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  • Melanie Wall
    I have a question about the parameter space of the rho parameter in a SAR model. Consider the following two dispersion matrices Sigma for Spatial Regression
    Message 1 of 1 , Nov 12, 1999
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      I have a question about the parameter space of the rho
      parameter in a SAR model.

      Consider the following two dispersion matrices Sigma for Spatial
      Regression Models. Let W be a symmetric neighbor matrix.

      CAR: Sigma = (I - rho W)^{-1}
      SAR: Sigma = [t(I - rho W) %*% (I - rho W)]^{-1}

      Everywhere I look, people discuss the restrictions on rho so that (I - rho
      W) is positive definite. It is very straightforward to show the
      conditions on rho that satisfy this condition as they are related to the
      eigenvalues of W. Obviously for the CAR model, we should be restricting
      ourself to (I -rho W) positive definite, but, in the SAR model there is no
      need to restrict (I - rho W) to be positive definite, instead all we need
      is invertibility (i.e. non singular). The problem with this is, it is not
      much of a restriction because all it requires is that rho does not = the
      inverse of any of the eigenvalues of W. Thus there is really no upper
      bound on what rho can be to ensure (I- rho W) to be invertible
      (but not positive definite). Has anyone ever seen any discussion of
      this?

      I have already run across two examples where if I use the CAR model in S+
      Spatial Stat's slm() function, I get a rho value that is less than
      1/(max(eigenvalue of W) (as it should be to ensure (I - rho W) positive
      definite). But then, when I run a SAR model using the exact same data I
      get a rho value outside the range that would ensure positive definiteness.
      That is (I - rho W) is invertible but not positive definite. It seems to
      me that it is hard to interpret an estimated rho if there is no upper
      bound on what it could be.

      Does anyone have any comments about this topic.

      Thanks
      MW

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


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