Chapter 4

Nonlinear Models and Their Applications

This chapter focuses on nonlinearity in financial data and nonlinear econometric models useful in analysis of financial time series. Consider a univariate time series xt, which, for simplicity, is observed at equally spaced time points. We denote the observations by {xt|t = 1, … , T}, where T is the sample size. As stated in Chapter 2, a purely stochastic time series xt is said to be linear if it can be written as

4.1

where μ is a constant, ψi are real numbers with ψ0 = 1, and {at} is a sequence of independent and identically distributed (iid) random variables with a well-defined distribution function. We assume that the distribution of at is continuous and E(at) = 0. In many cases, we further assume that or, even stronger, that at is Gaussian. If , then xt is weakly stationary (i.e., the first two moments of xt are time invariant). The ARMA process of Chapter 2 is linear because it has an MA representation in Eq. (4.1). Any stochastic process that does not satisfy the condition of Eq. (4.1) is said to be nonlinear. The prior definition of nonlinearity is for purely stochastic time series. One may extend the definition by allowing the mean of xt to be a linear function of some exogenous variables, including the time index and some periodic functions. But such a mean function can be handled easily by the methods discussed in Chapter 2, and we do not discuss it here. Mathematically, a purely stochastic time series model for xt is a function of an iid sequence consisting of the current and past shocks—that is,

4.2

The linear model in Eq. (4.1) says that f( · ) is a linear function of its arguments. Any nonlinearity in f( · ) results in a nonlinear model. The general nonlinear model in Eq. (4.2) is not directly applicable because it contains too many parameters.

To put nonlinear models available in the literature in a proper perspective, we write the model of xt in terms of its conditional moments. Let Ft−1 be the σ field generated by available information at time t − 1 (inclusive). Typically, Ft−1 denotes the collection of linear combinations of elements in {xt−1, xt−2, … } and {at−1, at−2, … }. The conditional mean and variance of xt given Ft−1 are

4.3

where g( · ) and h( · ) are well-defined functions with h( · ) > 0. Thus, we restrict the model to

where ϵt = at/σt is a standardized shock (or innovation). For the linear series xt in Eq. (4.3), g( · ) is a linear function of elements of Ft−1 and . The development of nonlinear models involves making extensions of the two equations in Eq. (4.3). If g( · ) is nonlinear, xt is said to be nonlinear in mean. If h( · ) is time variant, then xt is nonlinear in variance. The conditional heteroscedastic models of Chapter 3 are nonlinear in variance because their conditional variances evolve over time. In fact, except for the GARCH-M models, in which μt depends on and hence also evolves over time, all of the volatility models of Chapter 3 focus on modifications or extensions of the conditional variance equation in Eq. (4.3). Based on the well-known Wold decomposition, a weakly stationary and purely stochastic time series can be expressed as a linear function of uncorrelated shocks. For stationary volatility series, these shocks are uncorrelated but dependent. The models discussed in this chapter represent another extension to nonlinearity derived from modifying the conditional mean equation in Eq. (4.3).

Many nonlinear time series models have been proposed in the statistical literature, such as the bilinear models of Granger and Andersen (1978), the threshold autoregressive (TAR) model of Tong (1978), the state-dependent model of Priestley (1980), and the Markov switching model of Hamilton (1989). The basic idea underlying these nonlinear models is to let the conditional mean μt evolve over time according to some simple parametric nonlinear function. Recently, a number of nonlinear models have been proposed by making use of advances in computing facilities and computational methods. Examples of such extensions include the nonlinear state-space modeling of Carlin, Polson, and Stoffer (1992), the functional coefficient autoregressive model of Chen and Tsay (1993a), the nonlinear additive autoregressive model of Chen and Tsay (1993b), and the multivariate adaptive regression spline of Lewis and Stevens (1991). The basic idea of these extensions is either using simulation methods to describe the evolution of the conditional distribution of xt or using data-driven methods to explore the nonlinear characteristics of a series. Finally, nonparametric and semiparametric methods such as kernel regression and artificial neural networks have also been applied to explore the nonlinearity in a time series. We discuss some nonlinear models in Section 4.1 that are applicable to financial time series. The discussion includes some nonparametric and semiparametric methods.

Apart from the development of various nonlinear models, there is substantial interest in studying test statistics that can discriminate linear series from nonlinear ones. Both parametric and nonparametric tests are available. Most parametric tests employ either the Lagrange multiplier or likelihood ratio statistics. Nonparametric tests depend on either higher order spectra of xt or the concept of dimension correlation developed for chaotic time series. We review some nonlinearity tests in Section 4.2. Sections 4.3 and 4.4 discuss modeling and forecasting of nonlinear models. Finally, an application of nonlinear models is given in Section 4.5.