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-TGARCH MODEL: CAPTURING THE TAYLOR EFFECT 41 and, for n > 1,
⇢ (1)=#n 1⇢ (1). (2.4) n11
The su cient condition for the existence of the squared process autocorre- lations is #4 < 1, which implies #2 < 1. Under this condition,
↵ 12 + 12 3 12 ⌥ ⇢1(2)= 2 +(↵1+ 1) 1 + +(1 #4)
(2.5)
(2.6)
and, for n > 1,
where
⌥=[(↵1 + 1) 3 +2 1](1+2#1 +2#2 +#1#2)(1 #1)(1 #2)
n 1
⇢ (2)=#n 1⇢ (2)+(1 # ) ⇤ X#n i 1#i,
n214 21 i=1
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+ ⇣ ↵ 21 + 12 + ( ↵ 1 + 1 ) 1 3 + 12 1 ⌘ ( 1 + # 1 ) 2 ( 1 # 3 ) + 1 # 21 ( 1 # 2 ) ( 1 # 3 ) , 2 4 4
⇤ = 2#1 (1+#1) (1 #3) + [(↵1+ 1) 3 + 2 1] (1+2#1+2#2+#1#2) (1 #1) ,
= 4 (1 #1) (1 #2) (1 + #1)2 (1 #3) (1 #4) , and
=1+3#1 +5#2 +3#3 +3#1#2 +5#1#3 +3#2#3 +#1#2#3.
We note that the parameter ↵0 has no interference in the existence conditions nor in the autocorrelations expressions.
The expressions of the autocorrelations displayed here have been suitably adapted from [13]. These adaptations were crucial in obtaining the results pre- sented in the following section.
3. The Taylor property in the 1-TGARCH(1,1) model
In this section, we begin by establishing the presence of the Taylor property in the 1-TGARCH(1,1) model. Then, we explore the extent of the parameteriza- tions region that verifies it. Finally, considering the first-order autocorrelations and particular distributions for the generating process margins, we illustrate the influence of its kurtosis in the size of the referred region.
To develop this study we assume an additional condition on the moments of the generating process, more specifically, that 1/4 < < 1. This fundamen-
tal relation, first introduced by Gon¸calves, Leite and Mendes-Lopes [5], was further analysed and then characterized by Haas [12], as “will be satisfied for pratically any distribution that one can anticipate in the context of GARCH models”.
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