The SPATIALREG Procedure
Spatial Durbin Models
Unlike a SAR model, a spatial Durbin model (SDM) can account for exogenous interaction effects in addition to the endogenous interaction effects. Let 
denote the observation associated with a spatial unit 
for 
. For these spatial units, let 
be an 
spatial weights matrix of your choice. Further, assume that 
is a 
vector that denotes values of p regressors recorded for the spatial unit . Similarly, assume that 
is a 
vector that denotes values of q regressors measured at unit .
The SDM model can be described in vector form as (LeSage and Pace 2009)
where 
and 
with 
. Moreover, 
is an 
matrix where each row consists of 
, and 
is an 
matrix where each row consists of 
. In addition, 
and 
are and parameter vectors, respectively.
By letting 
and 
, you can rewrite the SDM model as
The log-likelihood function for the SDM model is
where 
.
For the SDM model, the gradients are
Both the SAR model and the SDM model account for endogenous interaction effects. However, in some cases there might be an interaction among error terms. In such cases, you might consider a spatial error model, which addresses spatial interaction among error terms.