Generalized Rouche´s theorem and its application to multivariate autoregressions

This paper proposes the matrix extension of Rouche´s theorem to investigate the location of zeros of polynomial matrices. The theorem is then applied to the Levinson-Wiggins-Robinson (LWR) algorithm in order to enumerate the zeros of a polynomial matrix at each step of the recursion and test the stability of fitted multivariate autoregressions. Extensive use is made of some important algebraic relations in the LWR algorithm, which are derived from the properties of symmetrizable matrices. In this paper, only a finite sequence of sample correlation matrices computed from obsereed data over a finite time interval is assumed to be given and, therefore, the spectral density matrix defined by its Fourier transform is not necessarily nonnegative definite.