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Semiparametric estimation for financial durations 235
2.2 Predictibility
We analyze the predictibility and specification of the model via density forecast, introduced
by Diebold et al. (1998) in the context of GARCH models and extensively used
by, among others, Bauwens et al. (2004) to compare different financial duration models.
Density forecast is more accurate than point, or even interval prediction, since it relies
on the forecasting performance of all the moments. The behavior of the out-of-sample
density function is evaluated via the probability integral transform. If the prediction is
correct, the probability integral transform over the out-of-sample should be i.i.d and
uniformly distributed. �
�
Let p
�
dj
� ¯dj−1, ¯yj−1;�ϑ1n, �φ�ϑ1n,�ξn
�
t ′ � �
j−1 , �ξn
for j = n + 1,...,m be a
sequence of one-step-ahead density forecasts given by the estimated model, and let
f � �
dj| ¯dj−1, ¯yj−1 be the sequence of densities defining the data generating process
of the duration process dj . n and m are the number of observations in-sample and
out-of-sample, respectively. Diebold et al. (1998) show that if the density is correctly
specified
� �
�
p dj � ¯dj−1,
�
¯yj−1;�ϑ1n, �φ�ϑ1n,�ξn
t ′ � �
j−1 ,�ξn = f � �
dj| ¯dj−1, ¯yj−1 .
To test this equality we use the probability integral transform
� dj
zj =
−∞
� �
�
p u
� ¯dj−1, ¯yj−1;�ϑ1n, �φ�ϑ1n,�ξn
�
t ′ � �
j−1 ,�ξn du.
If the one-step-ahead density forecast equals the density defining the data generating
process, z must be independent and uniformly distributed. This happens if (1) the predicted
density is correctly specified, (2) the dynamics are well captured, and (3) the
estimated seasonal component fits the out-of-sample seasonal pattern. If any of these
three elements fails, the integral probability transform is not independent and uniformly
distributed. Uniformity can be tested by using histograms based on the computed z
sequence. If the density is correctly specified, the histogram should be flat. Additionally,
the Spearman’s ρ of various centered moments of the z sequence may reveal some
dependency in z.
Last, a note on how we predict seasonality: Assuming a generalized gamma density,
the seasonal estimator is (13). It depends on (1) the current time-of-the-dayt ′
0
and duration
di that come from the same arrival times—see Eq. (5)—and (2) the estimated parameters
ˆϑ1, ˆξ. When forecasting, the parameters are fixed as they have been estimated using the
in-sample. But durations and the time-of-the-day are those of the out-of-sample. All this
translates into a forecasted seasonality that changes with the out-of-sample information
and adapts to changes in the intensity of the arrival times.
3 Illustration
3.1 Data and transformations
In this section we illustrate the method estimating the model for two duration processes—
price and volume—pertaining to two different stocks traded at NYSE. A price duration is
the minimum time interval required to witness a cumulative price change greater than a