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Modelling financial transaction price movements: a dynamic integer count data model 169
comes have to occur in the sample period to guarantee the identification and
estimation of the true dimension of the multinomial process. This creates a serious
limitation if the ACM is used for forecasting purposes. In the multinomial approaches,
as well as in the ordered response models, the number of parameters
increases with the outcome space. As long as one is not willing to categorize the
outcomes at the expense of a loss of information, both approaches are more suited
for the empirical analysis of financial markets which are characterized by a limited
number of discrete price changes.
In the following we propose a model that does not suffer from the drawbacks of
the discrete response models sketched above. We propose a dynamic model which
is based on a probability density function for an integer count variable, and which
can be interpreted as a count data hurdle model. Our integer count hurdle (ICH)
model is closely related to the components approach by Rydberg and Shephard
(2003), who suggest decomposing the process of transaction price changes into
three distinct processes: a binary process indicating whether a price change occurs
from one transaction to the next, a binary process indicating the direction of the
price change conditional on a price change having taken place, and a count process
for the size of the price change conditional on the direction of the price change. By
combining the above mentioned two binary processes into one trinomial ACM
model (no price change or price movement downwards or upwards), and using a
count process for the size of the price change based on a dynamic count data
specification, our approach is more parsimonious than the one proposed by
Rydberg and Shephard. The distribution of price changes used is that of a count
data hurdle model extended for the domain of negative integer counts. For both
components of the price process, the dynamics are modelled using a generalized
ARMA specification.
Our model exhibits a number of desirable features and can be extended in many
respects. The decomposition allows for a detailed analysis of the price direction
process and volatility as well as the analysis of tail behavior. Inclusion of contemporaneous
marks of the transaction price process as conditioning information
(e.g. transaction time and volume), can generate insights into the validity of various
hypotheses of market microstructure theory. In our empirical application of the
ICH model, we will analyze the distribution of price changes conditional on
transaction time and volume. Our model can also serve as a building block for the
joint process of transaction price and transaction times. In this sense, our approach
is more flexible than the competing risks ACD model by Bauwens and Giot (2003),
which focuses on the direction of the price process whereby neglecting information
on the size of the price changes.
For our empirical application of the ICH model, we use transaction data of the
stocks of the Halliburton Company (HAL) and Jack in the Box Inc. (JBX) traded at
the NYSE. Our sample period includes 35021 (HAL) and 4566 (JBX) transactions
observed from 1st to 30th March 2001. 1 The two stocks are chosen for reasons of
representativeness. HAL is a stock with medium market capitalization (about US
1 The data used stems from the NYSE Trade and Quote database. We have removed all trades
outside the regular trading hours and each day’s first trade, to circumvent contamination due to
the opening call auction at the NYSE. Besides, all trades are treated as split transactions, if they
exhibited exactly the same timestamp. In this case we have aggregated their volume to one
transaction and we have assigned the last price in the sequence to the aggregated transaction.