Data Analysis and Applications 1

7A Note on the Linear Approximation of TAR Models

The linear approximation of nonlinear time series models is not an easy task. In this chapter, we give a definition of linear process and we distinguish between linear approximation and linear representation of nonlinear models, briefly giving some examples that better clarify this distinction. The attention is here focused on the threshold autoregressive models whose linear approximation is discussed starting from a motivating example and some theoretical issues.

7.1. Introduction

The complexity of most nonlinear models often leads to evaluate if a linear representation or a linear approximation can be admitted for this class of models. In the presence of linear representation, the aim can be ascribed to the need to take advantage (under proper assumptions) of the large and strengthened literature developed in the linear domain (to cite the main references, Box and Jenkins 1976, Brockwell and Davies 1991) whereas linear approximations can be seen as a tool for model selection (or more generally to select candidate models for the data under analysis) to “filter” the dynamic relationship among variables such that the “purely” nonlinear component, obtained in output, can be properly examined.

Before showing the main advantages obtained from the linearization, it is useful to clarify when a stochastic process {X_t}, with t ∈ ℤ, is said to be linear.

Let {X_t} be a mean zero stationary process and let {e_t} be a sequence of white ...