GEOSTATS: Spatial autoregressive parameter restrictions
- 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
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?
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Department of Finance
Louisiana State University
Baton Rouge, LA 70803-6308
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