SPATIALREG Procedure
Spatial Autoregressive Models
The spatial autoregressive (SAR) model is useful for incorporating the spatial dependence in the dependent variable—that is, the endogenous interaction effect. Let 
denote the observation associated with a spatial unit 
for 
. For these spatial units, let an 
matrix 
with nonnegative elements be a spatial weights matrix. Further, let 
be a 
vector that denotes values of p regressors recorded for the spatial unit . The SAR model can be formulated as
where . Here 
is the spatial autoregressive coefficient and 
is a parameter vector. Moreover, 
is the 
th element of the matrix subject to 
. For the error term 
related to the spatial unit , it is assumed that 
.
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 
, with 
being an 
identity matrix. 
denotes the determinant of 
.
The gradients can be derived as follows:
For the matrix , 
, where 
is the ith diagonal element of .
A SAR model does not account for exogenous interaction effects. However, in practice, the value of the dependent variable y for a spatial unit might be affected by some independent exploratory variables of other spatial units as well. In such a case, you can use the SDM model instead.