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

228 J. M. Rodríguez-Poo et al.
this event are observed. We gather these characteristics in a k-dimensional vector
yi. Thus, the following set of observations is available
{(di,yi)}i=1,...,n.
One possible way to obtain information about the whole process is to assume that
the ith duration has a prespecified conditional parametric density di|Ii−1 ∼ p
� di| ¯di−1, ¯yi−1; δ � , where ( ¯di−1, ¯yi−1) is the past information and δ is a set of
finite-dimensional parameters. Under these conditions it is possible to estimate
the parameter vector δ through maximum likelihood techniques. But sometimes
assumptions about the knowledge of the whole conditional density are too strong
and the researcher prefers to make assumptions about some of its conditional
moments. Let
E[di| ¯di−1, ¯yi−1]=ψ(¯di−1, ¯yi−1; ϑ1) (1)
be the expectation of the ith duration conditional on the past filtration. The ACD
class of models consists of parameterizations of (1) and the assumption that
di = ψ(¯di−1, ¯yi−1; ϑ1)εi,
where εi is an i.i.d. random variable with density function p(εi; ξ)depending on a
set of parameters ξ and mean equal to one. The vector ϑ1 contains the parameters
that measure the dynamics of the scale and ξ contains the shape parameters. The
conditional log-likelihood function can be written as
Ln (d; ϑ1,ξ) =
n�
log p � di| ¯di−1, ¯yi−1; ϑ1,ξ � . (2)
i=1
If the conditional density is correctly specified and ϑ1 and ξ are finite-dimensional
parameters, then, under some standard regularity conditions, the maximum likelihood
estimators of the parameters of interest are consistent and asymptotically
normal (for conditions see Engle and Russell (1998)).
The specification described above is sometimes too simple and/or rigid since the
expected duration can vary systematically over time and can be subject to many
different time effects. One way to extend the above model is to decompose the
conditional mean into different effects. In standard time series literature a stochastic
process can be decomposed into a combination of cycle and trend, seasonality and
noise. This decomposition, with a long tradition in time series analysis, has already
been used in volatility (see, among others, Andersen and Bollerslev (1998)) and
duration analysis (e.g. Engle and Russell (1998)). Instead of (1), the following
nonlinear decomposition is proposed:
E[di| ¯di−1, ¯yi−1]=ϕ � ψ(¯di−1, ¯yi−1; ϑ1), φ( ¯di−1, ¯yi−1; ϑ2) � . (3)
Durations are modeled as a possibly nonlinear function of two components,
ψ(·; ϑ1) and φ(·; ϑ2), that represent dynamic and seasonal behavior, respectively.
The function ϕ(u, v) nests a great variety of models. ϕ(u, v) = (u × v) forms an