Multivariate Wold decompositions: a Hilbert A-module approach

Simone Cerreia-Vioglio (), Fulvio Ortu (), Federico Severino () and Claudio Tebaldi ()
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Simone Cerreia-Vioglio: Università Bocconi
Fulvio Ortu: Università Bocconi
Federico Severino: Université Laval
Claudio Tebaldi: Università Bocconi

Decisions in Economics and Finance, 2023, vol. 46, issue 1, No 2, 45-96

Abstract: Abstract 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.

Keywords: Multivariate stationary processes; Wold decomposition; Persistence heterogeneity; Hilbert A-modules (search for similar items in EconPapers)
JEL-codes: C18 C32 C50 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10203-023-00392-3

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