"Frontmatter". In: Analysis of Financial Time Series

100 CONDITIONAL HETEROSCEDASTIC MODELSwhere the estimated degrees of freedom is 6.51. Standard errors of the estimatesin Eq. (3.20) are close to those in Eq. (3.19). The standard error of the estimateddegrees of freedom is 1.49. Consequently, we cannot reject the hypothesis of using astandardized Student-t distribution with 5 degrees of freedom at the 5% significancelevel.3.4.2 Forecasting EvaluationSince the volatility of an asset return is not directly observable, comparing the forecastingperformance of different volatility models is a challenge to data analysts. Inthe literature, some researchers use out-of-sample forecasts and compare the volatilityforecasts σh 2(l) with the shock a2 h+lin the forecasting sample to assess the forecastingperformance of a volatility model. This approach often finds a low correlationcoefficient between ah+l 2 and σ h 2 (l). However,suchafinding is not surprisingbecause ah+l 2 alone is not an adequate measure of the volatility at time index h + l.Consider the 1-step ahead forecasts. From a statistical point of view, E(ah+1 2 | F h)= σh+1 2 so that a2 h+1 is a consistent estimate of σ h+1 2 . But it is not an accurate estimateof σh+1 2 because a single observation of a random variable with a known meanvalue cannot provide an accurate estimate of its variance. Consequently, such anapproach to evaluate forecasting performance of volatility models is strictly speakingnot proper. For more information concerning forecasting evaluation of GARCHmodels, readers are referred to Andersen and Bollerslev (1998).3.5 THE INTEGRATED GARCH MODELIf the AR polynomial of the GARCH representation in Eq. (3.14) has a unit root, thenwe have an IGARCH model. Thus, IGARCH models are unit-root GARCH models.Similar to ARIMA models, a key feature of IGARCH models is that the impact ofpast squared shocks η t−i = at−i 2 − σ t−i 2 for i > 0ona2 t is persistent.An IGARCH(1, 1) model can be written asa t = σ t ɛ t , σ 2t = α 0 + β 1 σ 2t−1 + (1 − β 1)a 2 t−1 ,where {ɛ t } is defined as before and 1 >β 1 > 0. For the monthly excess returns ofS&P 500 index, an estimated IGARCH(1, 1) model isr t = 0.0067 + a t ,a t = σ t ɛ tσ 2t = 0.000119 + 0.8059σ 2t−1 + 0.1941a2 t−1 ,where the standard errors of the estimates in the volatility equation are 0.0017,0.000013, and 0.0144, respectively. The parameter estimates are close to those ofthe GARCH(1, 1) model shown before, but there is a major difference between the