Fast MCMC estimation of multiple W-matrix spatial regression models and Metropolis–Hastings Monte Carlo log-marginal likelihoods

A strategy for MCMC estimation of a family of models involving multiple simultaneous dependence parameters is set forth that is capable of producing rapid estimates for problems involving a large number of observations. Simultaneous dependence parameters arise when dependence exists between dependent variable observations, with spatial and cross-sectional dependence being specific examples. The approach taken is to express the joint conditional distribution of the dependence parameters as a quadratic form, where the dependence parameters are outer vectors of the quadratic form and the inner matrix expressions of the quadratic form involve only sample data that allow these to be pre-calculated. During MCMC estimation, multiple evaluations of the joint conditional distribution of the dependence parameters can be carried out rapidly, allowing a block-sampling scheme. Block sampling of the dependence parameters is useful for imposing stability restrictions that arise for these parameters. In addition, a Taylor series approximation to the log-determinant expression that arises in the joint conditional distribution for the dependence parameters can be used to rapidly evaluate this term. The joint conditional distribution for the dependence parameters is obtained by analytically integrating out other model parameters, allowing Monte Carlo integration of the log-marginal likelihood, which can be used for model comparison. Timing results are discussed along with Monte Carlo evidence regarding performance as well as applied illustrations involving large sample sizes.