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specify the parametric form of the p.d.f. for the price direction and the absolute size
of the price changes.
2.1 Dynamics of the price direction
The parametric model for the direction of the transaction price change Di=j,(j=−1, 0, 1) is taken from the class of logistic ACM (autoregressive conditional multinomial)
models suggested by Russel and Engle (2002). In order to relate the
probability ji ¼ Pr ½Di¼jjF i 1Š
for the occurrence of price direction j to subsets
of F i 1 (and further explanatory variables), we use a logistic link function. This
leads to a multinomial logit model of the form:
ji ¼
exp ji
P 1
j¼ 1 exp ji
; j ¼ 1; 0; 1; (2.10)
where Λji represents a function of some subset of F i 1 to be specified below. As a
normalizing constraint, we use Λ0i=0, ∀i.
Due to the observed dynamic behavior of the transaction price changes
associated with the bid-ask bounce or the volatility clustering, one can expect that
the process of the price direction also exhibits serial dependence. In order to shed
light on this serial dependence, which has to be taken into account when modelling
the conditional distribution of the price direction, we define the following state
vector
xi ¼ ðx1i; x1iÞ
0 ð1; 0Þ
¼
0
if Yi < 0
ð0; 0Þ
0
if Yi ¼ 0
ð0; 1Þ
0
8
<
(2.11)
:
if Yi > 0;
and consider its corresponding sample autocorrelation matrix. For a lag length ‘,
this matrix is given by:
with
ðÞ¼D ‘
1
1
ðÞ¼ ‘
n ‘ 1
ðÞD ‘
1 ; ‘ ¼ 1; 2; ...; (2.12)
X n
ðxixÞ xi ‘ x
i¼‘þ1
ð Þ 0 :
R. Liesenfeld et al.
D denotes a diagonal matrix containing the standard deviations of x −1i and x 1i.
Figure 4 below depicts the cross-correlation function of up to 30 lagged
transactions. The significant, but not very large, first order cross-correlations provide
empirical support for the existence of a bid-ask bounce: The probability of a
price reduction is significantly (HAL) positively correlated with the price increase
in the previous period (upper right panel), and also significant (both stocks), a price
increase is more likely if a negative price change is observed for the previous
transaction (lower left panel). The cross-correlation effects turn out to be asymmetric,
in the sense that the correlation of a negative price change with a previous
positive one is smaller than the effect vice versa. For HAL we observe negative first