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Modelling financial transaction price movements: a dynamic integer count data model 181
and for HAL all but two correlations lie inside the 99% band. The means of
standardized residuals reported in Table 1 are close to zero, which should be expected
from a well specified model. However, the estimated variance–covariance
matrix of the standardized residuals deviate slightly from the identity matrix. This
may hint to a distributional misspecification or a misspecification of log–odds
ratios Λi, which is not fully compatible with the variation in the observed variation
of price change direction.
2.3 Dynamics of the size of price changes
In order to analyze the size of the non-zero price changes, we use a GLARMA
(generalized linear autoregressive moving average) model based on a truncated-atzero
Negative Binomial (Negbin) distribution. The choice of a Negbin in favor of a
Poisson distribution is motivated by the fact, that the unconditional distributions of
the non-zero price changes show over-dispersion for both stocks. For JBX (HAL)
the dispersion coefficient 7 is given by 3.770 (2.911). Moreover, note, that an atzero-truncated
Poisson distribution would allow only for under-dispersion.
Similar to the ACM model, the dynamic structure of this count data model rests
on a recursion on lagged observable variables. A comprehensive description of this
class of models can, for instance, be found in Davis et al. (2003). Note that the time
scale for absolute price changes (defined by transactions associated with non-zero
price changes) is different from the one of the ACM model for the direction of the
price changes, which is defined on the ticktime scale. Let u be a random variable
following a Negbin distribution with the p.d.f. 8
fðuÞ ¼
ð þ uÞ
ðÞ ðuþ1Þ þ !
!
! þ
u
; u ¼ 0; 1; 2;:::; (2.19)
with E(u)=ω > 0 and Var(u)=ω+ω 2 /κ. The overdispersion of the Negbin distribution
depends on parameter κ >0. As κ→∞, the Negbin collapses to a Poisson
distribution. The corresponding truncated-at-zero Negbin distribution is obtained
as h(u)=f (u)/[1−f (0)], (u = 1, 2, 3, ...), with f (0) = [κ/(κ+ω)] κ . This flexible class of
distributions will be used to model the size of non-zero price changes conditional
on filtration F i 1 and price direction D i. Thus, for S i∣S i >0,D i, F i 1 we assume
the following p.d.f.:
hsijDi; ð F i 1Þ
¼
ð þ siÞ
ðÞ ðsiþ1Þ with the conditional moments:
h
þ
i
!i
1
1
!i
!i þ
!i
ESijSi ½ > 0; Di; F i 1Š
¼ Si ¼
1 #i
7 Computed as variance over mean.
8 See, for example, Cameron and Trivedi (1998) (Ch. 4.2.2.).
si
; si ¼ 1; 2; ...;
(2.20)
(2.21)