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Semiparametric estimation for financial durations 243
knots are chosen ad hoc). For UniNW and UniSp we show the seasonal curve ˆφϑ1,ξ(t ′ 0 )
at a different scale than the other seasonal estimators. It is not possible to present them
at the same scale since it is a logarithmic function, see Eq. (13), and the argument of the
function is not the observed duration but the duration adjusted by the dynamics.
3.4 Predictibility
So far estimation results dovetail with the theory. We now check how well they predict the
probability distribution. Since density forecast is strongly dependent on the distributional
assumption, we only show results for ML (the exponential density does it worse than
the generalized gamma). Another reason for sticking to ML is because we want to study
the predictive capabilities of the model for different estimators of seasonality. Since the
density and the parametric component are the same, differences in density forecast may
exclusively come from the different specification of the seasonal component.
If εj ∼ GG � �
1, ˆγn, ˆνn then
� � � � �
dj ∼ GG ϕ ψ ¯dj−1, ¯yj−1; ˆϑ1n , ˆφ ˆϑ1n,ˆξn
t ′ �� �
−1
j−1 , ˆγn, ˆνn ,
for UniNW and UniSp. For the two-step methods, since ˜dj = dj ˆφ
�
t ′ �−1 j−1 , the seasonal
curve is outside the function ϕ(·), that is
or
˜dj =
ˆφ
dj
�
t ′ j−1
� � � �� �
−1
� ∼ GG ϕ ψ ¯dj−1, ¯yj−1; ˆϑ1n , ˆγn, ˆνn
� � � ��−1 �
dj ∼ GG ϕ ψ ¯dj−1, ¯yj−1; ˆϑ1n ˆφ t ′ � �
−1
j−1 , ˆγn, ˆνn .
Figures 7 and 8 show the density forecast results. The histograms for UniNW are
significantly better than for any other estimator. Although formal tests can be performed
to assess the accuracy of this statement, visual inspection of the figures reveals that
estimation in two steps (BiSp and BiNW) results in misspecification in the conditional
distribution of the durations—in line with Bauwens et al. (2004). Moreover, estimation in
one step with splines, UniSp, leads to similar results as the two step estimators. This fits
with the conclusions on estimation results that found similar estimates under BiSp and
UniSp. Second, for volume durations none of the specifications correctly captures the
dynamic component. This is expected as the dynamics are modeled in the same way for
all specifications and Bauwens et al. (2004) already show that a simple Log-ACD(1,1)
is unable to fit the dynamics.
The better performance of UniNW is a result of its flexibility, as already mentioned.
UniNW adapts to changes in the seasonal pattern. We can interpret the seasonal estimator
UniNW as a time-varying intercept that adapts the conditional expectation of the durations
to changes in seasonality. This is because UniNW is a pure nonparametric estimator
and follows the principle of let the data speak. This is not the case of the polynomial
spline (22). In fact, Eq. (22) not only depends implicitly on ˆϑ1n and ˆξn but also depends
on ˆθn and ˆδn. And these parameters are estimated using the in-sample to fit the in-sample