The role of sample cluster means in multilevel models: a view on endogeneity and measurement error issues

The paper explores some issues related to endogeneity in multilevel models, focusing on the case where the random effects are correlated with a level 1 covariate in a linear random intercept model. We consider two basic specifications, without and with the sample cluster mean. It is generally acknowledged that the omission of the cluster mean may cause omitted-variable bias. However, it is often neglected that the inclusion of the sample cluster mean in place of the population cluster mean entails a measurement error that yields biased estimators for both the slopes and the variance components. In particular, the contextual effect is attenuated, while the level 2 variance is inflated. We derive explicit formulae for measurement error biases that allow us to implement simple post-estimation corrections based on the reliability of the covariate. In the first part of the paper, the issue is tackled in a standard framework where the population cluster mean is treated as a latent variable. Later we consider a different framework arising when sampling from clusters of finite size, where the latent variable methods may have a poor performance, and we show how to effectively modify the measurement error correction. The theoretical analysis is supplemented with a simulation study and a discussion of the implications for effectiveness evaluation.