Modeling Clustered Heterogeneity: Fixed Effects, Random Effects and Mixtures

Although each statistical unit on which measurements are taken is unique typically there is not enough information available to account totally for its uniqueness. Therefore heterogeneity among units has to be limited by structural assumptions. One classical approach is to use random effects models, which assume that heterogeneity can be described by distributional assumptions. However, inference may depend on the assumed mixing distribution and it is assumed that the random effects and the observed covariates are independent. An alternative considered here, are fixed effect models, which let each unit have its own parameter. They are quite flexible but suffer from the large number of parameters. The structural assumption made here is that there are clusters of units that share the same effects. It is shown how clusters can be identified by tailored regularized estimators. Moreover, it is shown that the regularized estimates compete well with estimates for the random effects model, even if the latter is the data generating model. They dominate if clusters are present.