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
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 ...
