On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix

The recursive method proposed by Durbin (1960) for the fitting of autoregreseive schemes of successively increasing order is generalized to the fitting of multivariate autoregressions, and of schemes with rational spectral density function. It is also shown that an auto regression fitted from the Yule-Walker relations, even if of insufficient order, has the necessary stability properties. This property holds in the multivariate case, too, and is important in connexion with a problem arising in multivariate prediction: the approximate factorization of a spectral density matrix.