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

238 J. M. Rodríguez-Poo et al.
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Fig. 3 Intradaily seasonal patterns (middle line) for the days of the week and 95% confidence
bands (side lines) for Disney volume durations. The straight line represents the mean of each
curve
where I(·) is the indicator function and ⌊x⌋ is the integer part of x. We use a quartic
kernel with bandwidth 2.78sn −1/5 , where s is the sample standard deviation.
Figures 2 and 3 show the intradaily seasonality for each day of the week. They
also include pointwise confidence bands (see Bosq (1998), Theorem 3.1, p. 70) for the
different curves and, for comparison purposes, a straight line representing the mean
of each curve. Although a more formal test is needed, the hypothesis of differences
between the seasonal behavior over the days of the week is not supported. Confidence
bands increase through the day: Early in the morning, the bands are tighter than near
the closing. This indicates that the variance of the durations evolves through the day.
Indeed, Fig. 4 presents rolling means and standard deviations, in intervals of half an
hour, through the day. Each point represents the mean and standard deviation over half
an hour. They show an inverted U shape, i.e., as the day goes on, the variance increases
(with the exception of the half hour 13:00–13:30), widening the confidence bands with a
slight tightness near the closing. Incidentally, notice that for price durations the standard
deviation is above the mean while it is the opposite for volume durations. This is due to
the over and under dispersion, respectively.