Page 42 - Textos de Matemática Vol. 47

32 E. GONC¸ALVES AND C. M. MARTINS
Using an analogous procedure, we obtain
E(X2X2 ) = 4E +2 3E +2 3m E +4 2m E + 2E +2 m E +
and
E6 =
tt 1 1 2 13 14 5 16 + 2m2E(Xt2"2t ) + 2 m1m2E(Xt"t) + m2,
where
E1 =
E2 =
E3 =
E4 =
E5 =
ments of "t. Examples
In the following lines, we investigate the presence of the Taylor property in model (3.1), considering some non-negative distributions for the generator process, namely, the uniform distribution in ]0, ↵[, the exponential distribution in ]0,+1[ with mean ↵ and the Pareto distribution, Par(↵,✓), ✓ > 8. In all cases, ↵ is a non-negative parameter and the condition E(|ln"t|) < +1 is satisfied.
As in the case of the AR(1) model, the choice of these distributions takes into account the fact that the Taylor property seems to be related with the kurtosis value of the process. The uniform distribution is platykurtic with a constant kurtosis value equal to 1.8, while the exponential distribution is leptokurtic with a constant kurtosis value equal to 9. As we referred before, the kurtosis of the Pareto distribution is greater than 9.
We also point out that, in all cases, the condition 4 m4 < 1 and the values of ⇢X(1) and ⇢X2(1) can be written in terms of r = ↵ .
In each case, the value of the kurtosis of the process X given by (3.1) also depends on r = ↵ . We present the corresponding graphical representation. We point out that, in all these models, the leptokurtosis of the generator process implies the same property for the process X. In what concerns the Taylor property and kurtosis of X, comparisons are made separately between the first two distributions, uniform and exponential, and also between the Pareto
E(Xt2Xt2 1"2t "2t 1)
E(Xt2Xt 1"3t "t 1)
E(XtXt2 1"t"2t 1)
= 2m2E(Xt4"4t ) + 2 m3E(Xt3"3t ) + μ4E(Xt2"2t ),
= 2m3E(Xt3"3t ) + 2 m4E(Xt2"2t ) + m5E(Xt"t), = m1E(Xt3"3t ) + m2E(Xt2"2t ),
= m2E(Xt2"2t ) + m3E(Xt"t),
E(XtXt 1"2t "t 1)
E(Xt2"4t ) = 2m4E(Xt2"2t ) + 2 m5E(Xt"t) + m6,
E(Xt"3t ) = m3E(Xt"t) + m4.
Finally, the values of E(XtXt 1) and E(Xt2Xt2 1) appear in terms of the mo-