A tale of two “AR” models: a spatial analysis of Corsican second home incidence

Spatial autoregressive (AR) models can accommodate various forms of dependence among data with discrete support in a space, and hence are widely used in economics and social science. We examine the relationship between spatial (autoregressive) error models and conditional autoregressive models, considered to be the two main types of spatial AR models. This topic is likely incomplete in the literature and is often overlooked by econometricians. To further develop and broaden this topic, we demonstrate that spatial error and conditional autoregressive models can be made equivalent via hierarchal models, but have different variance-covariance matrices. We then propose a Bayesian approach, known as integrated nested Laplace approximations (INLA), to produce accurate estimates for these models and to speed up inferences. We also discuss how to interpret model coefficients, especially estimates of spatial latent effects. We illustrate the two AR models with the proposed methodology in an application to the second home incidence rates of Corsica, France in 2017. We find that both models can capture spatial dependence, but conditional autoregressive models perform slightly better and produce a higher spatial autocorrelation coefficient. We further illustrate estimates of latent effects by identifying several “hot spots” and “cold spots” in terms of second home incidence rates.