On the Relationship between Conditional (CAR) and Simultaneous (SAR) Autoregressive Models

TL;DR

This paper explores the fundamental relationships between conditional and simultaneous autoregressive models, demonstrating their equivalence and differences, and applying these insights to real-world spatial data analysis.

Contribution

It establishes the uniqueness of SAR models as representations of CAR models and shows how any multivariate Gaussian distribution can be expressed as either model, with practical applications.

Findings

A SAR model can be uniquely written as a CAR model.

Any multivariate Gaussian distribution can be represented as a CAR or SAR model.

Geostatistical models outperform first-order CAR models in likelihood optimization.

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.