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Multivariate linear time series models

Published online by Cambridge University Press:  01 July 2016

E. J. Hannan  and
L. Kavalieris
E. J. Hannan*
Affiliation:
The Australian National University
L. Kavalieris*
Affiliation:
The Australian National University
*
∗Postal address: Department of Statistics, IAS, Mathematical Sciences Building, The Australian National University, G.P.O. Box 4, Canberra ACT 2601, Australia.
∗Postal address: Department of Statistics, IAS, Mathematical Sciences Building, The Australian National University, G.P.O. Box 4, Canberra ACT 2601, Australia.
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Abstract

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.

The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.

The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

Keywords

ARMAX MODELSLINEAR SYSTEMSHANKEL MATRIXANALYTIC MANIFOLDCOORDINATESECHELON FORMSTATE-SPACE FORMALGORITHMAUTOREGRESSIONLEVIN SON–WHITTLE RECURSIONAICBICORDER DETERMINATIONSIMULATIONMARTINGALE DIFFERENCESLAW OF ITERATED LOGARITHMCENTRAL LIMIT THEOREM
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 3 , September 1984 , pp. 492 - 561
Copyright
Copyright © Applied Probability Trust 1984 

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