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1419GEOSTATS: simulating intrinsically stationary processes

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  • Hillman RJT
    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 J T Hillman

      Research Fellow
      Financial Econometrics Research Centre
      City University Business School
      Frobisher Crescent
      The Barbican
      EC2Y 8HB

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