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

246 J. M. Rodríguez-Poo et al.
Fig. 9 Intradaily seasonal curves, and confidence bands, for in-sample (solid lines) and out-ofsample
(dashed lines). Price durations in the left box and volume durations in the right box
seasonality. Hence the spline will fail to fit the out-of-sample seasonality if its pattern
has changed. This is not the case of UniNW as it does not depend on parameters, other
than ˆϑ1n and ˆξn, to capture the seasonality.
The empirical evidence supports this intuition. Figure 9 shows the in-sample and outof-sample
intradaily seasonalities. We observe that the out-of-sample seasonal patterns
shift down compared with the in-sample patterns. This is especially visible for volume
durations. If we use BiSp, BiNW, or UniSp to evaluate the out-of-sample density forecast,
they all fail because none may capture changes in seasonality. By contrast, UniNW
adapts quickly to changes in the out-of-sample and this explains why the density forecast
produced by UniNW is better than for any other estimator.
4 Conclusions
We propose a component model for the analysis of financial durations. The components
are dynamics and seasonality. The latter is left unspecified and the former is assumed
to fall within the class of (Log-)ACD models. Joint estimation of the parameters of
interest and the smooth curve is performed through a local (quasi-)likelihood method.
The resulting nonparametric estimator of the seasonal component shows a closed form
expression.
Although the methodology is applied to intradaily seasonality, it could also be
used to measure relations between any other two variables. This is particularly useful
when the relation is nonlinear and we are not sure about the functional link. So, for
instance, the time-of-the-day may be replaced by some volatility measure, spread, or
any other microstructure variables. Alternatively, the nonparametric estimator could be
multivariate, including the time-of-the-day and any other variable of interest.
The model is applied to the price and volume duration processes of two stocks traded
on the NYSE. We show that the proposed method produces better predictions, in terms
of densities, since it adapts quickly to changes in the seasonal pattern.
Acknowledgements The authors acknowledge financial support from the Université catholique
de Louvain (project 11000131), the Institute of Statistics at the Université Catholique de Louvain,
the Dirección General de Investigación del Ministerio de Ciencia y Tecnología under research
grants BEC2001-1121 and SEJ2005-05549/ECON as well as the Spanish Ministry of Education
(project PB98-0140), respectively. Work supported in part by the European Community’s Human