CSPATIALREG Procedure
Spatial Autoregressive Models
The spatial autoregressive (SAR) model is useful for incorporating spatial dependence in the dependent variable—that is, the endogenous interaction effect. Let 
denote the observation that is associated with a spatial unit 
for 
. For these spatial units, let an 
matrix 
with nonnegative elements be a spatial weights matrix and let 
be a 
vector that denotes values of p regressors that are recorded for the spatial unit . You can formulate the SAR model as
where , 
is the spatial autoregressive coefficient, 
is a parameter vector, 
is the 
th element of the matrix subject to 
, and the error term 
that is related to the spatial unit is assumed to follow 
for .
The SAR model is often described in vector form as
where 
, 
is an 
matrix where each row consists of 
, and 
.
The standard estimator for the SAR model is the maximum likelihood estimator (MLE). For the SAR model, the log-likelihood function is (Anselin 2001)
where 
, 
is an 
identity matrix, and 
denotes the determinant of 
.
You can derive the gradients as follows:
For the matrix , 
, where 
is the ith diagonal element of .
A SAR model does not account for exogenous interaction effects. In practice, the value of the dependent variable y for a spatial unit might also be affected by some independent exploratory variables of other spatial units. In such a case, you can use the spatial Durbin model instead.