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GEOSTATS: Re: Message from Internet

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  • Melanie Wall
    Kelley ... to a ... almost ... The cases where I was seeing this happen (i.e. estimated rho for SAR greater than estimated rho for CAR) was when I was using a
    Message 1 of 1 , Nov 19, 1999
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      Kelley

      > I have never encountered a situation such as you mention. Of course, I
      > usually restrict myself to row-stochastic matrices or matrices similar
      to a
      > row-stochastic matrix. In addition, positive spatial dependence is
      almost
      > guaranteed for my data.

      The cases where I was seeing this happen (i.e. estimated rho for SAR
      greater than estimated rho for CAR) was when I was using a
      non-row-stochastic matrix. In fact, for the same data, when I simply
      changed W from a matrix of zeroes and ones to a row-stochasic matrix, the
      values of rho for the SAR model fell into the region < 1, i.e. ensuring
      that (I - rho W) is positive definite. Also when that happen, as you
      suggested the rho for the SAR model was smaller than that for the CAR
      model as we may expect.

      >
      > By the way, did the SAR or CAR have the highest likelihood?
      >

      The answer to this question,

      using W with 0,1 entries
      SAR Log-likelihood = -219.39 (rho parameter > 1/(max(eigenW))
      CAR Log-likelihood = -215.6591

      using W with row-stochastic matrix (i.e. rows sum to one)
      SAR Log-likelihood = -212.1987 (rho parameter < 1)
      CAR Log-likelihood = -213.0049

      Is there some fundamental reason why one should not be using a W with 0,1
      entries and instead should favor the row-standardized W?


      -----------------------
      Melanie Wall
      Division of Biostatistics
      A460 Mayo Building Box 303
      420 Delaware Street S.E.
      Minneapolis, MN 55455
      (612)625-2138
      melanie@...

      On Sat, 13 Nov 1999, Robert K. Pace wrote:

      >
      > Kelejian and Robinson discuss this issue in:
      >
      > Kelejian and Robinson (1995), "Spatial Correlation: A Suggested Alternative
      > to the Autoregressive Model," in New Directions in Spatial Econometrics
      > edited by Luc Anselin and R. Florax, Springer
      >
      >
      > Of course, as you state, the variance-covariance matrix and the inverse
      > variance-covariance matrix are still p.d.
      >
      >
      > I have never encountered a situation such as you mention. Of course, I
      > usually restrict myself to row-stochastic matrices or matrices similar to a
      > row-stochastic matrix. In addition, positive spatial dependence is almost
      > guaranteed for my data.
      >
      >
      >
      > I do have one idea (possibly bad!). Consider the SAR variance-covariance
      > matrix that you mentioned with symmetric W.
      >
      > SAR: Sigma=inv((I-2 rho W + rho^2 W'W))
      >
      > Now W'W=WW by symmetry of W. The multiplication of an adjacency matrix by
      > an adjacency matrix (i.e., WW) captures the effect of neighbors of the
      > neighbors and by construction this has a positive weight in inv(Sigma) via
      > rho^2. A large value for rho may be allowing the model to differentially
      > weigh the nearby and far dependence. Hence, by respecifying W (picking via
      > max likelihood) one might be able to obtain values for rho more in the
      > conventional range.
      >
      >
      >
      > Ripley (1981) points out that for small values of rho, SAR will usually
      > produce autoregressive estimates approximately half that of CAR. This is
      > easy to see with the above formula as the squared rho term virtually
      > vanishes for small rho. In my work, SAR has always had a lower magnitude
      > autoregressive parameter than CAR.
      >
      >
      > By the way, did the SAR or CAR have the highest likelihood?
      >
      >
      >
      >
      >
      > Kelley Pace
      > 2164 CEBA Building
      > Department of Finance
      > Louisiana State University
      > Baton Rouge, LA 70803-6308
      > kelley@...
      > www.spatial-statistics.com
      > www.finance.lsu.edu/re
      > 225-388-6256
      > 225-334-1227 (FAX)
      >


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