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Modelling financial transaction price movements: a dynamic integer count data model 179
hypothesis of absence of serial and cross-correlation in v i can be tested by the
multivariate version of the Portmanteau statistic: 6
QL ð Þ ¼ n XL
‘¼1
tr v ðÞ ‘
0
vð0Þ 1 v ðÞ ‘ vðÞ 0
1
h i
; (2.18)
where v ðÞ¼ ‘
Pn i¼‘þ1 viv0 i ‘ ðn ‘ 1Þ.
Under the null hypothesis, Q(L) is
asymptotically χ 2 -distributed with degrees of freedom equal to the difference between
four times L and the number of parameters to be estimated.
The ML-estimation results for the pure time series specifications (obtained
using the optimization procedure of Berndt et al. (1974) (BHHH)) and the results
of the corresponding diagnostic checks are summarized in Table 1. The Schwarz
criterion suggests to select an ARMA(1,1)-specification for JBX and an ARMA
(2,2)-specification for HAL regardless whether symmetry on the coefficient matrices
is imposed or not. All coefficient estimates are at least significant at the 5%
level. The LR-test for symmetry of the coefficient matrices rejects the null of
symmetric responses for both stocks.
Our estimates reveal some differences in the dynamics of the price direction
variable for the two stocks. For the JBX stock the estimate of 0.912 (0.888 for the
non-symmetric specification) for the coefficient c1 (1) indicates a high degree of
persistence in the price direction variable. Independent of the direction of the price
movement, the probability of a price change is comparatively high if the probability
of a price change for the previous transaction was high. Because the probability
of a non-zero price change can be interpreted as a specific measure of price
volatility, this finding reflects a clustering of volatility. Since under the estimated
non-symmetric specification a12
(1) (1) (1) (1)
< a11 and a22 < a21 we can conclude that there
is no clear evidence for a bid-ask bounce. Hence, our estimates confirm the
explorative findings on the cross-correlations of the price direction variable
(Fig. 4) and the simple autocorrelation function of the transaction prices in Fig. 2.
The results for HAL are more along the lines of previous empirical findings on the
dynamics of the transaction price process. Since c1 (1) +c 1 (2) is positive and close to
unity, a high degree of persistence can be found as well. Moreover, there is now
evidence for a negative first-order serial cross-correlation for the price change
(1) (1) (1) (1)
indicators with a12 > a11 and a22 < a21 , indicating the existence of a bid-ask
bounce.
Our diagnostic checks show that most of the dynamics of the price direction
variable is captured for the JBX data, but not for the HAL. For JBX the generalized
Portmanteau statistic Q(30) with a value of 132.9 (121.1) does not reject the null
hypothesis of no cross-correlations at the 10% significance level for the symmetric
and the nonsymmetric specification. For the raw data xi, the corresponding value of
the Q(30) statistic was found to be 303.9. For HAL the generalized Portmanteau
statistics indicate that the standardized residuals are still cross-correlated, although
the model reduces the value of the Q(30) statistic quite substantially from value of
3590.0 for the raw data to 163.6 (non-symmetric specification).
Figure 5 depicts the cross-correlation functions of the standardized residuals
v−1i and v1i. For JBX all but one correlations lie within the 99% confidence band
6 See, for example, Lütkepohl (1993).