Vector Parametric Models and Methods

Abstract

In this chapter, we extend the univariate nonlinear parametric time series framework to encompass multiple, related time series exhibiting nonlinear behavior. Over the past few years, many multivariate (vector) nonlinear time series models have been proposed. Some of them are “ad - hoc”, with a special application in mind. Others are direct multivariate extensions of their univariate counterparts. Within the latter class, a definition of a multivariate nonlinear time series model is often proposed with the following objectives in mind. First, the definition should contain the most general linear vector model as a special case when the nonlinear part is not present. This is analogous to univariate nonlinear time series models embedding linear ones. Second, the definition should contain the most general univariate nonlinear model within its class of models. Also, a potential candidate for a multivariate nonlinear time series model should possess some specified properties in order to permit estimation of the unknown model parameters and allow statistical inference. Moreover, because one of the main uses of time series analysis is forecasting, it is reasonable to restrict consideration to models which are capable of producing forecasts.