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Semiparametric estimation for financial durations 233
family that we use. For the exponential distribution the link function is
1
θ =−
ϕ � ψ(¯di−1, ¯yi−1; ϑ1), φ(t ′ �.
i−1 ; ϑ2)
Since under this distribution E � di| ¯di−1, ¯yi−1,t ′ � �
i−1 =−θ−1 and V di| ¯di−1, ¯yi−1,
t ′ �
i−1 = θ 2 , Eq. (18) specifies the first order conditions for the maximization of the
log-likelihood function for exponentially distributed random variables.
In order to estimate the parameters of interest we maximize the quasi-log-likelihood
function
Qn (d,ϕ) =
n�
i=1
log q � di,ϕ � ψ(¯di−1, ¯yi−1; ϑ1), φ(t ′ ��
i−1 ; ϑ2)
with respect to ϑ1 and ϑ2. As already indicated in the likelihood context, if the seasonal
component is assumed to fall within the class of known parametric function, φ(ti−1; ϑ2),
the properties of the QML estimators of ϑ1 and ϑ2 are well known (see Engle and Russell
(1998) and Engle (2000)). But if the seasonal component is approximated nonparametrically,
then standard quasi-likelihood arguments do not hold. For the standard i.i.d. data
case, Severini and Staniswalis (1994) and Fan et al. (1995) propose consistent estimators
of both parametric and nonparametric parts. The statistical results presented in these
papers do not apply directly in our case since they assume independence of observations
but at the end of the subsection equivalent statistical results are shown for the dependent
case.
As in the likelihood case, let us define ˆφϑ1 (t′ 0 ), for fixed values of ϑ1, as the solution
to the following (smoothed) optimization problem
ˆφϑ1 (t′ 1
0 ) = arg sup
φ∈� nh
n�
K
i=1
�
t ′
0 − t ′ �
i
log q
h
� di,ϕ � ψ(¯di−1, ¯yi−1; ϑ1), φ ��
for t ′ 0 ∈[to,tc]. Then ˆφϑ1 (t′ 0 ) must fulfill the following first order condition
1
nh
n�
K
i=1
�
t ′
0 − t ′ �
i ∂
� �
log q di,ϕ ψ(¯di−1, ¯yi−1; ϑ1), ˆφϑ1
h ∂φ (t′ 0 )
��
= 0.
The estimator ofϑ1 is obtained as the solution to the following (unsmoothed) optimization
problem
ˆϑ1n = arg sup
n�
ϑ1∈�
i=1
� �
log q di,ϕ ψ(¯di−1, ¯yi−1; ϑ1), ˆφϑ1 (t′ i−1 )
��
,
and ˆϑ1n must fulfill the following first order condition
n�
i=1
∂
∂ϑ1
� �
log q di,ϕ ψ(¯di−1, ¯yi−1; ˆϑ1n), ˆφ ˆϑ1n (t′ i−1 )
��
= 0.
(19)