"Frontmatter". In: Analysis of Financial Time Series

THE GARCH MODEL 95ahead forecast can be written asTherefore,σ 2 h (l) = α 0[1 − (α 1 + β 1 ) l−1 ]1 − α 1 − β 1+ (α 1 + β 1 ) l−1 σ 2 h (1).σ 2 h (l) → α 01 − α 1 − β 1, as l →∞provided that α 1 + β 1 < 1. Consequently, the multistep ahead volatility forecasts ofa GARCH(1, 1) model converge to the unconditional variance of a t as the forecasthorizon increases to infinity provided that Var(a t ) exists.The literature on GARCH models is enormous; see Bollerslev, Chou, and Kroner(1992), Bollerslev, Engle, and Nelson (1994), and the references therein. The modelencounters the same weaknesses as the ARCH model. For instance, it respondsequally to positive and negative shocks. In addition, recent empirical studies of highfrequencyfinancial time series indicate that the tail behavior of GARCH modelsremains too short even with standardized Student-t innovations.3.4.1 An Illustrative ExampleThe modeling procedure of ARCH models can also be used to build a GARCHmodel. However, specifying the order of an GARCH model is not easy. Onlylower order GARCH models are used in most applications, say GARCH(1, 1),GARCH(2, 1), and GARCH(1, 2) models. The conditional maximum likelihoodmethod continues to apply provided that the starting values of the volatility {σt 2}areassumed to be known. Consider, for instance, a GARCH(1, 1) model. If σ1 2 is treatedas fixed, then σt2 can be computed recursively for a GARCH(1, 1) model. In someapplications, the sample variance of a t serves as a good starting value of σ1 2.Thefitted model can be checked by using the standardized residual ã t = a t /σ t and itssquared process.Example 3.3. In this example, we consider the monthly excess returns ofS&P 500 index starting from 1926 for 792 observations. The series is shown in Figure3.5. Denote the excess return series by r t . Figure 3.6 shows the sample ACF of r tand the sample PACF of rt 2.Ther t series has some serial correlations at lags 1 and3, but the key feature is that the PACF of rt 2 shows strong linear dependence. If anMA(3) model is entertained, we obtainr t = 0.0062 + a t + 0.0944a t−1 − 0.1407a t−3 , ˆσ a = 0.0576for the series, where all of the coefficients are significant at the 5% level. However,for simplicity, we use instead an AR(3) modelr t = φ 1 r t−1 + φ 2 r t−2 + φ 3 r t−3 + β 0 + a t .