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

232 J. M. Rodríguez-Poo et al.
where
V � t ′ 0 ,η� =
� K 2 (u)du
I � t ′ 0 ,η� �
∂
= E
∂φ
f(t ′ 0 )I � t ′ 0 ,η�,
�
�
log p (d; φ,η)2�
� t′ = t ′ �
0
and f(t ′ 0 ) is the marginal density function of t′ ,asn tends to infinity.
(16)
(17)
This theorem is very appealing since it allows us to make inference on the
parametric and nonparametric components. However, the result depends on the
correct specification of the conditional density function of the error term. In Section
2.1 we weaken certain assumptions about the error density function and show that
our result remains valid.
2.1 The GLM Approach
As pointed out in Engle and Russell (1998) and Engle (2000), it is of interest
to have estimation techniques available that do not rely on the knowledge
of the functional form of the conditional density function. Two alternative
approaches that allow for consistent estimation of the parameters of interest
without specifying the conditional density are quasi maximum likelihood techniques,
QML, (see Gouriéroux, Monfort and Trognon (1984)) and generalized
linear models, GLM, (see McCullagh and Nelder (1989)). In both approaches it
is assumed that di, conditional on ¯di−1 and ¯yi−1, depends on a scalar parameter
θ = h � ¯di−1, ¯yi−1,t ′ �
i−1 ; ϑ1,ϑ2 , and its distribution belongs to a one-dimensional
exponential family with conditional density
q � di| ¯di−1, ¯yi−1; θ � = exp (diθ − b(θ) + c(di)) ,
where b(·) and c(·) are known functions. The main difference between the QML
and the GLM approaches is simply a different parameterization. We adopt the GLM
approach for the sake of convenience. By adopting the GLM parametrization, it is
straightforward � to see that the ML estimator of θ solves the first order conditions
ni=1 �
di − b ′ (θ) � = 0. The ML estimator of θ can also be obtained from the
solution of the following equation
n�
i=1
�
di − ϕ � ψ(¯di−1, ¯yi−1; ϑ1), φ(t ′ i−1 ; ϑ2) �� ϕ ′ � ψ(¯di−1, ¯yi−1; ϑ1), φ(t ′ �
i−1 ; ϑ2)
V � ϕ � ψ(¯di−1, ¯yi−1; ϑ1), φ(t ′ �� = 0.
i−1 ; ϑ2)
(18)
The parameter of interest θ (the so-called canonical parameter) can be estimated without
specifying the whole conditional distribution function. It is only necessary to specify the
functional form of the conditional mean, ϕ(·), and of the conditional variance V(·), but
not the whole distribution.
The relationship between the predictors in Eq. (7) and the canonical parameter is
given by the link function. This function depends on the member of the exponential