"Frontmatter". In: Analysis of Financial Time Series

KURTOSIS OF GARCH MODELS 119where K ɛ is the excess kurtosis of the innovation ɛ t . Based on the assumption, wehave• Var(a t ) = E(σ 2t ) = α 0/[1 − (α 1 + β 1 )].• E(at 4) = (K ɛ + 3)E(σt 4 ) provided that E(σ4t ) exists.Taking the square of the volatility model, we haveσ 4t = α 2 0 + α2 1 a4 t−1 + β2 1 σ 4t−1 + 2α 0α 1 a 2 t−1 + 2α 0β 1 σ 2t−1 + 2α 1β 1 σ 2t−1 a2 t−1 .Taking expectation of the equation and using the two properties mentioned earlier,we obtainE(σ 4 1 ) = α 2 0 (1 + α 1 + β 1 )[1 − (α 1 + β 1 )][1 − α 2 1 (K ɛ + 2) − (α 1 + β 1 ) 2 ]provided that 1 >α 1 + β 1 ≥ 0and1− α 2 1 (K ɛ + 2) − (α 1 + β 1 ) 2 > 0. The excesskurtosis of a t , if it exists, is thenK a = E(a4 t )[E(a 2 t )] 2 − 3= (K ɛ + 3)[1 − (α 1 + β 1 ) 2 ]1 − 2α1 2 − (α 1 + β 1 ) 2 − K ɛ α12 − 3.This excess kurtosis can be written in an informative expression. First, considerthe case that ɛ t is normally distributed. In this case, K ɛ = 0, and some algebra showsthatK a (g)6α12 =1 − 2α1 2 − (α 1 + β 1 ) 2 ,where the superscript (g) is used to denote Gaussian distribution. This result has twoimportant implications: (a) the kurtosis of a t exists if 1 − 2α1 2 − (α 1 + β 1 ) 2 > 0, and(b) if α 1 = 0, then K a(g) = 0, meaning that the corresponding GARCH(1, 1) modeldoes not have heavy tails.Second, consider the case that ɛ t is not Gaussian. Using the prior result, we haveK a = K ɛ − K ɛ (α 1 + β 1 ) + 6α 2 1 + 3K ɛα 2 11 − 2α 2 1 − (α 1 + β 1 ) 2 − K ɛ α 2 1= K ɛ[1 − 2α 2 1 − (α 1 + β 1 ) 2 ]+6α 2 1 + 5K ɛα 2 11 − 2α 2 1 − (α 1 + β 1 ) 2 − K ɛ α 2 1