On the first-order bilinear time series model

Abstract

The paper investigates some properties of the first-order bilinear time series model: stationarity and invertibility. Estimates of the parameters are obtained by a modified least squares method and shown to be strongly consistent.

On White Noises Driven by Hidden Markov Chains

“…One of the most important problems is to know whether the partial derivatives of f è (å t aå tÀ1 ) with respect to è are integrable. Note that similar dif®culties arise with other non-linear models, such as bilinear models (see for example Tong, 1990, p. 312;Pham and Tran, 1981). However, in order to obtain con®dence intervals for the estimates of the parameters, it would be interesting to have estimators with known asymptotic distributions.…”

“…This lemma clearly has common features with numerous papers in the extensive literature on maximum likelihood estimation. For instance it can be considered as an adaptation of Lemma 4.4 given by Pham and Tran (1981) for the case of a bilinear process. Let V m (è 1 ) be the sphere with centre è 1 and radius 1am.…”

Stability analysis of the first-order periodic autoregressive diagonal bilinear model

“…So like the time-invariant case (see Pham and Tran [26]) we assume that { , ∈ ℤ} satis es the following:…”

Inference for a Special Bilinear Time‐series Model

“…The structure of model has been studied in the literature especially for some special cases. For example, Subba Rao considered model with ψ 1 =⋯= ψ q =0; Davis & Resnick studied the asymptotic behaviour of the correlation function for the simple bilinear model Y t = b Y t − 1 ϵ t − 1 + ϵ t ; Pham & Tran , Turkman & Turkman and Basrak et al , studied the model Y t = φ 1 Y t − 1 + b Y t − 1 ϵ t − 1 + ϵ t ; Zhang & Tong considered the model Y t = b Y t − 1 ϵ t + ϵ t . A sufficient condition for stationarity of the general model was obtained by Liu & Brockwell , which is far away from the necessary one as pointed out by Liu .…”

“…It is known that estimating the general bilinear model is quite challenging. Many different ideas have been proposed to solve this problem for some special cases of ; see Pham & Tran , Guegan & Pham , Wittwer , Liu , Kim & Billard , Kim et al , , Sesay & Subba Rao , Gabr and Hili . However, the asymptotic theory is either rarely established or only derived by assuming that ϵ t follows a normal distribution in these papers.…”