AR models of singular spectral matrices

Abstract

This paper deals with autoregressive models of singular spectra. The starting point is the assumption that there is available a transfer function matrix W(q) expressible in the form D -1(q)B for some tall constant matrix B of full column rank and with the determinantal zeros of D(q) all stable. It is shown that, even if this matrix fraction representation of W(q) is not coprime, W(q) has a coprime matrix fraction description of the form D̃ -1(q)[I m 0] T . It is also shown how to characterize the equivalence class of all autoregressive matrix fraction descriptions of W(q), and how canonical representatives can be obtained. A canonical representative can be obtained with a minimal set of row degrees for the submatrix of D̃(q) obtained by deleting the first m rows. The paper also considers singular autoregressive descriptions of nested sequences of W p(q), p = p 0, p 0+1, . . . , where p denotes the number of rows, and shows that these canonical descriptions are nested, and contain a number of parameters growing linearly with p.