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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