"Frontmatter". In: Analysis of Financial Time Series

128 NONLINEAR TIME SERIESApart from the development of various nonlinear models, there is substantialinterest in studying test statistics that can discriminate linear series from nonlinearones. Both parametric and nonparametric tests are available. Most parametric testsemploy either the Lagrange multiplier or likelihood ratio statistics. Nonparametrictests depend on either higher order spectra of x t or the concept of dimension correlationdeveloped for chaotic time series. We review some nonlinearity tests in Section4.2. Sections 4.3 and 4.4 discuss modeling and forecasting of nonlinear models.Finally, an application of nonlinear models is given in Section 4.5.4.1 NONLINEAR MODELSMost nonlinear models developed in the statistical literature focus on the conditionalmean equation in Eq. (4.3); see Priestley (1988) and Tong (1990) for summariesof nonlinear models. Our goal here is to introduce some nonlinear models that areapplicable to financial time series.4.1.1 Bilinear ModelThe linear model in Eq. (4.1) is simply the first-order Taylor series expansion of thef (.) function in Eq. (4.2). As such, a natural extension to nonlinearity is to employthe second-order terms in the expansion to improve the approximation. This is thebasic idea of bilinear models, which can be defined asx t = c +p∑φ i x t−i −i=1q∑m∑θ j a t− j +j=1i=1 j=1s∑β ij x t−i a t− j + a t , (4.4)where p, q, m, ands are non-negative integers. This model was introduced byGranger and Andersen (1978) and has been widely investigated. Subba Rao andGabr (1984) discuss some properties and applications of the model, and Liu andBrockwell (1988) study general bilinear models. Properties of bilinear models suchas stationarity conditions are often derived by (a) putting the model in a state-spaceform, and (b) using the state transition equation to express the state as a productof past innovations and random coefficient vectors. A special generalization of thebilinear model in Eq. (4.4) has conditional heteroscedasticity. For example, considerthe modelx t = µ +s∑β i a t−i a t + a t , (4.5)i=1where {a t } is a white noise series. The first two conditional moments of x t are(E(x t | F t−1 ) = µ, Var(x t | F t−1 ) = 1 +) 2 s∑β i a t−i σa 2 ,which are similar to that of the RCA or CHARMA model of Chapter 3.i=1