An Eigenvalue Problem of the theory of Arma Models and Multistep Iteration Procedures

Abstract

Anna-models are recursive equations of the kind

$${x_k} = {A_0} + \sum\limits_{i = 1}^p {{A_i}.{x_{k - i}}} + \sum\limits_{i = 0}^p {{B_i}.{z_{k - i}}} + \ldots $$

((1))

and are used to describe objects of unknown mathematical structure approximately. The main task is to determine appropriate (n,n)-matrices A_i, B_i. In order to guarantee that the n-vectors x_k form a stationary time series, it is required that the polynomial matrix

$${\lambda ^p}I - \sum\limits_{i = 1}^p {{\lambda ^{p - i}}} {A_i}$$

has only singular values A with absolute values | λ | < 1. Sometimes it is also required that Σλ^q-1B_i has only singular values within the unit circle. After having computed estimations A_i, B_i by means of concrete measured values x_k (k = 0,...,N) and some numerical method, f.e. the YULE-WALKER procedure, it is desirable to check — with as little effort as possible — whether the singular value condition mentioned above is satisfied and whether it is advisable to accept or to reject the computed results.