Identification of refined ARMA echelon form models for multivariate time series

In the present article, we are interested in the identification of canonical ARMA echelon form models represented in a ''refined'' form. An identification procedure for such models is given by Tsay (J. Time Ser. Anal. 10 (1989), 357-372). This procedure is based on the theory of canonical analysis. We propose an alternative procedure which does not rely on this theory. We show initially that an examination of the linear dependency structure of the rows of the Hankel matrix of correlations, with origin Ic in Z (i.e., with correlation at lag k in position (1,1)), allows us not only to identify the Kronecker indices n(1), ..., n(d), when k = 1, but also to determine the autoregressive orders p(1), ..., p(d), as well as the moving average orders q(1), ..., q(d) of the ARMA echelon form model by setting k > 1 and k < 1, respectively. Successive test procedures for the identification of the structural parameters n(i), p(i), and q(i) are then presented. We show, under the corresponding null hypotheses, that the test statistics employed asymptotically follow chi-square distributions. Furthermore, under the alternative hypothesis, these statistics are unbounded in probability and are of the form <N(delta)over tilde>{1 + o(p)(1)}, where <(delta)over tilde> is a positive constant and N denotes the number of observations. Finally, the behaviour of the proposed identification procedure is illustrated with a simulated series from a given ARMA model. (C) 1996 Academic Press, Inc.