On the Covariance Structure of Time Varying Bilinear Models

Abstract

In this paper, a class of bilinear time series models with time varying coefficients is considered. In this nonstationary and nonlinear framework, our aim is to study the structure of usual time series analysis tools, in particular the sample autocovariance function which has been developed for analyzing stationary linear time series. We use appropriately defined Markovian representations to derive a necessary and sufficient condition for the existence and uniqueness of a solution with bounded first and second order moments (BFSM). A more explicit sufficient condition for the existence of a BFSM solution is provided. An explicit expression of the autocovariance function is obtained. The existence of a weak time-varying ARMA representation of the bilinear model with time varying coefficients is shown. We also discuss the existence of higher order moments. Several subclasses of the model are shown to be quasi-stationary. Under this assumption of quasi-stationarity, the asymptotic distributions of the sample mean and sample covariances are obtained.