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in r 2 t : 96 {1 − α(L) − β(L)}r 2 t = α 0 + {1 − β(L)}ν t (3.22)ACTA WASAENSIA 55to one in financial data.Engle & Bollerslev (1986) define a GARCH process as Integrated in Variance (IGARCH),if α(L)+β(L) =1. Suchaspecification implies Persistence in Variance defined as 94lim sup |E(rt 2 |r 0 ,r −1 ,...) − E(rt 2 |r 1 ,r 0 ,...)| > 0 a.s. (3.21)t→∞such that shocks to the conditional variance persist indefinitely, which stands in contrastto their exponential decay in the conventional GARCH model. The IGARCHmodel has a strictly stationary solution, but implies infinite variance, which makesthe use of the sample autocorellation function for parameter estimation impossible 95 .Furthermore, as has been discussed already in sections 2.4 and 3.2.1, models implyinginfinite variance of returns may be safely ruled out based upon tail index values, whichfor financial returns have been found to be significantly larger than two.Both conventional GARCH and IGARCH models may be written as ARMA processeswith ν t = rt 2 − σ2 t denoting shocks in the conditional variance process. The polynomial{1 − α(L) − β(L)} has zeros outside the unit circle, unless it is integrated in variance,in which case it contains a unit root. This implies that the IGARCH may equivalentlybe written asφ(L)(1 − L)rt 2 = α 0 + {1 − β(L)}ν t (3.23)with zeros of the polynomial φ(L) ={1−α(L)−β(L)}(1−L) −1 outside the unit circle.Baillie et al. (1996) introduce the class of Fractionally Integrated Generalized AutoRegressiveConditionally Heteroskedasticity (FIGARCH)modelsasanintermediatemodel between conventional GARCH and IGARCH by replacing the first differenceoperator (1 − L) in (3.23) with the fractional differencing operator (1 − L) d defined as(1 − L) d ≡ {1 − dL + d(d − 1) L22!− d(d − 1)(d − 2)L33!94 see Bollerslev & Engle (1993).95 see Mikosch (2003a).96 see e.g. Baillie (1996) and Baillie, Bollerslev & Mikkelsen (1996).+ ···}, d ∈ [0, 1] (3.24)