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

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JBX HAL
Fig. 6 Autocorrelation function of the non-zero absolute price changes Si|Si > 0 for Jack in the
Box Inc. (JBX) and Halliburton p ffiffi Company (HAL). The dashed lines mark the approximate 99%
confidence interval 2:58 = en , where ñ is the number of non-zero price changes
VSi ½ jSi > 0; Di; F i 1Š ¼ 2 !i
¼ Si 1 #i
! 2 i
ð1 #iÞ
2 #i
1 #i
; (2.22)
where #i ¼ ð þ !iÞ
: In this specification, both mean and variance are monotonic
increasing functions of the variable ωi that is assumed to capture the variation
of the conditional distribution depending on Di and F i 1 .
As noted above, volatility clustering of asset returns is a well-known property,
which also occurs at high frequencies. This is confirmed by the autocorrelation
function of the nonzero absolute price changes shown in Fig. 6, which reveals a
significant autocorrelation of this specific volatility measure. For the less frequently
traded stock JBX, the correlations die out quicker than for HAL where
significant but small correlations can be observed even after more than 25 trades.
In order to take into account the dynamics of Si we follow Rydberg and
Shephard (2002) and impose a GLARMA structure on ln ωi as follows: 9
ln !i ¼ 0 d Di þ Pm
with i ¼ 0 þ S ð ; ; KÞþ
Pp
l¼1
l¼0
BlZ S i l þ i
l i lþ Pq
l¼1
l"i 1þ Pr
l¼1
l j"ilj: R. Liesenfeld et al.
(2.23)
Since it is a well-known stylized fact for high frequent financial data that there
exists intraday seasonality in price volatility, we introduce the seasonal component
S(ν, τ, K)=ν 0τ +∑ K k=1ν 2k −1sin (2π(2k−1)τ)+ν 2k cos (2π(2k)τ). This Fourier flexible
form is to capture intraday seasonality in the absolute price changes, where τ is
9 Similar to the alternative specification discussed in the context of the ACM model one could
also specify a dynamic latent process for ωi. See Zeger (1988) and Jung and Liesenfeld (2001) for
examples.