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On the general bilinear time series model

Published online by Cambridge University Press:  14 July 2016

Jian Liu  and
Peter J. Brockwell
Jian Liu*
Affiliation:
Colorado State University
Peter J. Brockwell*
Affiliation:
Colorado State University
*
∗ Present address: Department of Statistics, University of British Columbia, Vancouver, BC, Canada. Research carried out in partial fulfilment of Ph.D. requirements at Colorado State University.
∗∗ Present address: Department of Statistics, The University of Melbourne, Parkville, VIC 3052, Australia.
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Abstract

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

Keywords

CAUSALITYSTATIONARITYERGODICITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 3 , September 1988 , pp. 553 - 564
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported by NSF Grant No. MCS 8501763.

References

Bhaskara Rao, M., Subba Rao, T. and Walker, A. M. (1983) On the existence of some bilinear time series models. J. Time Series Anal. 4, 95–110.Google Scholar
Brockwell, P. J. and Davis, R. A. (1987) Time Series: Theory and Methods. Springer-Verlag, New York.Google Scholar
Gabr, M. M. and Subba Rao, T. (1981) The estimation and prediction of subset bilinear time series models with applications. J. Time Series Anal. 2, 155–171.Google Scholar
Pham-Dinh, T. and Tran, L. T. (1981) On the first order bilinear time series model. J. Appl. Prob. 18, 617–627.Google Scholar
Subba Rao, T. (1980) On the theory of bilinear time series models. J. R. Statist. Soc. B 43, 244–245.Google Scholar
Subba Rao, T and Gabr, M. M. (1984) An Introduction to Bispectral Analysis and Bilinear Time Series Models. Lecture Notes in Statistics 24, Springer-Verlag, New York.Google Scholar