  We study tick-by-tick financial returns belonging to the FTSE MIB index of
the Italian Stock Exchange (Borsa Italiana). We can confirm previously detected
non-stationarities. However, scaling properties reported in the previous
literature for other high-frequency financial data are only approximately
valid. As a consequence of the empirical analyses, we propose a simple method
for describing non-stationary returns, based on a non-homogeneous normal
compound Poisson process. We test this model against the empirical findings and
it turns out that the model can approximately reproduce several stylized facts
of high-frequency financial time series. Moreover, using Monte Carlo
simulations, we analyze order selection for this model class using three
information criteria: Akaike's information criterion (AIC), the Bayesian
information criterion (BIC) and the Hannan-Quinn information criterion (HQ).
For comparison, we also perform a similar Monte Carlo experiment for the ACD
(autoregressive conditional duration) model. Our results show that the
information criteria work best for small parameter numbers for the compound
Poisson type models, whereas for the ACD model the model selection procedure
does not work well in certain cases.
