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

Semiparametric estimation for financial durations 239
Fig. 4 Intradaily rolling means (solid lines) and standard deviations (dashed lines) for half hour
intervals. Boeing price durations in the left box and Disney volume durations in the right box
Fig. 5 Spearman’s ρ coefficients for serial dependence for price durations (left box), and volume
durations (right box)
Finally, Fig. 5 contains the Spearman’s ρ coefficients of serial dependence. They
indicate the presence of dependencies, justifying the dynamic component in the model.
3.3 Estimation results
Prior to estimation, we need to specify ϕ(u, v), the lag orders, and the form of g(·) in (4),
the conditional density of the error term and the nonparametric estimator φ(t ′
i−1 ).We
opt for a Log-ACD(1,1), which has been successfully used in the literature. As for the
density, we chose a generalized gamma as it is able to reproduce the features highlighted
earlier. We also provide the quasi maximum likelihood estimates using the exponential
density.
We estimate φ(t ′
i−1 ) in four different ways. Two of them are joint estimators, in the
sense that estimation is performed jointly with the parameters. One is Eq. (13) that we
denote by UniNW—standing for one step Nadaraya–Watson. The other is the polynomial
spline used by Engle and Russell (1995 and 1997) that we denote by UniSp—standing
for one step spline. We do not use their parametrization but
E � di| ¯di−1,t ′ �
�
� di−1
i−1 = ψi = exp ω + α ln + βψi−1 ,
)
φθ,δ(t ′ i−1 ) =
4�
j=1
θjt ′j−1
i−1 +
φθ,δ(t ′ i−1
G�
g=1
� � ′ 3
δg t i−1 − πg + , (22)