Multilevel Models Where the Random Effects Are Correlated with the Fixed Predictors

For small group sizes, the multilevel iterative generalised least squares (IGLS) estimator is biased and inconsistent where the random effects are correlated with the fixed predictors. Consistent estimates of the parameters of endogenous variables may be obtained using instrumental variables or conditioning on group level effects. In this paper we review various approaches to ensure consistency in panel data models and extend these to the general class of multilevel models. Further, by exploiting the iterative nature of the IGLS algorithm we derive an unbiased and consistent estimator based on conditioning on estimated group effects. The method proposed provides consistent estimation of the endogenous regression parameters of interest whilst retaining the properties of multilevel models via efficient estimation and full exploration of residual heterogeneity. The proposed estimator is termed conditioned iterative generalised least squares (CIGLS).