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On the general bilinear time series model. (English) Zbl 0654.60029

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 T. Pham-Dinh and L. T. Tran [ibid. 18, 617-627 (1981; Zbl 0466.62082)] and M. Bhaskara Rao, T. Subba Rao and A. M. Walker [J. Time Ser. Anal. 4, 95-110 (1983; Zbl 0543.62073)] 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.

MSC:

60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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