Page 96 - Textos de Matemática Vol. 47

86 I. PEREIRA, M. SCOTTO, AND R. NICOLETTE
Note that, A Y is a T -dimensional random vector with i-th component
T 1 0i1
[A Y]i = X0 Xj+tT +@Y↵jA XT+tT,
j=1 j=1
for i = 1, . . . , T . The components of B Z can be defined similarly.
The rest of the paper is organized as follows: In Section 2, we demonstrate the existence of a strictly ciclostationary PSETINAR(2; 1, 1)T process satisfying (1.5). Furthermore, the expression for the periodic mean is given. Parameter estimation is covered in Section 3. Finally, forecasting is discussed in Section 4.
2. Some properties of the PSETINAR model with two regimes
Let (Xt) be the PSETINAR(2; 1, 1)T process defined in (1.2). We first prove
thatthereexistsastrictlyciclostationaryPSETINAR(2;1,1)T processsatisfying
(1.2). Note that, since ↵(k) 2 (0,1) for j = 1,...,T and k = 1,2, and that j
P(Z(k) = 0) 2 (0,1), for k = 1,2, it follows by Lemma 3 in Franke and j +sT
Subba Rao (1995) that any solution (Yt) of (1.5) is an irreducible and aperiodic Markov chain on N0. Thus, the existence of a ciclostationary solution of (1.5) relies upon the largest eigenvalue of the A matrix in (1.5). The result is quoted below.
Proposition 2.1. Let (Yt) be the PSETINAR(2; 1, 1)T process defined in (1.5). If E||Mt|| < +1 and if the largest eigenvalue, say ⌘, of A is less than one, then there exists a strictly ciclostationary PSETINAR(2; 1, 1)T process satisfying (1.5).
Proof. Proposition B in Dion et al. (1995, p. 126) allows us to conclude that ↵(1)I(1) + ↵(2)I(2) < 1, j = 1, . . . , T , ⌘ < 1. (2.1)
Conditions in (2.1) imply that all roots of the characteristic polynomial of A lie inside the unit circle. Furthermore, if E||Zt|| < 1 it follows by Theorem 1 in Franke and Subba Rao (1995) that there exists a strictly ciclostationary PSETINAR(2; 1, 1)T process satisfying (1.5). ⇤
Without employing any distributional assumption on the periodic sequences Z(1) and Z(2), the periodic mean of the process is given in the next result. For
tt
jj jj
simplicity in notation we define
u(k1,k2,...,kj) :=EhXsT|X1+sT d 2r(k1),X2+sT d 2r(k2),...,Xj+sT d 2r(kj)i, 1,2,...,j 1 2 j
p(k1,k2,...,kj) :=PhX1+sT d 2r(k1),X2+sT d 2r(k2),...,Xj+sT d 2r(kj)i, 1,2,...,j 1 2 j