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Mathematical framework for pseudo-spectra of linear stochastic difference equations

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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.

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  • Andrés Bujosa Brun & Marcos Bujosa Brun & Antonio García-Ferrer, 2013. "Mathematical framework for pseudo-spectra of linear stochastic difference equations," Documentos de Trabajo del ICAE 2013-13, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico, revised May 2015.
    • Handle: RePEc:ucm:doicae:1313
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    More about this item

    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.;
    All these keywords.

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - 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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