Page 53 - Textos de Matemática Vol. 47

-TGARCH MODEL: CAPTURING THE TAYLOR EFFECT 43
It follows easily from the proof arguments above that the parameterizations
of the 1-TGARCH(1,1) model exhibiting the Taylor property are, at least, the
ones with ↵ ' . 1/4 and close or equal to zero, i.e., parameterizations 114 1
close to the AVARCH(1) model and near the boundary of existence of the autocorrelations. The focus in these particular parameterizations derives from prior works, namely He and Ter¨asvirta [13] and Gonc¸alves, Leite and Mendes- Lopes [5]. We will now show that this parameterizations set can be enlarged.
In order to do that, we set 1 = k↵1, where k is a real positive con-
stant. Considering the same hypotheses of Theorem 3.1 and following the same
reasoning of its proof, we first observe that, when (↵ , , ) converges to ✓ 1/4⇣ 2 ⌘1/4 1/4⇣ 2 ⌘1/4 ◆ 1 1 1
4 k4+1 ,k 4 k4+1 ,0 through values in C, then #4 ! 1. Moreover, for each n in N, we also have
and
since the denominators of the autocorrelations converge to nonzero numbers. Thus, taking into account these limits, the Taylor property holds for values of
⇢n(1) !"1+k· 1 ✓ 2 ◆1/4#n 2 1/4 k4 + 1
4
"1 + k2 1/2 ✓ 2 ◆1/2#n
⇢n (2) ! 2 · 4 k4 + 1 ,
k > 0 such that
1+k✓ 2 ◆ 1/4 1
1 + k2 k4 + 1 > 1/4 . (3.1)
14
As 1/4 < is equivalent to ( 1/4) 1 < 1, then the solutions k > 0 of 4114
(3.1) are also solutions of the following inequality 1+k ✓ 2 ◆ 1/4
1 + k2 k4 + 1 1, (k 1)2 k6 +2k5 +3k4 +8k3 +3k2 +2k 1 0.
(3.2)
(3.3)
which is equivalent to
Numerical analysis methods allow us to determine that if k 2 [0.2814, 3.5546] then it satisfies (3.3), thus also satisfies (3.1).
In view of the previous calculations, we can conclude that the Taylor prop- erty holds, at least, for parameterizations in a neighbourhood of the boundary of existence of the autocorrelations such that 0.2814↵1 < 1 < 3.5546↵1 (so ↵1 and 1 can be clearly di↵erent) and with 1 close or equal to zero, covering