recent developments in high frequency financial ... - Index of

172
2 The hurdle approach to integer counts
Consider a sequence of transaction prices {P(ti), i: 1→n} observed at times {ti,i: 1→n}. Let {Yi, i: 1→n} be a sequence of price changes, where Yi = P(ti)−P(ti−1) is an integer multiple of a fixed divisor (tick), then Yi 2Z . Our interest lies in
modelling the conditional distribution of the discrete price changes YijF i 1; where
F i 1 denotes the information set available at the time transaction i takes place. For
this, we generalize the hurdle approach proposed by Mullahy (1986) and Pohlmeier
and Ulrich (1995) for the Poisson and the negative binomial (Negbin) distribution,
respectively, to the domain of negative counts. The basic idea of this approach is to
decompose the overall process of transaction price changes into three components.
The first component determines the direction of the process (positive price change,
negative price change, or no price change) and will be specified as a dynamic
multinomial response model. Given the direction of the price change, count data
processes determine the size of positive and negative price changes, representing
the second and third component of our model. This yields the following structure
for the p.d.f. of YijF i 1 :
8
Pr½Yi¼yijFi1Š¼ < Pr½Yi < 0jF i 1ŠPr½Yi¼yijYi
< 0; F i 1Š
if yi < 0
Pr½Yi¼0jFi :
Pr½Yi > 0jF i
1Š
1ŠPr½Yi¼yijYi
> 0; F i 1Š
if
if
yi ¼ 0
yi > 0:
(2.1)
The process driving the direction of the price changes is represented by
Pr ½Yi < 0jF i 1Š;
Pr ½Yi¼0jF i 1Š
and Pr ½Yi > 0jF i 1Š;
while the two processes
for the size of the price changes conditional on the price direction, are defined
by Pr ½Yi¼yijYi < 0; F i 1Š
and Pr ½Yi¼yijYi > 0; F i 1Š:
Note that Pr ½Yi
¼ yij
Yi > 0; F i 1Š is a process defined over the set of strictly positive integers and
Pr ½Yi¼yijYi < 0; F i 1Š
is the corresponding p.d.f. for strictly negative counts.
This decomposition allows us to model the stochastic behavior of the transaction
price changes successively.
We follow Mullahy’s (1986) idea by modelling the size of positive price
changes as a truncated-at-zero count process. 3 Let f + (·) be the p.d.f. of a standard
count data distribution, then the p.d.f. for the size of positive price changes
conditional on the fact that the prices are positive is a truncated-at-zero count data
distribution:
Pr½Yi¼yijYi > 0; F i 1Š
¼ h þ ðyijF i 1Þ
¼ f þðyijF i 1Þ
ð Þ
1 f þ 0jF i 1
The process for the size of negative price jumps is treated in the same way:
1 f 0jF i 1
R. Liesenfeld et al.
(2.2)
Pr½Yi¼yijYi < 0; F i 1Š
¼ h ðyijF i 1Þ
¼ f yijF ð i 1Þ
; (2.3)
ð Þ
3 Alternatively, one could specify the p.d.f. of the transformed count Yi−1 conditional on Yi >0
using a standard count data approach. This approach was adopted by Rydberg and Shephard
(2003) in their decomposition model.