Page 52 - Textos de Matemática Vol. 47

42 J. LEITE
Theorem 3.1. The 1-TGARCH(1,1) model, with symmetric, centered and reduced generating process marginal distribution, such that 1/4 < < 1,
41
has a parameterization set, contained in the set defined by #4 < 1, for which
the Taylor property is verified.
Proof. Let us admit that ↵ is arbitrarily fixed and let us define the following n 0 o
3 12 ( ↵ 1 + 1 )
2 1,
setC= (↵, , )2 R+ 3:# <1 . 11104
We have
#1 = ↵1 + 1 1 + 1,
2
↵ 12 + 12 2
#2 = 2 + 1 + 1 (↵1 + 1) 1,
↵ 13 + 13 3 3 1 ↵ 12 + 12 #3= 2 3+ 1+ 2 +
and
#4 = 2 4 + 1 +2 1 ↵1 + 1 3 +3 1 ↵1 + 1 +2 1 (↵1 + 1) 1.
↵14+ 14 4 3 3 2 2 2 3
We start by observing that both ⇢n (1) and ⇢n (2), for each n 2 N, as func- tions of ↵1, 1 and 1, are continuous for all points of C.
Let us now consider that (↵ , , ) converges to ( 1/4 , 1/4 , 0) through 11144
values in C. Then, from expression (2.3), we can guarantee that ⇢1(1) ! 1/4, since the denominator (1 #2) 2(1 # ) converges to 1 2 6= 0.
Analogously, taking into account expression (2.5), we have ⇢ (2) ! 1/2, 14
141121
because (1 #4) converges to zero and, as the expression (1 #1)(1 #2) converges to (1 1/4)(1 1/2) 6= 0, the denominator does not converge
to zero. Therefore, recalling that > 1/4, we can state that 1/4 > 1414
1/2 and conclude that there exists a neighbourhood of ( 1/4, 1/4, 0) that, 444
intersected with C, only has points corresponding to parameterizations of the 1-TGARCH(1,1) that verify ⇢1(1) > ⇢1(2).
144
In a similar manner, for each n in N\ {1}, when (↵1, 1, 1) converges to ( 1/4, 1/4, 0) through values in C,
44 ⇣ 1⌘n ⇢ (1) = #n 1⇢ (1) ! 4 ,
n1114
⇤X ⇣⌘n
n 1 1
⇢ (2)=#n 1⇢ (2)+(1 # ) #n i 1#i ! 2 ,
n214 214 i=1
and so we can conclude that there exists a neighbourhood of ( 1/4, 1/4,0) 44
such that, intersected with C, its elements satisfy ⇢n (1) > ⇢n (2). ⇤