Handling Correlations Between Covariates and Random Slopes in Multilevel Models

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 (intercepts and slopes) and included covariates, which we refer to as cluster-level endogeneity, lead to bias when using standard random effects (RE) estimators such as (restricted) maximum likelihood. While the problem of correlations between unit-level covariates and random intercepts is well known and can be handled by fixed-effects (FE) 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 (PC) 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.