## 1419GEOSTATS: simulating intrinsically stationary processes

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• Nov 30, 1999
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Dear All,

I hope you can help me. I'm an econometrician analysing high-frequency
exchange rate data. We typically observe a transaction price at
irregular intervals, between .01 of a second and three hours depending
on the time of day. We can also observe other things like spreads,
liquidity etc.

I though it might be a good idea, given the irregular spacing of the
data (on the time-scale) to use some geo-stat methods. I've got Cressie,
plus some other papers, but there are still some outstanding questions.

1) Despite often reading claims that the variogram is defined for a
wider class of processes than the covariance (i.e. intrinsically
stationary processes), I haven't seen any convincing evidence that when
we simulate a non-covariance-stationary process that IS intrinsically
stationary, the variogram outperforms the covariance. I would imagine we
could demonstrate this in terms of measuring the dependence and through
kriging mean square erros. Do you know of any examples here people have
demonstrated this?

2) I have read that Fractional Brownian motion is not stationary, but is
intrinsically stationary. I thought FBM is stationary when -1/2<d<1/2.
Could someone clarify this?

What I am trying to so is simple.

Generate a univariate time series X(t(1)),X(t(2)),X(t(3))...X(t(N))
according to an intrinsically stationary process which isn't stationary
in the usual sense. Then demonstrate that the covariance fails where the
variogram succeeds.

Can anyone suggest an easily simulatable process for X(t) that would do
this for me?

Ultimately I would like to argue that intrinsic stationarity is a useful
concept for financial processes, but whilst it seems reasonable and
apparently un-tested to a large degree in the geo-stats I've seen, I'd
like some firm evidence of it's usefulness.

Any suggestions...

Robert

--
Robert J T Hillman
http://www.city.ac.uk/cubs/ferc/robert/index.html

Research Fellow
Financial Econometrics Research Centre
Frobisher Crescent
The Barbican
LONDON
EC2Y 8HB

tel: +44 (0) 171 477 8734 Direct Line
tel: +44 (0) 171 477 8611 Secretary
fax: +44 (0) 171 477 8881
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