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

Semiparametric estimation for financial durations 227
likelihood approach (see Severini and Wong (1992) and Severini and Staniswalis
(1994)).
In this paper we propose a new method to jointly estimate the parametric
dynamic and nonparametric seasonal components in an ACD framework. Estimation
is based on generalized profile likelihood techniques. To make inference
on the parameters of interest, we need to extend some previous results on i.i.d.
data, Severini and Wong (1992), to a dependent data setup. Our estimation method
presents several advantages against other methods in the literature: (1) It presents
closed form seasonal estimators that are very intuitive. The resulting nonparametric
estimator of the seasonal component is a simple transformation of the
Nadaraya–Watson estimator. (2) The statistical properties of both the parametric
and nonparametric estimators are well established. We present the asymptotic
distribution of both the nonparametric and the parametric estimator. This enables
us to make correct inference in the different components. (3) This methodology
provides a data driven method for computing the bandwidth. On the contrary,
polynomial spline techniques do not have a method for choosing the number and
location of nodes and the proportionality coefficients for the end-point restrictions.
(4) Multivariate extensions of the seasonal estimator are straightforward.
For example, considering a multivariate exponential distribution, the estimator
is easily adapted to the multivariate case. (5) The decomposition presented in
the paper can be easily extended to cope with other specifications that are frequently
used in the econometric analysis of tick-by-tick data. For example, to
capture nonlinear relationships between financial durations and market microstructure
variables (see, for instance, Spierdik et al. (2004), and references therein).
Likewise, it is also possible to replace the dependent variable by any other
tick-by-tick market microstructure variable and make it a function of its own
lags, through the parametric component, and any other variable through the nonparametric
component. In sum, although focused on durations and its nonlinear
dependency with the time-of-the-day, the potential applications of this model are
very ample.
As an illustration, we apply our method to price and volume durations of two
stocks traded on the NYSE. We show that the model is able to correctly capture
the seasonal pattern and it is able to adjust to changes on this pattern.
The structure of the paper is as follows. Section 2 develops a general ML
framework for analyzing tick-by-tick data, and proposes the new estimator for
seasonality. Its asymptotic properties are also analyzed. Second, it develops the
same method but in a generalized linear model (GLM) framework, which allows
us to use quasi maximum likelihood (QML). Section 3 is devoted to the empirical
application comparing the results with others existing in the literature. We use
density forecast to evaluate the out-of-sample goodness of fit. Section 4 concludes.
The assumptions and proofs of the main results are relegated to the Appendix.
2 Econometric model and estimators
Let ti be the time at which the ith event occurs and let di = ti − ti−1, where
ti−1