Page 41 - Textos de Matemática Vol. 47

THE TAYLOR PROPERTY IN AR AND BILINEAR PROCESSES 31
We point out that this figure stresses the fact that the presence of the Taylor property is stronger in the Pareto case, that is, its occurrence is more evident when the kurtosis of the model increases.
3. The Taylor property in first-order non-negative bilinear models
We consider the simple bilinear diagonal model
Xt = Xt 1"t 1 + "t, (3.1)
where > 0 is a real parameter and " = ("t , t 2 Z) a sequence of non-negative i.i.d. random variables.
In order to assure the strict and weak stationarity of the processes X = (Xt, t 2 Z) and X2 = (Xt2, t 2 Z), we suppose that E(ln |"t|) and m8 exist and that 4 m4 < 1 , with mi = E("it), i 2 N (Gon¸calves, Martins and Mendes- Lopes, 2015).
The nth moment of Xt, nX 4, can be expressed as n ✓n◆
where
E(Xn"n) = 1 Xn ✓n◆ n i m E(Xn i"n i), tt 1 nmni=1i n+i t t
E(Xn) = n i m E(Xn i"n i), tiitt
i=0
n  4.
It is easy to verify that E(Xt"t) = m2/(1 m1). The values E(Xtn"nt ), n = 1, 2, 3, are obtained recursively by using the previous equation; and finally, we getE(Xtn),forn4.Wenotethat 4m4 <1implies| nmn|<1,n=1,2,3, by Schwarz’s inequality.
In this context, the Taylor property is only studied for h = 1, that is, by analyzing if ⇢X (1) > ⇢X2 (1), where ⇢X (1) and ⇢X2 (1) denote, respectively, the autocorrelations of lag 1 of the processes X and X2. It is enough to evaluate E(XtXt 1) and E(Xt2Xt2 1) in order to obtain these autocorrelations. Using (3.1) and the stationarity of the involved processes, we have
E(XtXt 1) = E(Xt2"t) + E(Xt 1"t)
= E( 2X2 "2 " +2 X " "2 +"3)+E(X " ).
t 1 t 1 t t 1 t 1 t t t 1 t Taking into account the independence of the random variables "t, t 2 Z,
and the strict stationarity of the related processes, we have E(X2 "2 " ) = 2 2 2 t 1 t 1 t
m1E(Xt "t ) and E(Xt 1"t 1"t ) = m2E(Xt"t). Then
E(XtXt 1) = 3m1E(Xt2"2t ) + 2 2m2E(Xt"t) + m1E(Xt) + m3.