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

(and similarly for the imaginary part), where φ ∗ (u, v; j) = N −1 ∑ N
t=1 exp(i(uX t+j+vX t )), φ(u, v; j)=
(N − j) −1 ∑ N−j
t=1 exp(i(uX t+j + vX t )) and λ 2 = [ −1 1 1 ] . The required result then follows
from Slutzky’s theorem (e.g., Theorem 5.1.1. in Lehmann, 1999).
Simple algebra gives, for 0 ≤ j ≤ h,
⎛
N 1/2 E
∣ λ 2
⎝ Re ⎞ ⎛
φ∗ (s k , −s k ; j)
Reφ ∗ (s k , 0; j) ⎠ − λ 2
⎝ Re φ(s ⎞
k, −s k ; j)
Re φ(s k , 0; j) ⎠
Re φ ∗ (0, −s k ; j) Reφ(0, −s k ; j) ∣
⎛ j
∑
1 N (N−j) N t=1
= N 1/2 E
λ ⎜
cos(is k(X t+j − X t )) − 1 ∑ N
N−j t=N−j+1 cos(is(X ⎞
t+j − X t ))
j
∑
1 N 2 ⎝ (N−j) N t=1 cos(is kX t+j ) − 1 ∑ N
N−j t=N−j+1 cos(is ⎟
kX t+j ) ⎠
∣
j 1
∑ N
(N−j) N t=1 cos(−is kX t )) − 1 ∑ N
N−j t=N−j+1 cos(−is kX t ) ∣
≤ 6j(N − j) −1/2 ( N
N−j )1/2
The required result is obtained from 3j(N − j) −1/2 → 0 and N/(N − j) → 1 as N → ∞. Using
the same arguments, similar results can be obtained for the imaginary part. The conclusion of the
theorem then follows from Proposition B.3 above.
Proof of Theorem 2.2. Let g(·) be the function from R 2r×(h+1) to R 2r×h defined by
( [( ) ( ) ( )] ) T
ˆτ(s, 0) Re ˆτ(s, 1) Re ˆτ(s, h)
g
,
, . . . ,
0 Im ˆτ(s, 1) Im ˆτ(s, h)
[( ) ( ) ( )] T
Re Î(s, 1) Re Î(s, 2) Re Î(s, h)
=
Im Î(s, 1) ,
Im Î(s, 2) , . . .,
Im Î(s, h)
where for 0 < j ≤ h and ˆτ(0) ≠ 0, we have Re Î(s Re ˆτ(si,−si;j)
i, −s i ; j) =
ˆτ(s i,−s i;0)
and ImÎ(s i, −s i ; j) =
Im ˆτ(s i,−s i;j)
ˆτ(s i,−s i;0)
, for i = 1, . . .,r. By applying delta method and Theorem B.4 above, we can show
that
[( ) ( ) ( )] T
Re Î(s, 1) Re Î(s, 2) Re Î(s, h)
Im Î(s, 1) ,
Im Î(s, 2) , . . . ,
Im Î(s, h)
is asymptotically normal distributed with mean
( [( ) (
τ(s, 0) τ(s, 1)
g
,
0 0
[( ) ( I(1) I(2)
= ,
0 0
)
, . . . ,
) (
τ(s, h)
, . . . ,
0
( I(h)
0
)] T
)] T
)
and variance N −1 DMD T . Here the matrix M is as given in Proposition B.2 and D is the Jacobian
matrix of g(·). To obtain the elements of matrix D, we proceed as follows. First, note that the
codifference function at lag 0 is a real-valued function. Therefore, for 0 ≤ j ≤ h, and τ(0) ≠ 0,
we obtain Re I(j) =
Jacobian matrix D as
where
Re τ(j)
τ(0)
= I(j) and ImI(j) =
Im τ(j)
τ(0)
⎡
⎤
D 11 D 12 0 . . . 0
D 21 0 D 23 0
D = ⎢
⎣
.
.
.
. .. .
⎥
⎦
D h1 0 0 . . . D h(h+1)
[ ]
D
11
D l1 =
l1
0 r
0 r 0 r
= 0. It is straightforward to obtain the
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