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• ## GEOSTATS: simulating intrinsically stationary processes

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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
Message 1 of 2 , 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...

thanks in advance

Robert

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

Research Fellow
Financial Econometrics Research Centre
City University Business School
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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• For estimation/simulation with a finite search neighborhood, assuming an intrinsic hypothesis becomes quite tangential to the main point of interest -- i.e.,
Message 2 of 2 , Nov 30, 1999
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For estimation/simulation with a finite search neighborhood,
assuming an intrinsic hypothesis becomes quite tangential
to the main point of interest -- i.e., generating good estimates.

Assuming something is fBm is just that -- an assumption. It might even
be a valid one where variance increases without bounds, but still
other such concerns as the size of the search radius, number of
estimation control points, etc will still have an impact on overall
estimation quality.

Syed Abdul Rahman
Landmark Graphics
Jakarta, Indonesia

PS/- fBm is nonstationary but it's derivative -- fGn -- is. Easiest
simulation is to generate a Gaussian distribution using a random number
generator, then integrate to get fBm with Hurst exponent 0.5.

PS2/- _Sample_ covariance will never "fail" if calculated on
a simulated fBm trace. Does "fail" mean an inability to generate
"good" estimates, e.g. from a comparison of cross-validated
residuals?

----- Original Message -----
From: Hillman RJT <R.J.T.Hillman@...>
To: <ai-geostats@...>
Sent: Tuesday, November 30, 1999 10:02 PM
Subject: GEOSTATS: simulating intrinsically stationary processes

> 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...
>
> thanks in advance
>
> Robert
>
> --
> Robert J T Hillman
> http://www.city.ac.uk/cubs/ferc/robert/index.html
>
>
> Research Fellow
> Financial Econometrics Research Centre
> City University Business School
> 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
> --
> *To post a message to the list, send it to ai-geostats@....
> *As a general service to list users, please remember to post a summary
> of any useful responses to your questions.
> *To unsubscribe, send email to majordomo@... with no subject and
> "unsubscribe ai-geostats" in the message body.
> DO NOT SEND Subscribe/Unsubscribe requests to the list!
>

--
*To post a message to the list, send it to ai-geostats@....
*As a general service to list users, please remember to post a summary
of any useful responses to your questions.
*To unsubscribe, send email to majordomo@... with no subject and
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DO NOT SEND Subscribe/Unsubscribe requests to the list!
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