Estimating the Codifference Function of Linear Time Series Models ...

where (d k ) T = [d k 1 ,dk 2 , . . . ,dk r ], and the elements of dk i , i = 1, . . .r are
d k i (1, 1) = (ReΦ(0, −s i; k)) −1
d k i (2, 2) = (Re Φ(s i , 0; k)) −1
d k i (3, 3) = (Re Φ(s i , −s i ; k)) −1 )
and equal to 0, otherwise. The asymptotic variance-covariance matrix is obtained from (28) as
(( ) ( ))
Re ˆτ
lim N cov ∗ (s, −s; p) Re ˆτ
N→∞ Im ˆτ ∗ ,
∗ (s, −s; q)
(s, −s; p) Im ˆτ ∗ = λL p
(s, −s; q)
2 V pqL q 2 λT (35)
where
V pq =
( V
RR
pq
V IR
pq
Vpq
RI
Vpq
II
) ( cov(ReZ
p
= lim N N , ReZq N ) cov(ReZp N , ImZq N ) )
N→∞ cov(ImZ p N , ReZq N ) cov(ImZp N , ImZq N )
(36)
The matrix V pq can be obtained by applying Theorem 1 and Remark 2.6. in Hesse (1990). Its
elements can be derived in a similar way as obtaining variance-covariance matrix in Theorem 1 of
Hesse (1990). This is possible, because it can be shown that all elements of V pq (in the form of sum
of the absolute components) are finite. Therefore, one can apply the property of the sample mean
of ergodic processes (e.g., Theorem 7.1.1. in Brockwell and Davis, 1987). Notice that here in particular,
we obtain all elements of V pq with respect to cov(ReZ p N , ImZq N ) and cov(ImZp N , ReZq N )
are zeros. The elements of V pq with respect to cov(ReZ p N , ReZq N ) and cov(ImZp N , ImZq N ) can
be shown to be finite using identities (29)-(30) and applying a similar approach as obtaining eq.
(21) and (23), and further applying Theorem A.1, or sometimes, eq.(2.7) in Kokoszka and Taqqu
(1994) together with the similar steps as the proof of Theorem A.1. However, we omit details.
Proposition B.2 Let X t , t ∈ Z be the moving average process of order m, X t = ∑ m
j=0 c jǫ t−j ,
satisfying conditions C1 and C2. Then for h ∈ {1, 2, . . .}, s ∈ R, s ≠ 0
[(
Re ˆτ ∗ (s, 0)
Im ˆτ ∗ (s, 0)
) (
Re ˆτ
, . . . ,
∗ (s, h)
Im ˆτ ∗ (s, h)
where M is the covariance matrix
)]
is AN
([(
τ(s, 0)
0
) (
τ(s, h)
, . . .,
0
)] )
, N −1 M
M = [ λL p 2 V pqL q 2 λT] p,q=0,...,h
and the matrices λ,L k 2 , k = p, q and V pq are as given in Proposition B.1 above.
Proof.To show this relation, define vectors {Y t } by
where
where for j = 1, . . .,r
Y T t = (Z t ,Z t+1 , . . . ,Z t+h )
X k j = ⎛
⎝
⎛
Z t+k = ⎜
⎝
X k 1
X k 2
.
X k r
⎞
⎟
⎠
exp(−is j X t )
exp(is j X t+k )
exp(is j (X t+k − X t ))
By definition, {Z t+k } is m+k-dependent sequence and therefore {Y t } is m+h-dependent sequence.
Next define
ζ T t = (ξ t , ξ t+1 , . . . , ξ t+h )
⎞
⎠
13