Identification of a Multivariate AR Model

Abstract

This paper proposes a stationarity criterion for an autoregressive representation of the multivariable stationary stochastic process.
By using the properties of symmetrizable matrices, it is shown that block Toeplitz matrix of a normal equation, prediction error matrices and reflection coefficient matrices are closely related to each other.
By applying Lyapunov's theorem to block companion matrices, the stepwise stationarity criterion for the generalized Levinson's method is derived and its relation to the positive definiteness of the prediction error matrices is made clear.
Though Rouché's theorem, on the basis of Szegö's orthogonal polynomials, plays an important role in fitting a single channel autoregression, the theorem's extension to a multivariate version has never been attempted. In this paper Rouché's theorem is generalized to polynomial matrices and is applied directly to the recursion equations of the orthogonal polynomial matrices to test the stationarity of multivariate autoregression. The author's criteria may have an application in analyzing multiple input-output systems in the frequency domain.