MCMC sampling for a random intercepts model with non-independent residuals within and between cluster units

In this article, we discuss the effect of removing the independence assumptions between the residuals in two-level random effect models. We first consider removing the independence between the Level 2 residuals and instead assume that the vector of all residuals at the cluster level follows a general multivariate normal distribution. We demonstrate how this assumption can allow us to fit higher levels of clustering and school competition effects via an example from education. We then consider removing the assumption of independence between Level 1 residuals within clusters. We show how this extension can allow time series type models. Both normal and binary responses are considered.