Measurement error in multilevel models with sample clustermeans

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 omittedvariable 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 residual level 2 variance is inflated. After outlining a suitable framework, we derive explicit formulae for measurement error biases that allow to implement simple corrections. The theoretical analysis is supplemented with a simulation study and a discussion of the implications for effectiveness evaluation.