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

Semiparametric estimation for financial durations 229
ACD-type of model whereas ϕ(u, v) = exp(u + v) = exp(u) × exp(v) represents
a Log-ACD type of model with
ψ(¯di−1; ϑ1) = ω +
J�
αj ln di−j +
j=1
L�
βℓψi−ℓ. (4)
With respect to the seasonal component, φ(¯di−1, ¯yi−1,ϑ2), several alternatives
are available. In this class of models it is usually assumed that the seasonal term is
somehow related to ti. In order to make this dependence more explicit, we define
a rescaled time variable, t ′ i , such that
t ′ i =
�
ti −
ti
� ti−to
tc−to
ℓ=1
�
(tc − to) if ti >tc,
otherwise,
where ⌊x⌋ is the integer part of x. If the seasonal frequency is daily, to and tc
stand for the stock market opening and closing (in seconds), respectively; i.e.
to = 9.5 × 60 × 60 and tc = 16 × 60 × 60 if it opens at 09:30 and closes at
16:00. Note that defined in this way, t ′
i ∈[to,tc] is of bounded support and is the
time-of-the-day (in seconds) at which the event has occurred. By changing the
values of to and tc, other possible types of seasonality (hourly, weekly, monthly,
etc.) can be considered.
In the context of regularly spaced variables, several functional forms for
φ(·; θ2) have been proposed in literature. If the function is assumed to fall within
a known class of parametric functions then, by substituting (3) into (2), we obtain
the following log-likelihood function
Ln (d; ϑ1,ϑ2,ξ) =
(5)
n�
log p � di| ¯di−1, ¯yi−1,t ′ i−1 ; ϑ1,ϑ2,ξ � , (6)
i=1
and the parameters ϑ1, ϑ2, and ξ can be estimated as in the one component case
(2). If the error density is correctly specified then standard ML estimation methods
apply.
But if very little information is available about the functional form that relates
seasonality and time-of-the-day, the risk of misspecification in choosing φ(·; ϑ2)
is high. Consequently, it is worth the use of nonparametric methods to approximate
the unknown seasonal function. Let φ(t ′ i ) be the deterministic seasonal component
at time t ′ i . Given the proposed specification for the components, (3) becomes
� �
= ϕ ψ(¯di−1, ¯yi−1; ϑ1), φ(t ′ i−1 )� , (7)
E � di| ¯di−1, ¯yi−1,t ′ i−1
where ϑ1 and the function φ(·), evaluated at time points t ′ 1 ,...,t′ n , have to be esti-
mated. Following the standard approach (as in (2) or (6)), one would be tempted to
obtain estimators for ϑ1, ξ, φ(t ′ 1 ),...,φ(t′ n ), by choosing the values that maximize
the following log-likelihood function
Ln (d; ϑ1,ξ) =
n�
i=1
log p � di| ¯di−1, ¯yi−1; ϑ1,φ � t ′ � �
i−1 ,ξ . (8)