On the Existence of a General Multiple Bilinear Time Series

Abstract

A sufficient condition is derived for the existence of a strictly stationary solution of the general multiple bilinear time series equations (without assuming subdiagonality). The condition is shown to reduce to the condition of Stensholt and Tjostheim in the special case which they consider. Under this condition a solution is constructed which is shown to be casual in the sense we define, strictly 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, i.e. the multiple autoregressive moving-average (ARMA) model, the condition given here reduces to the well-known sufficient condition for the existence of a casual stationary solution.

Yule–Walker type estimators in periodic bilinear models: strong consistency and asymptotic normality

“…From the Condition (2.3), we deduce that the spectral radius of the matrix E A ⊗2 t is strictly less than one. Finally, we conclude as in Liu (1989) that S n (t) converges in mean and a.s. to some limit X t given by (2.2) which is a causal and ergodic solution of Eq. (2.1).…”

“…It is also strictly stationary with mean zero and covariance matrix Cov X t , Proof We use the same approach as Liu (1989) to prove the sufficiency part of Theorem 2.2. We first define the R s -valued processes S n (t) (n,t)∈Z×Z and n (t) (n,t)∈Z×Z as S n (t)= A t S n−1 (t − 1) + η t I {n≥0} and n (t) = S n (t) − S n−1 (t) for (n, t) ∈ Z × Z.…”

A Space‐Time Bilinear Model and its Identification

“…Stensholt and Tjùstheim (1987) and Liu (1989) extended some results for the univariate BL model to an MBL model. Liu (1989, Theorem 3.1) gives a suf®cient condition for the existence of a unique strictly stationary solution of general MBL models.…”

“…In order to investigate the relationship between the orders of an STBL model and their statistical characteristics, it is necessary that the model be stationary. Stensholt and Tjùstheim (1987) and Liu (1989) extended some results for the univariate BL model to an MBL model. Liu (1989, Theorem 3.1) gives a suf®cient condition for the existence of a unique strictly stationary solution of general MBL models.…”

On the ARCH model with random coefficients

“…The proof follows from standard arguments (see Liu [5]). First, define a R p -valued stochastic process which will be used in the proof to generated a strictly stationary solution of (3).…”