We propose a Bayesian generalized network autoregressive (BGNAR) model for multivariate time series whose component series are associated with the nodes of a known network. The proposed framework combines the parsimonious network structure of the generalized network autoregressive (GNAR) model with structured shrinkage and persistence priors adapted from Bayesian vector autoregressive (BVAR) modeling. We adapt Minnesota-type shrinkage to both own-lag and network-lag coefficients, with prior variances decreasing over temporal lags and, for network effects, neighborhood orders. A hierarchical prior on the own-lag coefficients allows information to be shared across nodes while retaining node-specific heterogeneity. We further adapt the sum-of-coefficients and dummy-initial-observation priors to the GNAR parameterization. Posterior inference is performed using a Gibbs sampler. Simulation studies show that BGNAR can use the same deliberately over-specified temporal and neighborhood structure across datasets without dataset-specific BIC order selection, while maintaining forecasting accuracy comparable to BIC-selected GNAR and outperforming the unrestricted BVAR benchmark in the settings considered. The structured prior regularizes weakly supported coefficients toward zero within this fixed model. Posterior distributions for the dynamic coefficients and posterior predictive distributions for future observations provide direct quantification of parameter and predictive uncertainty. An application to a wind-speed network demonstrates that BGNAR achieves point-forecast performance comparable to GNAR while additionally providing posterior inference on own-lag and network-lag effects and posterior predictive uncertainty.

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