Template-type: ReDIF-Paper 1.0
Author-Name: Andrés Bujosa Brun
Author-Workplace-Name: Dpto. de Matemáticas, ETSI Telecomunicación. Universidad Politécnica de Madrid. Spain
Author-Name: Marcos Bujosa Brun
Author-Email: marcos.bujosa@ccee.ucm.es
Author-Homepage: https://www.ucm.es/fundamentos-analisis-economico2/marcos-bujosa
Author-Person: pbu154
Author-Workplace-Name: Departamento de Fundamentos del Análisis Económico II (Economía Cuantitativa). Universidad Complutense de 
	Madrid.
Author-Workplace-Homepage: https://www.ucm.es/fundamentos-analisis-economico2
Author-Name: Antonio García-Ferrer
Author-Workplace-Name: Dpto. de Análisis Económico: Economía cuantitativa. Universidad Autónoma de Madrid. Spain
Title: Mathematical framework for pseudo-spectra of linear stochastic difference equations
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.
Classification-JEL: C00, C22.
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.
Length: 13 pages 
Creation-Date: 2013-04-07
Revision-Date: 2015-05 
Number: 2013-13 
X-File-Ref: http://america.sim.ucm.es/repec/ucm/ref/doicae1313.txt 
File-URL: https://eprints.ucm.es/id/eprint/20699/1/1313.pdf
File-Format: Application/pdf
Handle: RePEc:ucm:doicae:1313