Analysis of Financial Time Series, Third Edition

4.1 Nonlinear Models

Most nonlinear models developed in the statistical literature focus on the conditional mean equation in Eq. (4.3); see Priestley (1988) and Tong (1990) for summaries of nonlinear models. Our goal here is to introduce some nonlinear models that are applicable to financial time series.

4.1.1 Bilinear Model

The linear model in Eq. (4.1) is simply the first-order Taylor series expansion of the f( · ) function in Eq. (4.2). As such, a natural extension to nonlinearity is to employ the second-order terms in the expansion to improve the approximation. This is the basic idea of bilinear models, which can be defined as

4.4

where p, q, m, and s are nonnegative integers. This model was introduced by Granger and Andersen (1978) and has been widely investigated. Subba Rao and Gabr (1984) discuss some properties and applications of the model, and Liu and Brockwell (1988) study general bilinear models. Properties of bilinear models such as stationarity conditions are often derived by (a) putting the model in a state-space form (see Chapter 11) and (b) using the state transition equation to express the state as a product of past innovations and random coefficient vectors. A special generalization of the bilinear model in Eq. (4.4) has conditional heteroscedasticity. For example, consider the model

4.5

where {at} is a white noise series. The first two conditional moments of ...