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Modelling financial transaction price movements: a dynamic integer count data model 183
the intraday trading time standardized on [0, 1] and ν is a 2K+1 dimensional
parameter vector.
In Eq. (2.23) the standardized absolute price change, "i ¼ ðSiSiÞ Si ; drives
the extended ARMA process in λi. Similar to the dynamics structure in the
EGARCH model of Nelson (1991) we allow λi to depend on the innovation term,
and additionally on its absolute value to be able to capture a wide range of possible
news impact functions. The vector Di ¼ ðDi; Di 1; ...Di lÞ
0 contains information
about the contemporaneous and lagged price directions with a corresponding
coefficient vector β d =(β d0, β d1 ,...,β dl)′ . The inclusion of D i ∈ {−1, 1} is to
capture potential differences in the behavior of the absolute price changes
depending on the direction of the price process. Thus, a negative βd0 implies that
with negative price changes (Di = −1) large absolute price changes are more likely
to occur than with positive price changes (D i = 1). If one interprets the absolute
price changes as an alternative volatility measure, this asymmetry reflects a kind of
leverage effect, where downward price movements imply a higher volatility than
upward movements. 10 The inclusion of lagged terms D i − l l =1, 2, . . . allows for a
dynamic variant of the leverage effect. Z i S denote further explanatory variables,
which will be discussed below.
2.4 Empirical results for the GLARMA model
For estimation of parameters of the GLARMA(p,q) model, given by Eqs. (2.20)
to (2.23), we maximize the log-likelihood L2 ¼ Pn i¼1 1 0
i ln hsijDi; ð F i 1Þ
using the BHHH algorithm. We use the Schwarz information criterion to determine
the optimal order of p and q. The diagnostic checks are based on the standardized
residuals
ei ¼ Si ^ Si
^si
(2.24)
For a correctly specified model, the residuals evaluated at the true parameter
values should be uncorrelated in the first two moments with E [e i]=0andE [e i 2 ] =1.
The ML-estimates of the pure time series component of the GLARMA model
for two specifications allowing for a plain (ζ l = 0) and a more flexible news impact
curve (ζ l ≠ 0) are given in Table 2. Since the coefficients of the seasonal component
S(ν, τ, K) are jointly significant for all specifications estimated, we refrain from
reporting the results for the models without any seasonal component in the absolute
price change variable. The diurnal seasonalities of the absolut price changes for
JBX and HAL are depicted in Fig. 7. We observe a standard intraday volatility (or
market activity) seasonality pattern: A high volatility when trading starts, which
declines until lunch-time (after around 9,000 s, 12:00 EST), a slight recovery in the
early afternoon, and another decrease when the European stock markets close (after
around 16,200 s, 14:00 EST). As for the price direction variable for the JBX data, a
parsimonious GLARMA(1,1)-specification turns out to be best specification, while
10 See, for instance, Nelson (1991).