Multivariate Wold Decompositions

The Wold decomposition of a weakly stationary time series extends to the multivariate case by allowing each entry of a weakly stationary vectorial process to linearly depend on the components of a vector of shocks. Since univariate coefficients are replaced by matrices, we propose a modelling approach based on Hilbert A-modules defined over the algebra of squared matrices. The Abstract Wold Theorem for Hilbert A-modules, that we prove, delivers two orthogonal decompositions of vectorial processes: the Multivariate Classical Wold Decomposition, which exploits the lag operator as isometry, and the Multivariate Extended Wold Decomposition, where a scaling operator is employed. The latter enables us to disentangle the heterogeneous levels of persistence of a weakly stationary vectorial process. Hence, the persistent components of the macro-financial variables into consideration are related to the overlapping of different sources of randomness with specific persistence. We finally provide a simple application to V AR models.