate
inference within a reasonable computational time, thus improving its
performance. The performance of the VB methods is evaluated using simulated and
real datasets.
