Page 45 - Textos de Matemática Vol. 47

0 40
0 0.06
D
K
10 9
THE TAYLOR PROPERTY IN AR AND BILINEAR PROCESSES 35
0.8
r
0.8 r
0 9
Θ
Θ
50
50
Figure 6. Graphs for K = KX(r) (on the left) and D=⇢X(1) ⇢X2(1) (on the right) in the case "t ⇠ Par(↵,✓), 0 < r < 0.85, 9  ✓  50.
As can be seen in this figure, the Taylor property is now achieved for all considered parameterizations of model (3.1) and the process X is always lep- tokurtic since KX(r) > 3. These graphical representations also suggest that the presence of the Taylor property is stronger for higher values of the kurtosis of the process X. In fact, as function of ✓, the di↵erence ⇢X (1) ⇢X2 (1) seems to increase when KX(r) increases, for all values of r that satisfy the condition 4m4 < 1. This situation strongly contributes to conjecture that the Taylor property and leptokurtosis are highly related in time series.
4. Conclusion
The studies here presented confirm that linear and bilinear models may mimic the Taylor e↵ect. We note that in the linear case we were able to study the subject in a more complete way than in the bilinear class. In fact, a nec- essary and su cient condition assuring the presence of the Taylor property in the linear class for all lag h in the functions ⇢|X| and ⇢X2 was derived and we were able to conclude its presence in all models with symmetric or right-skewed distributed generator processes.
New examples were studied and we still point out the strong relation between the Taylor property and the kurtosis of the process. In fact, in the platykurtic cases studied, the property is not present or is present in a very soft way, becoming more visible as the kurtosis goes away from the reference value of 3. In these cases, the presence of the property is not significant, in the sense