Ch. 21. On the theory of discrete and continuous bilinear time series models

Abstract

Large number of time series we come across are neither linear nor Gaussian. During the last two decades, several nonlinear time series models have been developed, their statistical properties have been investigated. Nonlinear models often produce better forecasts than linear models. One of the models which received considerable attention is the bilinear model. In this chapter we review the recent developments associated with this model. We discuss both discrete time and continuous time versions of the model. Higher-order moments and cumulants play an important role in the identification of non-linear models, and we discuss various relationships associated with higher-order cumulants for both univariate and multivariate time series. We consider the estimation of the bispectral density function and its use in detection of linearity of the time series. The estimation of univariate and multivariate bilinear models is discussed. We introduce a long memory bilinear model. We also introduce nonstationary models. We study the effect of nonlinearity on spurious regression and cointegration.