© 2014 AERA. This article discusses estimation of multilevel/hierarchical linear models that include cluster-level random intercepts and random slopes. Viewing the models as structural, the random intercepts and slopes represent the effects of omitted cluster-level covariates that may be correlated with included covariates. The resulting correlations between random effects and included covariates, which we refer to as “cluster-level endogeneity,” lead to bias when using standard random effects e…

Read more© 2014 AERA. This article discusses estimation of multilevel/hierarchical linear models that include cluster-level random intercepts and random slopes. Viewing the models as structural, the random intercepts and slopes represent the effects of omitted cluster-level covariates that may be correlated with included covariates. The resulting correlations between random effects and included covariates, which we refer to as “cluster-level endogeneity,” lead to bias when using standard random effects estimators such as maximum likelihood. While the problem of correlations between unit-level covariates and random intercepts is well known and can be handled by fixed-effects estimators, the problem of correlations between unit-level covariates and random slopes is rarely considered. When applied to models with random slopes, the standard FE estimator does not rely on standard cluster-level exogeneity assumptions, but requires an “uncorrelated variance assumption” that the variances of unit-level covariates are uncorrelated with their random slopes. We propose a “per-cluster regression” estimator that is straightforward to implement in standard software, and we show analytically that it is unbiased for all regression coefficients under cluster-level endogeneity and violation of the uncorrelated variance assumption. The PC, RE, and an augmented FE estimator are applied to a real data set and evaluated in a simulation study that demonstrates that our PC estimator performs well in practice.