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A Note on the Pseudo-Spectra and the Pseudo-Covariance Generating Functions of ARMA Processes

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Abstract

Although the spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not the case for non-stationary stochastic processes. In this paper, the algebraic foundations of the spectral analysis of non-stationary ARMA processes are established. For this purpose the Fourier Transform is extended to the field of fractions of polynomials. Then, the Extended Fourier Transform pair pseudo-covariance generating function / pseudo-spectrum, analogous to the Fourier Transform pair covariance generating function / spectrum,is defined. The new transform pair is well defined for stationary and non-stationary ARMA processes. This new approach can be viewed as an extension of the classical spectral analysis. It is shown that the frequency domain has some additional algebraic advantages over the time domain.

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  • Andrés Bujosa & Marcos Bujosa & Antonio García Ferrer, 2002. "A Note on the Pseudo-Spectra and the Pseudo-Covariance Generating Functions of ARMA Processes," Documentos de Trabajo del ICAE 0203, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
  • Handle: RePEc:ucm:doicae:0203
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    1. Marcos Bujosa & Antonio García Ferrer & Peter Young, 2002. "An ARMA Representation of Unobserved Component Models under Generalized Random Walk Specifications: New Algorithms and Examples," Documentos de Trabajo del ICAE 0204, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
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    1. Bujosa, Marcos & Garcia-Ferrer, Antonio & Young, Peter C., 2007. "Linear dynamic harmonic regression," Computational Statistics & Data Analysis, Elsevier, vol. 52(2), pages 999-1024, October.
    2. Marcos Bujosa & Antonio García Ferrer & Peter Young, 2002. "An ARMA Representation of Unobserved Component Models under Generalized Random Walk Specifications: New Algorithms and Examples," Documentos de Trabajo del ICAE 0204, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.

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