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Contributions To Evolutionary Spectral Theory

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  • Guy Mélard
  • Annie Herteleer‐de Schutter

Abstract

. The purpose of this paper is to discuss several fundamental issues in the theory of time‐dependent spectra for univariate and multivariate non‐stationary processes. The general framework is provided by Priestley's evolutionary spectral theory which is based on a family of stochastic integral representations. A particular spectral density function can be obtained from the Wold—Cramér decomposition, as illustrated by several examples. It is shown why the coherence is time invariant in the evolutionary theory and how the theory can be generalized so that the coherence becomes time dependent. Statistical estimation of the spectrum is also considered. An improved upper bound for the bias due to non‐stationarity is obtained which does not rely on the characteristic width of the process. The results obtained in the paper are illustrated using time series simulated from an evolving bivariate autoregressive moving‐average process of order (1, 1) with a highly time‐varying coherence.

Suggested Citation

  • Guy Mélard & Annie Herteleer‐de Schutter, 1989. "Contributions To Evolutionary Spectral Theory," Journal of Time Series Analysis, Wiley Blackwell, vol. 10(1), pages 41-63, January.
    • Handle: RePEc:bla:jtsera:v:10:y:1989:i:1:p:41-63
    DOI: 10.1111/j.1467-9892.1989.tb00014.x
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    Cited by:

    1. Peter Robinson, 2007. "On Discrete Sampling Of Time-Varyingcontinuous-Time Systems," STICERD - Econometrics Paper Series 520, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    2. D. M. Nachane, 2018. "Time-varying spectral analysis: theory and applications," Indian Economic Review, Springer, vol. 53(1), pages 3-27, December.
    3. Rajae Azrak & Guy Melard, 1993. "Exact maximum likelihood estimation for extended ARIMA models," ULB Institutional Repository 2013/13802, ULB -- Universite Libre de Bruxelles.
    4. Dahlhaus, R., 1996. "On the Kullback-Leibler information divergence of locally stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 139-168, March.
    5. Fryzlewicz, Piotr & Nason, Guy P., 2004. "Smoothing the wavelet periodogram using the Haar-Fisz transform," LSE Research Online Documents on Economics 25231, London School of Economics and Political Science, LSE Library.
    6. Robinson, Peter, 2007. "On discrete sampling of time-varying continuous-time systems," LSE Research Online Documents on Economics 6795, London School of Economics and Political Science, LSE Library.
    7. Fryzlewicz, Piotr & Nason, Guy P., 2006. "Haar-Fisz estimation of evolutionary wavelet spectra," LSE Research Online Documents on Economics 25227, London School of Economics and Political Science, LSE Library.

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