Bayesian estimation of the multilevel/hierarchical spatially autocorrelated random intercept model

Multilevel or Hierarchical models are statistical models that allow for parameter estimation at more than one level. Many empirical research questions in regional science involve hierarchically--structured data, e.g counties nested within states. Standard normal linear models are ill--suited to estimate these models because they ignore by assumption this nesting of the data. One type of hierarchical model is the random intercept models; it allow for the separate estimation of an intercept for each group at the second level of the hierarchy, e.g. states in a model with counties nested within states. To date, little work has been done on capturing spatial dynamics in hierarchical models; for instance the idea that the random intercept at the second (or group) level in the hierarchical model might follow a spatial process. In this paper we introduce a model that considers the hierarchical random intercept term as following a spatial autoregressive process. Intuitively, if we consider the upper level random intercept term in this way, we are able to test whether or not there is spatial dependence in the random intercepts between the upper level groupings (states for instance). Using now familiar Bayesian computational techniques, this paper derives and illustrates the hierarchical spatially autocorrelated random intercept model that includes covariate information at the second level of the hierarchy. This model has a number of attractive potential applications, for instance to examine crime patterns within neighbourhoods nested within police divisions, or looking at individual labour market outcomes, while nesting individuals within interlinked local labour market areas.