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

Modelling financial transaction price movements: a dynamic integer count data model 195
Acknowledgements For helpful comments and suggestions we like to thank Bernd Fitzenberger,
Nikolaus Hautsch, Neil Shephard, Gerd Ronning, Timo Teräsvirta and Pravin Trivedi. The work
of the second and third co-author is supported by the Friedrich Thyssen Foundation and the
European Community’s Human Potential Programme under contract HPRN-CT-2002-00232,
Microstructure of Financial Markets in Europe (MICFINMA), respectively.
Appendix
This appendix shows that under a correctly specified model for Yi, the ui’s drawn
from the uniform distributions (4.3) follow a uniform distribution on the interval
[0, 1]. 15 Consider a discrete random variable Y with support Z , and let u be a
continuous random variable with the following conditional uniform distribution
y2
u U u l y ; uu y ; (A.1)
l u
where the boundaries are uy=Pr(Y≤y−1), uy =Pr(Y≤ y) (for ease of notation we
ignore the index i for the variables u and Y). Then, the c.d.f. of the unconditional
distribution of u is
Pr ðucÞ ¼ X
Pr ðucjY¼yÞPr ðY¼yÞ; c 2 ½0; 1Š;
(A.2)
with
PrðucjY¼yÞ ¼ c uly u u y
u l y
I ul y ;uu ½ yÞ
c ð ÞþI uu y ;1
½ Š
ðÞ c
(A.3)
Pr ðY¼yÞ ¼ u u y u l y ; (A.4)
where I AðÞis z an indicator function which is 1 if z ∈ A and zero for z ∉ A. Inserting
Eqs. (A.3) and (A.4) into Eq. (A.2), we obtain
PrðucÞ ¼ X
ðÞþ uuy u l y I uu y ;1 ½ Š
c ðÞ
n o
: (A.5)
y2
c u l y I ul y ;uu ½ yÞ
c
Assuming that c ∈ [u j l , uj u ], j ∈ Δ, we find
PrðucÞ ¼ cI ½0;1ŠðÞ c PrðYj1Þþ... þ PrðYj3Þ PrðYj4Þ þ PrðYj2Þ PrðYj3Þ þ PrðY ðÞ;
j 1Þ
PrðYj2Þ ¼ cI ½0;1Š c
which represents the c.d.f. of a uniform distribution on the interval [0, 1].
(A.6)
15 This technique of continuization is widely used to describe the properties of the p.d.f. of
discrete random variables, see e.g. Stevens (1950) and Denuit and Lambert (2005).