On the relation between Global VAR Models and Matrix Time Series Models with Multiple Terms

TL;DR

This work analyzes the relationship between matrix time series models ($MAR_j(p)$) and Global VAR models (GVAR). It shows that GVAR can be viewed as a special case of $MAR_j(p)$ when star variables are constructed with a Kronecker-structured weight matrix, and that the resulting system admits a reduced-form VAR via a left-hand-side operator $\mathcal{G}_0$. Identifiability of the $MAR_j(p)$ parameters is addressed using the magical rearrangement $\mathcal{R}$ and a blockwise Kronecker decomposition, providing practical routes to estimation. An alternating estimation strategy is proposed to jointly recover regional dynamics and cross-regional weights, offering a middle ground between the parsimonious GVAR and the flexible VAR and enabling more robust weight specification and testing. The approach enhances the interpretability and estimation efficiency for high-dimensional VARs in multi-regional settings, benefiting macroeconomic analyses with cross-region spillovers.

Abstract

Matrix valued time series (MaTS) and global vector autoregressive (GVAR) models both impose restrictions on the general VAR for multidimensional data sets, in order to bring down the number of parameters. Both models are motivated from a different viewpoint such that on first sight they do not have much in common. When investigating the models more closely, however, one notices many connections between the two model sets. This paper investigates the relations between the restrictions imposed by the two models. We show that under appropriate restrictions in both models we obtain a joint framework allowing to gain insight into the nature of GVARs from the viewpoint of MaTS.