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.