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

226 J. M. Rodríguez-Poo et al.
version of the ACD. Grammig and Mauer (2000) use a Burr distribution in the
ACD model. Zhang et al. (2001) introduce a threshold ACD (TACD). Fernandes
and Grammig (2001) introduce the augmented ACD model, a very general model
that covers almost all the existing ones. Drost and Werker (2004) provide a method
for obtaining efficient estimators of the ACD model with no need to specify the
distribution. Meitz and Terasvirta (2005) introduce smooth transition ACD models
and testing evaluation procedures.Alternative models are the stochastic conditional
duration (SCD) model of Bauwens and Veredas (2004) and the stochastic volatility
duration (SVD) model of Ghysels et al. (2004), both based on latent variables.
In the application of most of the above studies, durations show a strong
intradaily seasonality, i.e., an inverted U shape pattern along the day, as it is shown
in Figs. 2 and 3 for price and volume durations of two stocks traded at the NYSE.
The study of seasonality for regularly spaced variables, i.e., observed at equidistant
periods of time, is well known. It has focused mainly on the intradaily behavior
of volatility either on stock markets or on foreign exchange markets—see, among
others, Baillie and Bollerslev (1990), Bollerslev and Domowitz (1993), Andersen
and Bollerslev (1997 and 1998) and Beltratti and Morana (1999). All these articles
used data sampled at different frequencies: hourly, half-hourly, every 15 min or
every 5 min.
Durations do not fit into this category as they are themselves the main characteristic
of irregularly spaced data. For tick-by-tick data, Engle and Russell
(1998) introduce a method for dealing with intradaily seasonality. It consists of
decomposing the expected duration into a deterministic part, that depends on the
time-of-the-day at which the duration starts, and a stochastic part. The deterministic
component accounts for the seasonal effect whereas the stochastic component
models the dynamics. If both components are assumed to belong to a parametric
family of functions, the two sets of parameters can be jointly estimated by maximum
likelihood (ML) techniques. And, under standard regularity conditions, ML
estimators of the parameters of interest are consistent and asymptotically normal
(see Engle and Russell, 1998; corollary, p. 1135).
In many situations the researcher does not have enough information to fully
specify the seasonality functions or seasonality itself is not the main subject
of analysis but it has to be taken into account. In these cases, the choice of a
particular parametric function can be delicate. An alternative is to approximate
the seasonal component through nonparametric functions (mainly Fourier series,
spline functions, or other types of smoothers) and then estimate the parameters
of the dynamic component using maximum likelihood techniques. Unfortunately,
standard maximum likelihood for finite dimensional parameters, in the presence of
infinite “incidental” parameters, may yield inconsistency and slow rates of convergence
(see, for examples of inconsistency: Kiefer and Wolfowitz (1956); Grenander
(1981); and Shen and Wong (1994) and for examples of slow rates of convergence
Birgé and Massart (1994)). In order to solve this problem many alternative solutions
have been proposed. If the seasonal component is estimated through splines,
Fourier series, neural networks, or wavelets, then the method of sieve extremum
estimation can be used to make inference on the seasonal term. Chen and Shen
(1998) give sufficient conditions for splines and Fourier series under regularly
spaced dependent data. If, instead, other methods such as kernels or local polynomials
are used, then the sieve method is no longer valid and other nonsieve ML
estimation methods are needed. Among others, the so-called generalized profile