"Frontmatter". In: Analysis of Financial Time Series

90 CONDITIONAL HETEROSCEDASTIC MODELSforecast of σ 2 h+1 is σ 2 h (1) = α 0 + α 1 a 2 h +···+α ma 2 h+1−m .The 2-step ahead forecast isσ 2 h (2) = α 0 + α 1 σ 2 h (1) + α 2a 2 h +···+α ma 2 h+2−m ,and the l-step ahead forecast for σ 2 h+l isσ 2 h (l) = α 0 +where σh 2(l − i) = a2 h+l−iif l − i ≤ 0.m∑i=1α i σh 2 (l − i), (3.9)3.3.4 ExamplesIn this subsection, we illustrate ARCH modeling by considering two examples.Example 3.1. We first apply the modeling procedure to build a simpleARCH model for the monthly log stock returns of Intel Corporation. The sampleACF and PACF of the squared returns in Figure 3.1 clearly show the existence ofconditional heteroscedasticity. Thus, it is unnecessary to perform any statistical teststo confirm the need of ARCH modeling, and we proceed to identify the order of anARCH model. The sample PACF in the lower right panel of Figure 3.1 indicates thatan ARCH(3) model might be appropriate. Consequently, we specify the modelr t = µ + a t , a t = σ t ɛ t , σ 2t= α 0 + α 1 a 2 t−1 + α 2a 2 t−2 + α 3a 2 t−3for the monthly log returns of Intel stock. Assuming that ɛ t are iid standard normal,we obtain the fitted modelr t = 0.0196 + a t , σ 2t = 0.0090 + 0.2973a 2 t−1 + 0.0900a2 t−2 + 0.0626a2 t−3 ,where the standard errors of the parameters are 0.0062, 0.0013, 0.0887, 0.0645,and 0.0777, respectively. While the estimates meet the general requirement of anARCH(3) model, the estimates of α 2 and α 3 appear to be statistically nonsignificantat the 5% level. Therefore, the model can be simplified.Dropping the two nonsignificant parameters, we obtain the modelr t = 0.0213 + a t , σ 2t = 0.00998 + 0.4437a 2 t−1 , (3.10)where the standard errors of the parameters are 0.0062, 0.00124, and 0.0938, respectively.All the estimates are highly significant. Figure 3.4 shows the standardized