CSPATIALREG 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 that is 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 that are recorded for the spatial unit and assume that 
is a 
vector that denotes values of q regressors that are measured at unit .
The SDM can be described in vector form as (LeSage and Pace 2009)
where 
, 
with 
, 
is an 
matrix where each row consists of 
, 
is an 
matrix where each row consists of 
, and 
and 
are and parameter vectors, respectively.
By letting 
and 
, you can rewrite the SDM as
The log-likelihood function for the SDM is
where 
.
For the SDM, the gradients are
Both the SDM and the SAR 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.