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Time‐Reversibility, Identifiability And Independence Of Innovations For Stationary Time Series

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  • F. J. Breidt
  • R. A. Davis

Abstract

. Weiss (J. Appl. Prob. 12 (1975) 831–36) has shown that for causal autoregressive moving‐average (ARMA) models with independent and identically distributed (i.i.d.) noise, time‐reversibility is essentially unique to Gaussian processes. This result extends to quite general linear processes and the extension can be used to deduce that a non‐Gaussian fractionally integrated ARMA process has at most one representation as a moving average of i.i.d. random variables with finite variance. In the proof of this uniqueness result, we use a time‐reversibility argument to show that the innovations sequence (one‐step prediction residuals) of an ARMA process driven by i.i.d. non‐Gaussian noise is typically not independent, a result of interest in deconvolution problems. Further, we consider the case of an ARMA process to which independent noise is added. Using a time‐reversibility argument we show that the innovations of the ARMA process with added independent noise are independent if and only if both the driving noise of the process and the added noise are Gaussian.

Suggested Citation

  • F. J. Breidt & R. A. Davis, 1992. "Time‐Reversibility, Identifiability And Independence Of Innovations For Stationary Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 13(5), pages 377-390, September.
    • Handle: RePEc:bla:jtsera:v:13:y:1992:i:5:p:377-390
    DOI: 10.1111/j.1467-9892.1992.tb00114.x
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    Cited by:

    1. Francesco Giancaterini & Alain Hecq & Claudio Morana, 2022. "Is Climate Change Time-Reversible?," Econometrics, MDPI, vol. 10(4), pages 1-18, December.
    2. Davis, Richard A. & Song, Li, 2020. "Noncausal vector AR processes with application to economic time series," Journal of Econometrics, Elsevier, vol. 216(1), pages 246-267.
    3. Frédérique Bec & Alain Guay & Heino Bohn Nielsen & Sarra Saïdi, 2022. "Power of unit root tests against nonlinear and noncausal alternatives," THEMA Working Papers 2022-14, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.

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