"Frontmatter". In: Analysis of Financial Time Series

THE GARCH MODEL 93other softwares available, including Eviews, Scientific Computing Associates (SCA),and S-Plus.3.4 THE GARCH MODELAlthough the ARCH model is simple, it often requires many parameters to adequatelydescribe the volatility process of an asset return. For instance, consider themonthly excess returns of S&P 500 index. An ARCH(9) model is needed for thevolatility process. Some alternative model must be sought. Bollerslev (1986) proposesa useful extension known as the generalized ARCH (GARCH) model. For alog return series r t , we assume that the mean equation of the process can be adequatedlydescribed by an ARMA model. Let a t = r t − µ t be the mean-corrected logreturn. Then a t follows a GARCH(m, s) model ifa t = σ t ɛ t , σ 2t = α 0 +m∑α i at−i 2 +i=1s∑j=1β j σ 2t− j , (3.13)where again {ɛ t } is a sequence of iid random variables with mean 0 and variance 1.0,α 0 > 0, α i ≥ 0, β j ≥ 0, and ∑ max(m,s)i=1(α i + β i ) m and β j = 0for j > s. The latter constraint on α i + β i implies thatthe unconditional variance of a t is finite, whereas its conditional variance σt2 evolvesover time. As before, ɛ t is often assumed to be a standard normal or standardizedStudent-t distribution. Equation (3.13) reduces to a pure ARCH(m) model if s = 0.To understand properties of GARCH models, it is informative to use the followingrepresentation. Let η t = at 2 − σt 2 so that σt 2 = at 2 − η t . By plugging σt−i 2 = at−i 2 −η t−i (i = 0,...,s) into Eq. (3.13), we can rewrite the GARCH model asmax(m,s) ∑at 2 = α 0 + (α i + β i )at−i 2 + η t −i=1s∑β j η t− j . (3.14)It is easy to check that {η t } is a martingale difference series [i.e., E(η t ) = 0andcov(η t ,η t− j )=0for j ≥ 1]. However, {η t } in general is not an iid sequence. Equation(3.14) is an ARMA form for the squared series at 2 . Thus, a GARCH model canbe regarded as an application of the ARMA idea to the squared series at 2 . Using theunconditional mean of an ARMA model, we havej=1E(a 2 t ) = α 01 − ∑ max(m,s)i=1(α i + β i )provided that the denominator of the prior fraction is positive.The strengths and weaknesses of GARCH models can easily be seen by focusingon the simplest GARCH(1, 1) model with