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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
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.

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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