Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time Series

Abstract

The aim of this paper to give a multidimensional version of the classical one-dimensional case of smooth spectral density. A spectral density with smooth eigenvalues and H∞ eigenvectors gives an explicit method to factorize the spectral density and compute the Wold representation of a weakly stationary time series. A formula, similar to the Kolmogorov–Szego formula, is given for the covariance matrix of the innovations. These results are important to give the best linear predictions of the time series. The results are applicable when the rank of the process is smaller than the dimension of the process, which occurs frequently in many current applications, including econometrics.