On the relationship between conditional (CAR) and simultaneous (SAR) autoregressive models

Spatial Statistics
By: , and 

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Abstract

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.

Additional publication details

Publication type Article
Publication Subtype Journal Article
Title On the relationship between conditional (CAR) and simultaneous (SAR) autoregressive models
Series title Spatial Statistics
DOI 10.1016/j.spasta.2018.04.006
Volume 25
Year Published 2019
Language English
Publisher Wiley
Contributing office(s) Core Research Center, Colorado Water Science Center, Fort Collins Science Center, Coop Res Unit Seattle
Description 18 p.
First page 68
Last page 85