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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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    1. Bell, William R & Hillmer, Steven C, 1984. "Issues Involved with the Seasonal Adjustment of Time Series: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 2(4), pages 343-349, October.
    2. Marcos Bujosa & Antonio García‐Ferrer & Aránzazu Juan, 2013. "Predicting Recessions with Factor Linear Dynamic Harmonic Regressions," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 32(6), pages 481-499, September.
    3. J. Haywood & G. Tunnicliffe Wilson, 1997. "Fitting Time Series Models by Minimizing Multistep‐ahead Errors: a Frequency Domain Approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(1), pages 237-254.
    4. Bell, William R & Hillmer, Steven C, 1984. "Issues Involved with the Seasonal Adjustment of Economic Time Series," Journal of Business & Economic Statistics, American Statistical Association, vol. 2(4), pages 291-320, October.
    5. Harvey, A C & Todd, P H J, 1983. "Forecasting Economic Time Series with Structural and Box-Jenkins Models: A Case Study," Journal of Business & Economic Statistics, American Statistical Association, vol. 1(4), pages 299-307, October.
    6. Harvey, A C & Todd, P H J, 1983. "Forecasting Economic Time Series with Structural and Box-Jenkins Models: A Case Study: Response," Journal of Business & Economic Statistics, American Statistical Association, vol. 1(4), pages 313-315, October.
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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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