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

Intraday stock prices, duration, and volume 259
be chosen. The choice of K in smoothing problems is usually not crucial (see e.g.,
Wand and Jones 1995) but the choice of H is important. For K we use the standard
M-variate normal distribution
KðÞ¼ x ð2Þ 1=2 exp kxk 2.
2 :
We take H=hIM where IM is the M-dimensional identity matrix. Other, possibly
non-symmetric kernels may be useful here (for instance, the gamma kernels in
Chen 2000), although we stay with the Gaussian kernel as the theoretical properties
for this specific method has been established for symmetric kernels only. As we are
effectively using only one bandwidth for multiple regressors, our regressors are
always scaled to a common variance before the estimator is implemented.
To obtain the optimal value of h, we adapt the bootstrap bandwidth selection
method suggested by Hall et al. (1999). This approach exploits the fact that, as we
are estimating distribution functions, there is limited scope for highly complicated
behavior. First, a simple parametric model is fitted to the data and used to obtain an
estimate ^Fðyx j Þ. We use
Yt ¼ a0 þ a1X1t þ ...þ aMXMt þ aMþ1X 2 1t þ ...þ a2MX 2 Mt þ "t (2)
and assume that ɛt is heteroskedastic, depending on the square of lagged price
changes. We then simulate from this model to obtain a bootstrap sample {Y1*, Y2*,..., YT*}using the actual observations { x1,x2,...xT}. For each bootstrap sample
(and for a given value of h), we compute a bootstrap estimate ~F hðyjxÞ. We choose
h to minimize
X ~F * h
y x j ð Þ ^Fðyx j Þ
where the summation is over all bootstrap replications and over values of y for any
given x. We checked the sensitivity of our estimates to the choice of the
bandwidth, and we note that we obtained very similar results over a wide range of
bandwidth values.
Computation of the weights w t is carried out using the Lagrange Multiplier
method. The Lagrangian is
L ¼ XT
t¼1
log ðwtÞ 0
X T
t¼1
wt 1
which gives the first-order conditions
1
wt
0
X M
m¼1
! X M
m¼1
X T
m
t¼1
wtðXm;txmÞKHðXtxÞ; mðXm;txmÞKHðXtxÞ ¼ 0; 8m ¼ 1; ...; M (3)