Multivariate Wold decompositions: a Hilbert A-module approach

Orthogonal decompositions are essential tools for the study of weakly stationary time series. Some examples are given by the classical Wold decomposition of Wold (A study in the analysis of stationary time series, Almqvist & Wiksells Boktryckeri, Uppsala, 1938) and the extended Wold decomposition of Ortu et al. (Quant Econ 11(1):203–230, 2020), which permits to disentangle shocks with heterogeneous degrees of persistence from a given weakly stationary process. The analysis becomes more involved when dealing with vector processes because of the presence of different simultaneous shocks. In this paper, we recast the standard treatment of multivariate time series in terms of Hilbert A-modules (where matrices replace the field of scalars) and we prove the abstract Wold theorem for self-dual pre-Hilbert A-modules with an isometric operator. This theorem allows us to easily retrieve the multivariate classical Wold decomposition and the multivariate version of the extended Wold decomposition. The theory helps in handling matrix coefficients and computing orthogonal projections on closed submodules. The orthogonality notion is key to decompose the given vector process into uncorrelated subseries, and it implies a variance decomposition.