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Isometric operators on Hilbert spaces and Wold decomposition of stationary time series

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  • Federico Severino

    (Bocconi University)

Abstract

Wold Theorem plays a fundamental role in the decomposition of weakly stationary time series. It provides a moving average representation of the process under consideration in terms of uncorrelated innovations, whatever the nature of the process is. From an empirical point of view, this result enables to identify orthogonal shocks, for instance in macroeconomic and financial time series. More theoretically, the decomposition of weakly stationary stochastic processes can be seen as a special case of the Abstract Wold Theorem, that allows to decompose Hilbert spaces by using isometric operators. In this work we explain this link in detail, employing the Hilbert space spanned by a weakly stationary time series and the lag operator as isometry. In particular, we characterize the innovation subspace by exploiting the adjoint operator. We also show that the isometry of the lag operator is equivalent to weak stationarity. Our methodology, fully based on operator theory, provides novel tools useful to discover new Wold-type decompositions of stochastic processes, in which the involved isometry is no more the lag operator. In such decompositions the orthogonality of innovations is ensured by construction since they are derived from the Abstract Wold Theorem.

Suggested Citation

  • Federico Severino, 2016. "Isometric operators on Hilbert spaces and Wold decomposition of stationary time series," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 39(2), pages 203-234, November.
  • Handle: RePEc:spr:decfin:v:39:y:2016:i:2:d:10.1007_s10203-016-0181-5
    DOI: 10.1007/s10203-016-0181-5
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    References listed on IDEAS

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    2. Fulvio Ortu & Andrea Tamoni & Claudio Tebaldi, 2013. "Long-Run Risk and the Persistence of Consumption Shocks," The Review of Financial Studies, Society for Financial Studies, vol. 26(11), pages 2876-2915.
    3. Andrew B. Abel, 2003. "The Effects of a Baby Boom on Stock Prices and Capital Accumulation in the Presence of Social Security," Econometrica, Econometric Society, vol. 71(2), pages 551-578, March.
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    Cited by:

    1. Simone Cerreia-Vioglio & Fulvio Ortu & Federico Severino & Claudio Tebaldi, 2023. "Multivariate Wold decompositions: a Hilbert A-module approach," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 45-96, June.

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    More about this item

    Keywords

    Isometry; Hilbert space; Wold decomposition; Stationary time series;
    All these keywords.

    JEL classification:

    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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