Page 100 - Textos de Matemática Vol. 47

90 I. PEREIRA, M. SCOTTO, AND R. NICOLETTE
are obtained by maximizing the conditional log-likelihood function
⇣⌘
↵ˆ(1) ↵(1) 11
PP
N 1 T (1) (1) (2) (2)
`(✓) =
From the partial derivatives of first-order we obtain the set of systems
s=0 j=1
8
ln pj xj 1+sT,xj+sT,↵j Ij +↵j Ij , j .
=0 ,
> > > >
N 1
I(k) X (k)
j (x x↵) ↵(k)(1 ↵(k)) j+sT j 1+sT j
> <
p
⇣j j s=0 ⌘ x ,x 1,↵(1)I(1) +↵(2)I(2),
> >
j
j j 1+sT j+sT j j j j j p ⇣x ,x ,↵(1)I(1) +↵(2)I(2), ⌘
>
j j 1+sT j+sT j j j j j
> >:
P pj
s=0 jjjj
N 1 ⇣ (1) (1) (2) (2)
⌘
xj 1+sT ,xj+sT 1,↵j Ij
+↵j Ij , j pj ⇣xj 1+sT ,xj+sT ,↵(1)I(1)+↵(2)I(2), j ⌘
= N
for k = 1,2 and j = 1,...,T. In order to solve those systems numerical proce- dures have to be employed. Note, however, that the CML-estimates for the ’s, are readily available from those for the ↵’s through the following expression
1 N 1 X⇣⌘
s=0
xj+sT ↵ˆ(k) xj 1+sT , j=1,...,T. N j,CML
ˆj,CML =
The following result establishes the consistency and the asymptotic distribution
of the CML-estimators.
Theorem 3.2. Let (Xt ) be the PSETINAR(2; 1, 1)T model in (1.1). The CML-
estimators are asymptotically normal, i.e,
↵ˆ ( 2 ) ↵ ( 2 ) 2
11 M0...0
3 ˆ1 1 6 1 7
pn . !d N(0,I 1), where I=6 0 M2 ... 0 7 , . 4 . . ... . 5
↵ˆ ( 1 ) ↵ ( 1 )
T T 00...MT
ˆT T
↵ˆ ( 2 ) ↵ ( 2 ) TT