"Frontmatter". In: Analysis of Financial Time Series

Analysis of Financial Time Series. Ruey S. TsayCopyright © 2002 John Wiley & Sons, Inc.ISBN: 0-471-41544-8CHAPTER 4Nonlinear Modelsand Their ApplicationsThis chapter focuses on nonlinearity in financial data and nonlinear econometricmodels useful in analysis of financial time series. Consider a univariate time seriesx t , which, for simplicity, is observed at equally spaced time intervals. We denotethe observations by {x t | t = 1,...,T }, whereT is the sample size. As stated inChapter 2, a purely stochastic time series x t is said to be linear if it can be written asx t = µ +∞∑ψ i a t−i , (4.1)i=0where µ is a constant, ψ i are real numbers with ψ 0 = 1, and {a t } is a sequenceof independent and identically distributed (iid) random variables with a welldefineddistribution function. We assume that the distribution of a t is continuousand E(a t ) = 0. In many cases, we further assume that Var(a t ) = σa2 or, evenstronger, that a t is Gaussian. If σa 2 ∑ ∞i=1ψi 2 < ∞, then x t is weakly stationary (i.e.,the first two moments of x t are time-invariant). The ARMA process of Chapter 2 islinear because it has an MA representation in Eq. (4.1). Any stochastic process thatdoes not satisfy the condition of Eq. (4.1) is said to be nonlinear. The prior definitionof nonlinearity is for purely stochastic time series. One may extend the definitionby allowing the mean of x t to be a linear function of some exogenous variables,including the time index and some periodic functions. But such a mean function canbe handled easily by methods discussed in Chapter 2, and we do not discuss it here.Mathematically, a purely stochastic time series model for x t is a function of an iidsequence consisting of the current and past shocks—that is,x t = f (a t , a t−1 ,...). (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 modelin Eq. (4.2) is not directly applicable because it contains too many parameters.126