Mathematical framework for pseudo-spectra of linear stochastic difference equations

Andrés Bujosa Brun, Marcos Bujosa and Antonio García-Ferrer
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Andrés Bujosa Brun: Dpto. de Matemáticas, ETSI Telecomunicación. Universidad Politécnica de Madrid. Spain
Antonio García-Ferrer: Dpto. de Análisis Económico: Economía cuantitativa. Universidad Autónoma de Madrid. Spain

No 2013-13, Documentos de Trabajo del ICAE from Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico

Abstract: Although spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not always so for some non-stationary cases. Here, we establish a rigorous mathematical extension of the classic Fourier spectrum to the case in which there are AR roots in the unit circle, ie, the transfer function of the linear time-invariant filter has poles on the unit circle. To achieve it we: embed the classical problem in a wider framework, the Rigged Hilbert space, extend the Discrete Time Fourier Transform and defined a new Extended Fourier Transform pair pseudo-covariance function/pseudo-spectrum. Our approach is a proper extension of the classical spectral analysis, within which the Fourier Transform pair auto-covariance function/spectrum is a particular case. Consequently spectrum and pseudo-spectrum coincide when the first one is defined.

Keywords: Spectral analysis; Time series; Non-stationarity; Frequency domain; Pseudo-covariance function; Linear stochastic difference equations; Rigged Hilbert space; Partial inner product; Extended Fourier Transform. (search for similar items in EconPapers)
JEL-codes: C00 C22 (search for similar items in EconPapers)
Pages: 13 pages
Date: 2013-04-07, Revised 2015-05
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