Page 50 - Textos de Matemática Vol. 47

40 J. LEITE
generated by Xt 1,Xt 2,.... In addition, we consider that the process Z is a sequence of independent and identically distributed random variables with zero mean and unit variance.
The process X follows a first-order power threshold GARCH model, de-
noted -TGAR⇢CH(1,1), if for every t 2 Z, Xt = Zt t
, (2.1) =↵+↵ X+ + X +
t 0 1 t 1 1 t 1 1t 1 where >0,↵0 >0,↵1 0, 1 0, 1 0andZt isindependentofthe Xt 1. Z is called the generating process of X. If 1 = 0, we say that X follows a -TARCH(1) model.
Regarding this specification of t, first proposed by Ding, Granger and En- gle [2], it is relevant to notice that it allows the model to capture the leverage e↵ect, being enough that ↵1 6= 1, and is more flexible since it does not estab- lish a priori the power value. Thus, it incorporates, for example, the spec- ificationsofGARCH( =2and↵1 = 1),AVGARCH( =1and↵1= 1), TGARCH ( = 1) and power-GARCH (↵1 = 1) models. In Gon¸calves, Leite and Mendes-Lopes [6] a detailed probabilistic analysis is presented, which in- cludes strict stationarity and stationarity up to -order of a more general version of this model, namely considering the -TGARCH(p,q) model with 6= 0 and assuming even less restrictive conditions for the generating process.
As previously mentioned, the analysis of the Taylor property requires both ⇢n (1) = corr (|Xt| , |Xt n|) and ⇢n (2) = corr ⇣|Xt|2 , |Xt n|2⌘ analytical ex-
pressions. Let
He and Ter¨asvirta [13] proved that, for 2 {1,2} and k a positive integer, # ,k < 1 is a necessary and su cient condition for the existence of E ⇣|Xt| k⌘.
Ling and McAleer [14] note that He and Ter¨asvirta’s proofs actually hold for any > 0 and that # ,k < 1 also guarantees the strict stationarity of X. However, only for = 1 both autocorrelations of interest were obtained in [13].
# ,k =E⇢h↵1 Zt+ + 1 Zt + 1ik . (2.2)
Thus, in the following, let us assume that X is a 1-TGARCH(1,1) (also writ- ten TGARCH(1,1)) process such that the generating process margins have zero mean, unit variance and a symmetric distribution with finite fourth moment.
To simplify our notation, let #k = #1,k and k = E ⇣|Zt|k⌘.
If #2 < 1, X is strictly and weakly stationary and ⇢n (1) exists for all n, with
⇢1(1)=#1 + 1 1 #21 21 1 (2.3) (1 #21) 21 (1 #2)