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

140
The innovation term ei is computed from the integrated intensity function
associated with the most recently observed process. Hence,
0
1
ei ¼ XK
k¼1
@1
Z k t
Nk ðÞ ti tk Nk ðÞ ti 1
k
ðs; F sÞds
Ay k i
: (5)
Under fairly weak assumptions, the integrated intensity function corresponds to an
i.i.d. standard exponential variate. 7 Therefore, e i is a random mixture of
exponential variates. For this reason, weak stationarity of the model depends on
the eigenvalues of the matrix B. If these lie inside the unit circle, the process e i is
weakly stationary.
Since the intensity function has left-continuous sample paths, Ψ i also has to be a
left-continuous function and predetermined, at least instantaneously before the
arrival of a new event. Therefore, Ψi is known instantaneously after the occurrence
of ti−1 and does not change until ti. Then kðt;F tÞ changes between ti−1 and ti only
k k
as a deterministic function of time according to the functions l0 (t) and s (t).
The baseline intensity function l 0 k (t) is specified in terms of the backward re-
currence times xkðÞ¼t t t k
M k , k =1,...,K, of all processes and may be specified
ðÞ t
using a Weibull-type parameterization depending on the parameters ω k k
and pr ,
k
0 t
YK
ðÞ¼exp !k
r¼1
x r ðÞ t
pkr 1 ; p k r > 0 : (6)
Moreover, the seasonality functions s k (t) are parameterized as linear spline
functions given by 8
s k ðÞ¼1 t þ XS
j¼1
k
j t j 1
ft> jg
; (7)
where τj, j=1,..., S, denote the S nodes within a trading day and νj k the corresponding
parameters.
As a result, by denoting W as the data matrix consisting of all points and
explanatory variables and denoting θ as the parameter vector of the model, the loglikelihood
function of the multivariate ACI model is given by
ln LðW; Þ ¼ XK
X n
k¼1 i¼1
Z ti
ti 1
k
ðs; F sÞds
þ y k i ln k ti;F ti ; (8)
where t 0=0 and n denotes the number of points of the pooled process. Under correct
specification of the model, the resulting k-type ACI residuals
^" k i ¼
Z k ti t k i 1
^ k ðs;FsÞds A. D. Hall, N. Hautsch
7 See Brémaud (1981) and Bowsher (2002).
8 In order to identify the constant ω k , s k (t) is set to one at the beginning of a trading day.