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

and for p ≠ q
cov(Re(φ 3 (s i , p)), Re(φ 3 (s j , q))) = 2e −σα (|s i| α +|s j| α +|s i−s j| α )
+ 2e −σα (|s i| α +|s j| α +|s i+s j | α) − 4e −σα (2|s i| α +|2s j| α )
cov(Im(φ 1 (s i , p)), Im(φ 1 (s j , q))) = cov(sin(−s i X t ), sin(−s j X t )) = 1 |s i−s j| α
2 {e−σα − e −σα |s i+s j | α }
cov(Im(φ 1 (s i , p)), Im(φ 2 (s j , q))) = cov(Im(φ 2 (s i , p)), Im(φ 1 (s j , q)))
= −cov(Im(φ 1 (s i , p)), Im(φ 1 (s j , q)))
cov(Im(φ 2 (s i , p)), Im(φ 2 (s j , q))) = cov(Im(φ 1 (s i , p)), Im(φ 1 (s j , q)))
cov(Im(φ 3 (s i , p)), Im(φ 3 (s j , q)))
= cov(sin(s i (X t+p − X t )), sin(s j (X t+q − X t ))) + cov(sin(s i (X t+p+q − X t+q )), sin(s j (X t+q − X t )))
+ cov(sin(s i (X t+p+q − X t+p )), sin(s j (X t+p − X t ))) + c pq
Im
where
yielding for p = q
c pq
Im = ⎧
⎨
⎩
cov(Im(φ 3 (s i , k)), Im(φ 3 (s j , k)))
0 if p = q
cov(sin(s i (X t+q − X t+q−p )), sin(s j (X t+q − X t ))) if q > p
cov(sin(s i (X t+p − X t )), sin(s j (X t+p − X t+p−q ))) if p > q
= 1 2 e−2σα |s i−s j| α − 1 2 e−2σα |s i+s j| α + e −σα (|s i| α +|s j| α +|s i+s j | α) − e −σα (|s i| α +|s j| α +|s i−s j | α )
and cov(Im(φ 3 (s i , k)), Im(φ 3 (s j , k))) = 0 for p ≠ q. The other elements are all zeros. The elements
of L k 2 are as given in (34), where the elements of d k i , i = 1, . . .r are
As from (45) we obtain
and
then the (i,j)-th element of m RR
kk
d k i (1, 1) = (Re Φ(0, −s i; k)) −1 = e σα |s i| α
d k i (2, 2) = (Re Φ(s i , 0; k)) −1 = e σα |s i| α
d k i (3, 3) = (Re Φ(s i , −s i ; k)) −1 = e 2σα |s i| α )
m RR
kk = cov(Re ˆτ(s, k), Re ˆτ(s, k)) = (I r ⊗ λ 1 )d k V RR
kk dk (I r ⊗ λ T 1 )
m II
kk = cov(Im ˆτ(s, k), Im ˆτ(s, k)) = (I r ⊗ λ 1 )d k V RR
kk d k (I r ⊗ λ T 1 )
and mII
kk
is obtained from
m RR
kk (i, j) = λ 1 d k i V RR
kk (i, j)d k jλ T 1
and
m II
kk (i, j) = λ 1d k i V kk II (i, j)dk j λT 1
which therefore after a simple algebra, we obtain
m RR
kk (i, j) = eσα (|s i| α +|s j| α −|s i−s j| α ) { 1
2 eσα (|s i| α +|s j | α −|s i−s j| α) − 1
+ e σα (|s i| α +|s j| α −|s i+s j| α ) { 1
2 eσα (|s i| α +|s j| α −|s i+s j | α) − 1
}
}
+ 1
{
m II
kk (i, j) = (|s i| α +|s j| α −|s i−s j| α ) 1 (|s i| α +|s j| α −|s i−s j | α) }
eσα 2 eσα − 1
{ }
+ e σα (|s i| α +|s j| α −|s i+s j| α )
1 − 1 (|s i| α +|s j| α −|s i+s j | α )
2 eσα
The required result follows directly from (44).
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