SPATIALREG Procedure
Spatial Durbin Autoregressive Moving Average Models
You can also accommodate exogenous interaction effects in the SARMA model. The term spatial Durbin autoregressive moving average (SDARMA) model is used to refer to such an extension of the SARMA model. Let 
denote the observation associated with the spatial unit 
for 
. For these spatial units, let 
and 
be two spatial weights matrices. Further, let 
be a 
vector that denotes values of p regressors recorded at unit . Similarly, let 
be a 
vector that denotes values of q regressors measured at unit .
The SDARMA model can be described in vector form by using the following two-stage formulation,
where 
and 
, with 
. Moreover, 
is an 
matrix that has 
in each row, 
is an 
matrix that has 
in each row. In addition, 
is a parameter vector.
By letting 
and 
, the SDARMA model can be written as
The log-likelihood function for the SDARMA model is
where 
and 
.
For the SDARMA model, the gradients are
