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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