Page 99 - Textos de Matemática Vol. 47

PERIODIC THRESHOLD PROCESSES 89
Theorem 3.1 it follows that the CLS-estimators are strongly consistent. Fur- thermore, in proving asymptotic normality we have to check first conditions (A)-(C) in Monteiro et al. (2012, p. 2725). To this extent, note first that condi- tions (A) and (B) follow easily by the arguments given in Monteiro et al. (2012). Finally, note that each block :j of matrix V in (3.3) is defined as
where
:j
=
" q(1)m(1) j j,2
0
0 q(1)u(1) # j j
q(2) m(2) q(2) u(2) , j j,2 j j
q(1) u(1)
jj jj
q(2) u(2) 1
u(1) :=E[Xj 1+sT|Xj 1+sT d rj] u(2) :=E[Xj 1+sT|Xj 1+sT d >rj]
jhijhi m(1) := E Xi |Xj 1+sT d  rj m(2) := E Xi |Xj 1+sT d > rj
j,i j 1+sT j,i j 1+sT q(1) := P [Xj+sT d  rj] q(1) := P [Xj+sT d > rj].
jj
Noticing that det| :j| =6 0, 8j = 1,...,T lead us to conclude that V is in- vertible and condition (C) is thus fulfilled. Thus, Theorem 3.2 of Klimko and Nelson (1978) is thereby satisfied implying that the CLS-estimators are asymp- totically normal. This concludes the proof. ⇤
3.2. Conditional maximum likelihood estimators. For a fixed value of x0, the conditional likelihood function for the PSETINAR(2; 1, 1)T takes the form
N 1 T QQ
s=0j=1 ⇣
N 1 T QQ
s=0 j=1
L(✓) := =
Pj (Xj+sT = xj+sT |Xj 1+sT = xj 1+sT )
⌘ ⇣⌘ x m
(1) (1) (2) (2)
pj xj 1+sT,xj+sT,↵j Ij +↵j Ij , j
with
= pj (xj 1+sT , xj+sT )(1) + pj (xj 1+sT , xj+sT )(2) andM⇤ :=min(xj 1+sT,xj+sT).
The CML-estimators
✓ˆCML :=(↵ˆ(1) ,↵ˆ(2) , ˆ1,CML,...,↵ˆ(1) ,↵ˆ(2) , ˆT,CML), 1,CML 1,CML T,CML T,CML
pj ⇣xj 1+sT,xj+sT,↵(1)I(1) +↵(2)I(2), j⌘= jj jj
M⇤ 2
j XX xj 1+sT (k)m (k) (xj 1+sT m) j+sT (k)
Cm ↵j 1 ↵j (xj+sT m)!Ij
= e
⌘ pj xj 1+sT , xj+sT , ↵(1)I(1), j + pj xj 1+sT , xj+sT , ↵(2)I(2), j
⇣ m=0k=1 ⌘
jj jj
⇣ ⌘