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

240 J. M. Rodríguez-Poo et al.
where (·) + = max (·, 0), G is the number of knots, and πg is the gth knot. θj,j =
1, ··· , 4, and δg, g = 1, ··· ,G, are the parameters of the spline to be estimated. Further
details on the derivation of this form can be found in Eubank (1988, p 354). Knots are
set at every hour with an additional knot in the last half hour, as in Engle and Russell
(1998).
The two remaining estimators are two steps estimators, in the sense that they are
not estimated jointly with the finite dimensional parameters but independently: First the
seasonal curve is estimated, then durations are adjusted by seasonality and finally the
parameters are estimated using the deseasonalized durations. This procedure is often
followed in the literature (see, for instance, Engle and Russell (1998) and Bauwens and
Giot (2000)). The first of these two estimators is the plain Nadaraya–Watson estimator
ˆφ(t ′ 0 ) =
1 �ni=1 nh K
1
nh
� ni=1 K
�
t ′
0−t ′ i
h
�
t ′
0−t ′ i
h
�
di
� , (23)
which we denote by BiNW—standing for two steps and Nadaraya–Watson. The second
two-step estimator is the one used by Engle and Russell (1998); it is the seasonal component
that is estimated by averaging the durations over 30 min intervals, and smoothing
the resulting piece-wise constant function via cubic splines. We denote this estimator by
BiSp.
To construct pointwise confidence bands for the seasonal curve in UniNW, we need
to estimate the empirical counterparts of Eqs. (16) and (17):
�φ�ϑ1,�ξ (t′ 0 ) ± z1− α �
�
�
�
�K�
2
2 2
�f(t ′ 0 )�I(�φ�ϑ1,�ξ ,t′ 0 ),
(24)
where
�I(�φ�ϑ1,�ξ ,t′ 0 ) =
�f(t ′ 1
0 ) =
nh
1 �ni=1 nh K
n�
K
i=1
�
t ′
0−t ′ �
i
h
⎡
⎣�γ
� t ′ 0 − t ′ i
h
⎛�
⎝
1 �ni=1 nh K
�
,
di �
exp ψ(�ϑ1)+�φ�ϑ 1 ,�ξ (t′ 0 )
�
�
t ′
0−t ′ i
h
� �γ
⎞⎤
2
−�ν ⎠⎦
� ,
z α
1− is the
2 α 2 -quantile of the standard normal distribution, and �K�2 2 is a known constant
that depends on the kernel. For the quartic kernel, �K�2 2 = 5 7 . We can also compute
consistent estimators of the variance–covariance matrix of the parameters. For the other
three cases, no results in this direction are available and, therefore, the standard errors
we present for these three cases have unknown properties. Nonetheless, we present them
for the sake of comparison.
Table 2 presents the estimation results of the four specifications for price and volume
durations. Comparing UniNW with the other specifications we conclude: The estimated
parameters of the dynamic component are very similar under the exponential and the
generalized gamma distribution but the standard deviations are smaller under the generalized
gamma distribution. This supports the theory of ML and GLM in the sense that the