Page 97 - Textos de Matemática Vol. 47

PERIODIC THRESHOLD PROCESSES 87
= {1, 2}(, where rj denotes the regime corresponding to the
the process takes the form
for k1, k2, . . . , kj period j, i.e.,
, t2N0. (2.2) Lemma 2.2. Let (Xt ) be the PSETINAR(2; 1, 1)T process in (1.5). The mean of
r(1), Xj+sT d rj rj = j
r(2), Xj+sT d >rj j
P2 P2 P2 h (k1) (k2) (kj) (k1,k2,...,kj) ! (k1,k2,...,kj)i E[Xt] = ··· ↵1 ↵2 ···↵j ⇥u1,2,...,j ⇥p1,2,...,j
fort=j+sT,j=1,...,T ands2N0.
3. Parameters estimation
In this section, we consider the parameter estimation of the PSETINAR(2; 1, 1)T process. In particular, the conditional least squares (CLS) and conditional max- imum likelihood methods are adopted. For this purpose, let (X1, . . . , Xn) be a sequence of r.v’s satisfying (1.4) and denote by
✓ := (↵(1),↵(2), ,...,↵(1),↵(2), ), 111TTT
the vector of unkown parameters. Recall that Rt is assumed to be known. 3.1. Conditional least squares estimators. The CLS-estimators
,↵ˆ(2) , ˆT,CLS), T,CLS
k1 =1k2 =1 kj =1
Pj P2 P2 ( k l + 1 ) ( k j ) k l + 1 , . . . , k j
+ l ··· ↵l+1 ···↵j ⇥pl+1,...,j , l=1 kl+1 =1 kj =1
✓ˆCLS := (↵ˆ(1) ,↵ˆ(2) , ˆ1,CLS,...,↵ˆ(1)
1,CLS 1,CLS
are obtained by minimizing the expression
T,CLS
◆2
with N and T denoting the number of complete cycles and number of periods,
N 1 T ✓
Q(✓):= XX Xj+sT gj(✓j,Xj+sT 1)
s=0 j=1
respectively. Moreover, ✓j := ⇣↵(1),↵(2), j⌘ and the function gj takes the form jj
gj(✓j,Xj+sT 1)=↵(1)Xj+sT 1I(1) +↵(2)Xj+sT 1I(2) + j. jjjj
Solving the systems of 8the form @Q
> > > >: ( 1 ) = 0 >< @ ↵ j
> @↵j
@Q = 0, (2)
j = 1,...,T,
(3.1)
@Q =0 @ j