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Modelling financial transaction price movements: a dynamic integer count data model 177
first component of the likelihood of the overall model, takes on the familiar form
presented below:
L1 ¼ Xn
i¼1
i ln 1i þ 0 i ln 0i þ þ i ln 1i : (2.15)
Concerning the coefficient matrix Al, our empirical analysis will be based on
two alternative specifications: an unrestricted one, and one including symmetry
restrictions as suggested by Russel and Engle (2002). In particular, we impose the
(1) (1)
symmetry restriction a12=a21. This implies that the marginal effect of a negative
price change on the conditional probability of a future positive price change is of
the same size as the marginal effect of a positive change on the probability of a
future negative change. Moreover, we impose the symmetry restriction a (1)
11 = a (1)
22
which guarantees that the impact of a negative change on the probability of a future
negative change is the same as the corresponding effect for positive price changes.
The symmetry of impacts on the conditional price direction probabilities will also
be imposed for all lagged values of the probabilities and normalized state variables.
Following Russel and Engle (2002), we set in the model with symmetry restrictions
c2 (l) =0, ∀l, which implies that shocks in the log–odds ratios vanish at an exponential
(l)
rate determined by the diagonal element c1 . This simplifies the ARMA specification
(2.13) for the symmetric model to:
¼
1
2
; Cl ¼
l
c ðÞ
1
0
0
c l
!
; Al ¼
ðÞ
1
l
a ðÞ
1
a l ðÞ
2
a l ðÞ
2
a l ðÞ
1
!
: (2.16)
Although the reasoning behind these restrictions seems appropriate due to the
explorative evidence of the state variable xi, the validity of these restrictions can, of
course, be easily tested by standard ML based tests.
2.2 Empirical results for the ACM model
In search of the best specification, we use the Schwarz information criterion (SIC)
to determine the order of the ARMA process. For the selected specification, its
standardized residuals will be subject to diagnostic checks. For the estimates of
the conditional expectations and the variances ^E½xijF i 1Š
¼ ^i and ^V½xijF i 1Š
¼
diagð^iÞ ^i ^0 i , respectively, the standardized residuals can be computed as:
vi ¼ ðv1i; v1iÞ
0 1=2
¼ ^V xijF i 1
h
i
xi ^E xijF i 1 ; (2.17)
where ^V½xijF i 1Š
1=2
is the inverse of the Cholesky factor of the conditional
variance. For a correctly specified model, the standardized residuals evaluated at
the true parameter values should be serially uncorrelated in the first two moments
with the following unconditional moments: E[vi]=0 and E[vivi′ ]=I. The null