Commercial Statistical Packages by Daniel A. Griffith. One area of spatial analysis that has received considerable treatment during the past two ... The data preprocessing involves calculating then eigenvalues hi of matrix W, and calculating ...
Research Notes and Comments Estimating Spatial Autoregressive Model Parameters with Commercial Statistical Packages by Daniel A. Griffith One area of spatial analysis that has received considerable treatment during the past two decades is spatial autoregressive models. Useful and readable introductions to this material for the statistically literate researcher are provided by Cliff and Ord (1981), Upton and Fingleton (1985),and Griffith (1987).The basic statistical problem in spatial autoregressive modeling arises from a dependence of observations. Capturing these dependencies greatly reduces the residual variance and strengthens the inferential basis affiliated with a model. One of the hurdles most quantitative spatial analysts encounter when dealing with spatial modeling is the lack of appropriate commercial computer software for estimating parameters of spatial autoregressive models. While time series routines are available in packages such as MINITAB, SPSS, and SAS, spatial series routines are not. Those researchers who have calibrated spatial models in the past have done so by doing extensive personal programming using libraries such as IMSL. But with a minimum of data preprocessingthese former packages can be tricked into calibrating spatial autoregressive models. This paper will outline one method of doing so using MINITAB and SAS. The simultaneous spatial autoregressive (SAR) model will be used to illustrate the procedure. PROPERTIES OF THE SAR MODEL
Consider a random variate Y having a geographic distribution over a set of n areal units whose configuration is depicted by matrix C. Let matrix C be converted to its stochastic counterpart, matrix W. The SAR model has as its covariance matrix
d[(I
- pW)T(I - pW)]-l
.
If, as is commonly done, a map is viewed as being a sample of size one from a multivariate normal distribution, then the maximum likelihood estimates of the SAR parameters fl (for a set of fixed effects X variate), u 2 , and pare given by
G2
=
(Y - xg)T ( I -
pW) T ( I - pW)
(Y - xgyn,
Daniel A. Griffith is in the department of geography, State University of N e w York at Buffalo. Geographical Analysis, Vol. 20, No. 2 (April 1988)@ 1988 Ohio State University Press
Submitted 7/87. Revised version accepted 10/87.
Research Notes and Comments /
S = [ X T ( I - pW) T(I - pW)
177
X ] - ' X T ( I - pW) T(I - pW)Y , and
This last expression is what prevents OLS from being used to estimate p. is a constant. In fact, it can be
For any givenvalueof p,
rewritten as the product of two identical terms, namely:
Furthermore, (I - pW)(Y - X/3)=Y - pWY - Xf3 + pWXB = E, SO that the second term in the expression is a sum of squared errors term. Hence, the minimization in question is equivalent to one for the model:
Y/K = pWY/K
+ XB/K
-
pWXB/K
+ E* .
(2)
In the form defined by equation (2), the minimization problem MIN: ( E * ) T E * becomes MIN: (Y/K - pWY/K - XB/K - P W X B / K ) ~X ( Y / K - pWY/K - XB/K - pWX@/K) , which is equivalent to expression ( l ) ,once the constraint Ip 1
