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

with λ is as given as (27), and the elements of the matrix L k 2 and the covariance matrix V kk will
be given below. Let us denote
and
V RR
kk (i, j) = [cov(Re φ p(s i , k), Re φ q (s j , k)))] p,q=1,2,3
V II
kk(i, j) = [cov(Imφ p (s i , k), Im φ q (s j , k)))] p,q=1,2,3
as the (i, j)-th block elements of Vkk
RR and VII
kk
, respectively. Using identities (31)-(33) (and the
identities for imaginary part afterwards) in p.12, we can obtain their components as follows
cov(Re(φ 1 (s i , p)), Re(φ 1 (s j , q))) = cov(cos(−s i X t ), cos(−s j X t ))
= 1 2 {e−σα |s i+s j| α + e −σα |s i−s j| α } − e −σα (|s i| α +|s j| α )
cov(Re(φ 1 (s i , p)), Re(φ 2 (s j , q))) = cov(Re(φ 2 (s i , p)), Re(φ 1 (s j , q))) = cov(Re(φ 1 (s i , p)), Re(φ 1 (s j , q)))
cov(Re(φ 2 (s i , p)), Re(φ 2 (s j , q))) = cov(Re(φ 1 (s i , p)), Re(φ 1 (s j , q)))
cov(Re(φ 1 (s i , p)), Re(φ 3 (s j , q)))
= cov(cos(−s i X t ), cos(s j (X t+q − X t ))) + cov(cos(−s i X t+q ), cos(s j (X t+q − X t )))
= e −σα (|s j| α +|s i−s j| α) + e −σα (|s j| α +|s i+s j| α) − 2e −σα (|s i| α +|2s j| α )
cov(Re(φ 3 (s i , p)), Re(φ 1 (s j , q)))
= cov(cos(−s j X t ), cos(s i (X t+p − X t ))) + cov(cos(−s j X t+p ), cos(s i (X t+p − X t )))
= e −σα (|s i| α +|s i−s j| α) + e −σα (|s i| α +|s i+s j| α) − 2e −σα (|s j| α +|2s i| α )
cov(Re(φ 2 (s i , p)), Re(φ 3 (s j , q)))
= cov(cos(s i X t+q ), cos(s j (X t+q − X t ))) + cov(cos(s i X t+p ), cos(s j (X t+p+q − X t+p )))
= e −σα (|s j| α +|s i−s j| α) + e −σα (|s j| α +|s i+s j| α) − 2e −σα (|s i| α +|2s j| α )
cov(Re(φ 3 (s i , k)), Re(φ 2 (s j , k)))
= cov(cos(s j X t+k ), cos(s i (X t+k − X t ))) + cov(cos(s j X t+k ), cos(s i (X t+2k − X t+k )))
= e −σα (|s i| α +|s i−s j | α) + e −σα (|s i| α +|s i+s j| α) − 2e −σα (|s j| α +|2s i| α )
cov(Re(φ 3 (s i , p)), Re(φ 3 (s j , q)))
= cov(cos(s i (X t+p − X t )), cos(s j (X t+q − X t ))) + cov(cos(s i (X t+p+q − X t+q )), cos(s j (X t+q − X t )))
+ cov(cos(s i (X t+p − X t )), cos(s j (X t++p+q − X t+p ))) + c pq
Re
where
yielding for p = q
c pq
Re = ⎧
⎨
⎩
0 if p = q
cov(cos(s i (X t+q − X t+q−p )), cos(s j (X t+q − X t ))) if q > p
cov(cos(s i (X t+p − X t )), cos(s j (X t+p − X t+p−q ))) if p > q
cov(Re(φ 3 (s i , p)), Re(φ 3 (s j , q))) = 1 2 e−2σα |s i+s j | α + 1 2 e−2σα |s i−s j| α − 3e −σα (2|s i| α +|2s 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 )
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