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Modelling financial transaction price movements: a dynamic integer count data model 173
where f − (·) denotes the p.d.f. of a standard count data model. Combining the single
components leads to the following p.d.f. for the transaction price changes:
h
i h
i
Pr½Yi¼yjF i i 1Š
¼ Pr½Yi < 0jF i 1ŠhðyjF
i i 1Þ
Pr½Yi¼0jF i 1Š
h
Pr½Yi > 0jF i 1Šhþ
i þ
i
ðyijF i 1Þ
;
i 0 i
(2.4)
where i ¼ 1fYi0g are binary variables indicating
positive, negative, or no price change for transaction i.
A more parsimonious distribution results if one assumes that h − (·) and h + (·)
arise from the same parametric family of probability density functions. Based on
this assumption, the stochastic behavior of positive and negative price movements
can be summarized in a conditional p.d.f. for the absolute price changes Si ≡∣Yi∣ conditional on the price direction:
Pr½Si¼sijSi > 0; Di; F i 1Š
¼ hsijDi; ð F i
8
< 1 if Yi < 0;
1Þ
with
Di ¼
:
0
1
if
if
Yi ¼ 0;
Yi > 0;
(2.5)
where h(·) is the p.d.f. of a truncated-at-zero count data model. For the
parsimonious specification, the p.d.f. for a transaction price change is:
Pr½Yi¼yijF i 1Š
¼ Pr½Yi < 0jF i 1Š
i Pr½Yi¼0jF i 1Š
0 i Pr½Yi > 0jF i 1Š
þ
i
½hðjyijjDi; F i 1ÞŠ
1 0 ð i Þ
: ð2:6Þ
In this case, the resulting sample log-likelihood function of the ICH-model
consists of two additive components:
where:
L ¼ Xn
i¼1
ln Pr ½Yi ¼ yijF i 1Š
¼ Xn
L1;i þ
i¼1
Xn
L2;i; (2.7)
i¼1
L1;i¼ i ln Pr½Yi < 0jF i 1Šþ
0 i ln Pr Yi ½ ¼ 0jF i 1Šþ
þ i ln Pr½Yi > 0jF i 1Š
L2;i ¼ 1
0
(2.8)
i ln hðjyijjDi; F i 1Þ:
(2.9)
The component ∑L 1,i is the log-likelihood of the multinomial process
determining the direction of prices, while ∑L 2,i is the log-likelihood of the
truncated-at-zero count process for the absolute size of the price change. If there
are no parametric restrictions across the two likelihoods, we can maximize the
complete likelihood (2.7) by separately maximizing its components (2.8) and (2.9).
This reduces the computational burden considerably. In the following, we now