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Multivariate Wold Decompositions

Author

Listed:
  • Simone Cerreia-Vioglio
  • Fulvio Ortu
  • Federico Severino
  • Claudio Tebaldi
Abstract
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.

Suggested Citation

  • Simone Cerreia-Vioglio & Fulvio Ortu & Federico Severino & Claudio Tebaldi, 2017. "Multivariate Wold Decompositions," Working Papers 606, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    • Handle: RePEc:igi:igierp:606
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    File URL: https://repec.unibocconi.it/igier/igi/wp/2017/606.pdf
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    References listed on IDEAS

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    Cited by:

    1. Bandi, F.M. & Perron, B. & Tamoni, A. & Tebaldi, C., 2019. "The scale of predictability," Journal of Econometrics, Elsevier, vol. 208(1), pages 120-140.

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