> I have never encountered a situation such as you mention. Of course, I

to a

> usually restrict myself to row-stochastic matrices or matrices similar

> row-stochastic matrix. In addition, positive spatial dependence is

almost

> guaranteed for my data.

The cases where I was seeing this happen (i.e. estimated rho for SAR

greater than estimated rho for CAR) was when I was using a

non-row-stochastic matrix. In fact, for the same data, when I simply

changed W from a matrix of zeroes and ones to a row-stochasic matrix, the

values of rho for the SAR model fell into the region < 1, i.e. ensuring

that (I - rho W) is positive definite. Also when that happen, as you

suggested the rho for the SAR model was smaller than that for the CAR

model as we may expect.

>

The answer to this question,

> By the way, did the SAR or CAR have the highest likelihood?

>

using W with 0,1 entries

SAR Log-likelihood = -219.39 (rho parameter > 1/(max(eigenW))

CAR Log-likelihood = -215.6591

using W with row-stochastic matrix (i.e. rows sum to one)

SAR Log-likelihood = -212.1987 (rho parameter < 1)

CAR Log-likelihood = -213.0049

Is there some fundamental reason why one should not be using a W with 0,1

entries and instead should favor the row-standardized W?

-----------------------

Melanie Wall

Division of Biostatistics

A460 Mayo Building Box 303

420 Delaware Street S.E.

Minneapolis, MN 55455

(612)625-2138

melanie@...

On Sat, 13 Nov 1999, Robert K. Pace wrote:

>

> 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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