Multivariate Spatio-Temporal Regression with Penalized Model Selection and an Empirical Application
This paper develops the statistical foundations of a multivariate general nesting spatio-temporal (MGNST) regression framework for analyzing spatial, temporal, and cross-equation dependence among responses. Four parameter matrices represent spatial lag dependence, spatial error dependence,temporal autoregression, and contemporaneous error covariance. Their off-diagonal elements allow dependence to propagate within and across responses. Matrix restrictions yield eleven specifications encompassing multivariate spatial autoregressive models, multivariate spatial error models, vector autoregressive models with exogenous variables, and independent spatio-temporal regressions as special cases. We establish identifiability conditions using instrumental-variable rank conditions and introduce penalized likelihood estimation and information criteria based on effective degrees of freedom. Monte Carlo experiments examine three data-generating models, three sample sizes, and three levels of spatial dependence. Correct-selection rates under penalized AIC generally increase with sample size and the strength of spatial dependence, while parameter recovery improves as the sample size increases. For socioeconomic data from 198 municipalities in Japan's Kansai region, penalized AIC selects the full MGNST model, whereas penalized BIC selects a response-wise independent spatial error model. Despite selecting models of different complexity, both criteria support temporal persistence and spatial error dependence. The pAIC-selected MGNST model reduces strong spatial autocorrelation in the responses to negligible residual levels, demonstrating its usefulness for identifying and comparing multivariate spatio-temporal dependence structures.