  This paper establishes asymptotic results for the maximum likelihood and
restricted maximum likelihood (REML) estimators of the parameters in the nested
error regression model for clustered data when both of the number of
independent clusters and the cluster sizes (the number of observations in each
cluster) go to infinity. Under very mild conditions, the estimators are shown
to be asymptotically normal with an elegantly structured covariance matrix.
There are no restrictions on the rate at which the cluster size tends to
infinity but it turns out that we need to treat within cluster parameters (i.e.
coefficients of unit-level covariates that vary within clusters and the within
cluster variance) differently from between cluster parameters (i.e.
coefficients of cluster-level covariates that are constant within clusters and
the between cluster variance) because they require different normalisations and
are asymptotically independent.
