Cluster effects and simultaneity in multilevel models

For small group sizes, the Generalised Least Squares (CLS) estimator in multilevel models is biased and inconsistent when the random cluster effects are correlated with the regressors. A fixed effects approach, conditioning on the cluster effects, provides consistent estimates for the slope parameters. The two estimators are equivalent when group sizes are large. The same results obtain for two stage estimation procedures that allow for some of the regressors to be simultaneously determined with the variable. The GLS and fixed effects estimators are applied to data on hospital utilization in the UK, allowing for health authority district effects. These data and models have been used by Smith et al. (1994) for developing a formula for the regional distribution of the UK National Health Service revenues.