"Frontmatter". In: Analysis of Financial Time Series

94 CONDITIONAL HETEROSCEDASTIC MODELSσ 2t = α 0 + α 1 a 2 t−1 + β 1σ 2t−1 , 0 ≤ α 1,β 1 ≤ 1,(α 1 + β 1 ) 0, thenE(a 4 t )[E(a 2 t )] 2 = 3[1 − (α 1 + β 1 ) 2 ]1 − (α 1 + β 1 ) 2 − 2α 2 1> 3.Consequently, similar to ARCH models, the tail distribution of a GARCH(1, 1) processis heavier than that of a normal distribution. Third, the model provides a simpleparametric function that can be used to describe the volatility evolution.Forecasts of a GARCH model can be obtained using methods similar to those ofan ARMA model. Consider the GARCH(1, 1) model in Eq. (3.15) and assume thatthe forecast origin is h. For 1-step ahead forecast, we haveσ 2 h+1 = α 0 + α 1 a 2 h + β 1σ 2 h ,where a h and σh 2isare known at the time index h. Therefore, the 1-step ahead forecastσ 2 h (1) = α 0 + α 1 a 2 h + β 1σ 2 h .For multistep ahead forecasts, we use a 2 t = σ 2t ɛ2 t and rewrite the volatility equationin Eq. (3.15) asWhen t = h + 1, the equation becomesσ 2t+1 = α 0 + (α 1 + β 1 )σ 2t + α 1 σ 2t (ɛ2 t − 1).σ 2 h+2 = α 0 + (α 1 + β 1 )σ 2 h+1 + α 1σ 2 h+1 (ɛ2 h+1 − 1).Since E(ɛ 2 h+1 −1 | F h) = 0, the 2-step ahead volatility forecast at the forecast originh satisfies the equationIn general, we haveσ 2 h (2) = α 0 + (α 1 + β 1 )σ 2 h (1).σh 2 (l) = α 0 + (α 1 + β 1 )σh 2 (l − 1), l > 1. (3.16)This result is exactly the same as that of an ARMA(1, 1) model with AR polynomial1 − (α 1 + β 1 )B. By repeated substitutions in Eq. (3.16), we obtain that the l-step