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THE TAYLOR PROPERTY IN AR AND BILINEAR PROCESSES 33
distributions, according to the values of the kurtosis of the process X, that is also, in this last case, a function of ✓.
Generator process with uniform distribution in ]0,↵[
Inthiscase,thecondition 4m4 <1isequivalentto0<r<p4 5'1.495. For these values of r, the graphs of the di↵erence ⇢X(1) ⇢X2(1) and of KX(r) are presented in Figure 4.
15
10
0.04
0.02
- 0.02
0.2 0.4 0.6 0.8
1.0 1.2 1.4
0.2 0.4
0.6 0.8
1.0 1.2 1.4
Figure 4. Graphs for KX (r) (on the left) and ⇢X (1) ⇢ 2 (1)
p (ontheright)inthecase"t ⇠U(]0,↵[),0<r< 4 5.
X
From Figure 4 (on the right), we can see that the Taylor property is present for values of r in the interval ]1.1868987, p4 5[. So, for a fixed ↵, the Taylor property is achieved for parameterizations of model (3.1) such that
2 #1.1868987, p4 5". ↵↵
From Figure 4 (on the left), we observe that the kurtosis of this model is an increasing function of r and that the model is leptokurtic for r > 0.8 (approx.). We also observe that the Taylor property occurs for large values of the kurtosis, namely for KX (r) > 7.403 (' KX (1.1868987)).
Generator process with exponential distribution with mean ↵ 41
Thecondition m4 <1isnowequivalentto0<r< p4 24 '0.4518.
In this case, model (3.1) presents the Taylor property for parameterizations
such that
2 0, 0.0695566 [ 0.1437879, p1 . ↵ ↵424↵
