"Frontmatter". In: Analysis of Financial Time Series

106 CONDITIONAL HETEROSCEDASTIC MODELSthe 2-step ahead forecast, Eq. (3.29) givesσ 2 h+2 = σ 2α 1h+1 exp[(1 − α 1)α 0 ] exp[g(ɛ h+1 )].Taking conditional expectation at time h, wehaveˆσ 2 h (2) =ˆσ 2α 1h(1) exp[(1 − α 1 )α 0 ]E h {exp[g(ɛ h+1 )]},where E h denotes a conditional expectation taking at the time origin h. The priorexpectation can be obtained as follows:E{exp[g(ɛ)]} =∫ ∞−∞= exp= expexp[θɛ + γ(| ɛ |− √ 2/π)] f (ɛ)dɛ(−γ √ 2/π) [∫ ∞∫ 0+ e (θ−γ)ɛ 1√ e −ɛ2 /2 dɛ−∞ 2π(−γ √ 2/π0e (θ+γ)ɛ 1√2πe −ɛ2 /2 dɛ])[]e (θ+γ)2/2 (θ + γ)+ e (θ−γ)2/2 (γ − θ) ,where f (ɛ) and (x) are the probability density function and CDF of the standardnormal distribution, respectively. Consequently, the 2-step ahead volatility forecastisˆσ 2 h (2) =ˆσ 2α 1h×[(1) exp (1 − α 1 )α 0 − γ √ ]2/π[exp{(θ + γ) 2 /2}(θ + γ)+ exp{(θ − γ) 2 /2}(γ − θ)Repeating the previous procedure, we obtain a recursive formula for j-step aheadforecastˆσ h 2 ( j) =ˆσ 2α 1h( j − 1) exp(ω)[]× exp{(θ + γ) 2 /2}(θ + γ)+ exp{(θ − γ) 2 /2}(γ − θ) ,].where ω = (1 − α 1 )α 0 − γ √ 2/π. The values of (θ + γ) and (θ − γ) can beobtained from most statistical packages. Alternatively, accurate approximations tothese values can be obtained by using the method in Appendix B of Chapter 6.For illustration, consider the AR(1)-EGARCH(1, 0) model of the previous subsectionfor the monthly log returns of IBM stock. Using the fitted EGARCH(1, 0)model, we can compute the volatility forecasts for the series. At the forecast origint = 864, the forecasts are ˆσ 2 864 (1) = 6.05 × 10−3 , ˆσ 2 864 (2) = 5.82 × 10−3 ,