SPATIALREG Procedure
Spatial Autoregressive Moving Average Models
The spatial autoregressive moving average (SARMA) model, like the SMA model, can account for spatial dependence among the error terms. In addition, the SARMA model enables you to account for spatial dependence in the dependent variable, as the SAR model does. Let 
denote the observation associated with the spatial unit 
for 
. For these spatial units, let the 
matrices 
and 
with nonnegative elements be two spatial weights matrices. In practice, and can be identical. Further, it is assumed that 
is a 
vector that denotes values of p covariates recorded at unit .
The SARMA model can be described in the vector form by using the following two-stage formulation (LeSage and Pace 2009),
where 
and 
, with 
. Moreover, 
is an 
matrix that consists of 
in each row. In addition, 
is a parameter vector, and 
is an 
identity matrix.
The log-likelihood function for the SARMA model is
where 
, 
and 
denotes the matrix determinant operator.
For the SARMA model, the gradients are
