Page 98 - Textos de Matemática Vol. 47

88 I. PEREIRA, M. SCOTTO, AND R. NICOLETTE
we obtain the following set of CLS-estimators
8
>
>
> > ↵ˆ ( 1 )
> j,MQC
> j,MQC P
=
◆2 Xj+sT 1Ij
N
Xj+sT Xj+sT 1Ij Xj+sT
s = 0 s = 0 s = 0
N 1 (1) N 1 N 1 PPP
(1) Xj+sT 1Ij
> >
N 1 ✓N 1
N P X2 I(1) P Xj+sT 1I(1)
s=0 j+sT 1 j s=0 j N 1 (2) N 1 N 1
><P PP > N Xj+sT Xj+sT 1Ij Xj+sT >↵ˆ(2) = s=0 s=0 s=0
(2)
>: ˆj,MQC =N 1 PXj+sT ↵(1) PXj+sT 1I(1) ↵(2) PXj+sT 1I(2) jjjj
s=0 s=0 s=0
for j = 1,...,T. The consistency and asymptotic distribution of the CLS-
estimators are established in the result given below.
Theorem 3.1. The CLS-estimators are strongly consistent and asymptotically
◆2 >✓◆
✓P
,
N 1
> (2) (2)
s=0 s=0
> N 1 N 1 N 1
> >
N 1
(N Xj2+sT 1Ij Xj+sT 1Ij
normal, i.e.,
pˆd 1 1
n(✓ ✓) !N(0,V WV ), (3.2)
where V and W are square matrices of order 3T defined by blocks of 3 ⇥ 3 given by
26 : 1 0 . . . 0 37 26 ⌦ : 1 0 . . . 0 37
6 0 :2 ... 0 7 6 0 ⌦:2 ... 0 7
V=64 . . ... . 75 and W=64 . . ... . 75, (3.3)
where
0 0 ... :T 0 0 ... ⌦:T (k,l):j := E @ gj(✓j,Xj+sT 1) @ gj(✓j,Xj+sT 1) ,
@✓k:j @✓l:j
:=EU2 @ g(✓,X ) @ g(✓,X
) ,
k,l 2 {1,2,3}; j = 1,...,T and ✓j := (✓1:j,✓2:j,✓3:j) ⌘ ⇣↵(1),↵(2), j⌘, are
⌦
are the elements of the matrices :j and ⌦:j , j = 1, . . . , T , respectively, with
(k,l):j j+sT @✓a:j j j j+sT 1 @✓l:j j j j+sT 1 jj
the parameters associated to j-th period.
Proof. Consistency and asymptotic normality can be proved using the results of Klimko and Nelson (1978, sec.3). It is easily checked that all regularity con- ditions by Klimko and Nelson (1978, p. 634) are satisfied, and thus, by their