  We clarify relationships between conditional (CAR) and simultaneous (SAR)
autoregressive models. We review the literature on this topic and find that it
is mostly incomplete. Our main result is that a SAR model can be written as a
unique CAR model, and while a CAR model can be written as a SAR model, it is
not unique. In fact, we show how any multivariate Gaussian distribution on a
finite set of points with a positive-definite covariance matrix can be written
as either a CAR or a SAR model. We illustrate how to obtain any number of SAR
covariance matrices from a single CAR covariance matrix by using Givens
rotation matrices on a simulated example. We also discuss sparseness in the
original CAR construction, and for the resulting SAR weights matrix. For a real
example, we use crime data in 49 neighborhoods from Columbus, Ohio, and show
that a geostatistical model optimizes the likelihood much better than typical
first-order CAR models. We then use the implied weights from the geostatistical
model to estimate CAR model parameters that provides the best overall
optimization.
