A close look at the spatial structure implied by the CAR and SAR models

Modeling spatial interactions that arise in spatially referenced data is commonly done by incorporating the spatial dependence into the covariance structure either explicitly or implicitly via an autoregressive model. In the case of lattice (regional summary) data, two common autoregressive models used are the conditional autoregressive model (CAR) and the simultaneously autoregressive model (SAR). Both of these models produce spatial dependence in the covariance structure as a function of a neighbor matrix W and often a 1xed unknown spatial correlation parameter. This paper examines in detail the correlation structures implied by these models as applied to an irregular lattice in an attempt to demonstrate their many counterintuitive or impractical results. A data example is used for illustration where US statewide average SAT verbal scores are modeled and examined for spatial structure using di4erent spatial models. c © 2003 Elsevier B.V. All rights reserved.