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

230 J. M. Rodríguez-Poo et al.
Standard ML techniques do not apply directly since the estimation of the parameter
vector, ϑ1 and ξ, does not necessarily provide consistent estimators in the
presence of infinite dimensional nuisance parameters. In order to implement valid
inference not only on the parameter estimators of the dynamic component but on
the estimated seasonal curve as well, we propose the generalized profile likelihood
approach. It has been introduced, in an i.i.d. context, by Severini and Wong
(1992). The basic idea of this method is to estimate the nonparametric function φ(·)
by maximizing a local (and hence smoothed) likelihood function (see Staniswalis
(1989)), and simultaneously estimate the parameter vector ϑ1 and ξ by maximizing
the unsmoothed likelihood function. For a given value of the time-of-the-day,
t ′ 0 ∈[to,tc], and fixed values of ϑ1 and ξ, we estimate φ(t ′ 0 ) as the solution of the
problem
ˆφϑ1,ξ(t ′ 1
0 ) = arg sup
φ∈� nh
n�
K
i=1
�
t ′
0 − t ′ �
i
log p
h
� di| ¯di−1, ¯yi−1; ϑ1,φ,ξ � , (9)
where K(·) is a kernel function and h is the corresponding bandwidth. Note also
that all estimators depend on ϑ1 and ξ. Then, ˆφϑ1,ξ(t ′ 0 ) must fulfill the first order
condition
n�
�
1 t ′
0 − t
K
nh
′ �
i ∂
�
log p di| ¯di−1, ¯yi−1; ϑ1, ˆφϑ1,ξ(t
h ∂φ ′ �
0 ), ξ = 0. (10)
i=1
Given the above estimates for the nonparametric part, a simple ML estimation for
ϑ1 and ξ is performed
and
�
ˆϑ1n ˆξn
�T = arg sup
sup
n�
ϑ1∈� ξ∈�
i=1
�
ˆϑ1n ˆξn
�
must fulfill the first order condition
n�
i=1
∂
log p
T ∂ (ϑ1 ξ)
�
log p di| ¯di−1, ¯yi−1; ϑ1, ˆφϑ1,ξ(t ′ �
i−1 ), ξ
(11)
�
di| ¯di−1, ¯yi−1; ˆϑ1n, ˆφ ˆϑ1n,ˆξn (t′ i−1 ), �
ˆξn = 0. (12)
The procedure is implemented as follows: (1) For a given t ′ 0
and ξ, and find the ˆφϑ1,ξ(t ′ 0 ) that fulfills condition (9). Repeat it for all t′ i
, fix the values of ϑ
. (2) Plug
the vector ˆφϑ1,ξ(t ′ i ) into (11). The log-likelihood is hence concentrated on ϑ and
ξ, which can be easily estimated. (3) Given the estimators ˆϑn and ˆξn come back to
(1). (4) Iterate until (10) and (12) are fulfilled. This procedure is computationally
intensive as the optimization in (1) has to be done n times. It would be significantly
alleviated if we would have a closed form expression for ˆφϑ1,ξ(t ′ 0 ). This is the case
for log p being one of the log-likelihoods that are typically assumed for financial
durations.
As an example, assume that p(·) in the log-likelihood function (2) is the
generalized gamma density function, i.e., εi ∼ GG (1,γ,ν) then
�
di ∼ GG ϕ � ψ � � �� �
¯di−1,
−1
¯yi−1; ϑ1 ,φϑ1,ξ ,γ,ν .
� t ′ i−1