CSPATIALREG Procedure
Conditional Autoregressive Models
Let 
denote the observation that is associated with a spatial unit 
for 
, and let 
be a 
vector that denotes values of p regressors recorded at unit .
The conditional autoregressive (CAR) model is defined by the set of full conditional distributions as
where 
and 
is an 
matrix whose 
th element is 
.
Denote 
. Under the conditions that 
is symmetric and 
is positive definite, the joint distribution of 
is well defined as
where 
is an 
matrix in which each row consists of 
and 
.
The log-likelihood function for the CAR model takes the form
Often, it is considered that 
, where 
is a spatial weights matrix, and
where 
, in which 
is the sum of entries in the ith row of the matrix for .
Let 
be defined as
The gradients for the CAR model can be derived as
where 
.