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Semiparametric estimation for financial durations 231
Assuming that ϕ(u, v) = exp(u+v), the seasonal component is estimated through
the following expression
ˆφϑ1,ξ(t ′ 1
0 ) =
γν log
⎧
⎪⎨
⎪⎩
1
nh
� ni=1 K
�
t ′
0−t ′ �
i
h
�
1 �ni=1 nh K
di
exp � ψ(¯di−1,¯yi−1;ϑ1) �
�
t ′
0−t ′ �
i
h
� γ
⎫
⎪⎬
. (13)
⎪⎭
This closed form estimator can be plugged into (11), reducing the simultaneous
optimization problem to a standard ML procedure. Two useful particular cases
are nested in this estimator. If ν = 1, we obtain the nonparametric estimator
when p(·) is a Weibull density function. And ν = γ = 1 corresponds to the
exponential density. In all cases, the nonparametric seasonal curve is estimated
by a transformation of the Nadaraya–Watson nonparametric regression estimator
of the duration—adjusted by the dynamic component—on the time-of-the-day at
time t ′ 0 .
The results available in the earlier literature were obtained for independent
observations and hence do not hold for tick-by-tick data. The following Theorem
shows the equivalent statistical results that allows us to make correct inference
about the unknown parameters of the Log-ACD model (the proof is given in the
Appendix and Eq. (8) is simplified to log p (d; φ,η)):
Theorem 1: Let η = (ϑ1 ξ) T , and ˆηn be the vector of corresponding parametric
estimates. Under conditions (L.1), (L.2), (A.1) to (A.3), (B.1), and (B.3) to (B.6)
stated in the Appendix
where
and
√ �
n ˆηn − η � �
→d N 0,I −1
�
η (φ,η) , (14)
� �
∂
∂
Iη (φ,η) = E log p (d; φ,η) log p (d; φ,η)
∂η ∂ηT � �
∂
∂
− E log p (d; φ,η) log p (d; φ,η)
∂η ∂φ
�
∂2 �−1
× E log p (d; φ,η)
∂φ2 � �
∂
∂
× E log p (d; φ,η) log p (d; φ,η)
∂φ ∂ηT √ �
nh ˆφ ˆηn (t′ 0 ) − φ(t′ 0 )
�
→d N � 0,V � t ′ 0 ,η�� , (15)