recent developments in high frequency financial ... - Index of

234 J. M. Rodríguez-Poo et al.
As an example, set ϕ(u, v) = exp(u+v). Then, the quasi-likelihood function corresponds
to the log-likelihood function from an exponential distribution and a Log-ACD
representation, i.e.,
n�
�
�ψ(¯di−1, − ¯yi−1; ϑ1) + φϑ1 (t′ i−1 )� di
+
exp � ψ(¯di−1, ¯yi−1; ϑ1) + φϑ1 (t′ i−1 )�
�
,
i=1
and, after some derivations, the estimator (19) takes the explicit form
ˆφϑ1 (t′ ⎧
⎪⎨
1 � �
ni=1 t ′
0−t nh K
0 ) = log
⎪⎩
′ �
i di
h exp � ψ(¯di−1,¯yi−1;ϑ1) �
1 � �
ni=1 t ′
0−t nh K
′ ⎫
⎪⎬
�
i ⎪⎭
h
.
The above expressions are obtained by assuming a seasonal component for a given period
(e.g., daily). But it is also possible to extend this method to cover several seasonal effects.
For example, we may be interested in looking at whether the seasonal patterns for each
day of the week are different and test the differences using confidence bands. For sth
day of the week, we have for the exponential density and the Log-ACD representation,
ˆφs(t ′
⎧
⎪⎨
� 1 ⌊n/5⌋ �n nh j=1 i=1
0 ) = log
⎪⎩
K
�
t ′
0
−t ′ � �
i I ⌊ h
t−to
�
di + 1⌋=js
tc−to exp{ψ( ¯d i−1 ,¯y i−1 ;ϑ1)} � 1 ⌊n/5⌋ �n nh j=1 i=1 K � t0−t � �
i I ⌊ h
t−to
⎫
⎪⎬
�
+ 1⌋=js
⎪⎭
tc−to
for s = 1,...,5. I(·) is the indicator function and ⌊x⌋ is the integer part of x.
The following Theorem shows the equivalence of statistical results for making correct
inference (notation is simplified, as in Theorem 1, and the proof is given in theAppendix):
Theorem 2:
Appendix then,
Under conditions (A.1) to (A.5), and (B.1) to (B.6), provided in the
√ � � �
n ˆϑ1n − ϑ1 →d N 0,� −1
�
, ϑ1
(20)
where
and
where
�ϑ1 =−E
V � t ′ 0 ; η� =
�
∂ 2
∂ϑ1∂ϑ T 1
log q � d,ϕ � ψ(¯d, ¯y; ϑ1), φ(t ′ ) ���
,
√ �
nh ˆφ ˆϑ1 (t′ 0 ) − φ(t′ 0 )
�
→d N � 0,V � t ′ 0 ; η�� , (21)
� K 2 (u)du
×
f(t ′ 0 )
�� d−ϕ(ψ(ϑ1),φ) ∂
E V (ϕ(ψ(ϑ1),φ)) ∂φϕ (ψ(ϑ1),
�2 ��t
φ) ′ = t ′ �
0
�
∂
E × ∂φϕ (ψ(ϑ1), φ) 2 � � ,
2
�t ′ = t ′
0
1
V0(ϕ(ψ(ϑ1),φ))
and f(t ′ 0 ) is the marginal density function of t′ ,asn tends to infinity.