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GEOSTATS: Spatial autoregressive parameter restrictions

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  • Robert K. Pace
    Kelejian and Robinson discuss this issue in: Kelejian and Robinson (1995), Spatial Correlation: A Suggested Alternative to the Autoregressive Model, in New
    Message 1 of 1 , Nov 12 10:27 PM
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      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
      225-334-1227 (FAX)
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