  Hierarchical autocorrelation in the error term of linear models arises when
sampling units are related to each other according to a tree. The residual
covariance is parametrized using the tree-distance between sampling units. When
observations are modeled using an Ornstein-Uhlenbeck (OU) process along the
tree, the autocorrelation between two tips decreases exponentially with their
tree distance. These models are most often applied in evolutionary biology,
when tips represent biological species and the OU process parameters represent
the strength and direction of natural selection. For these models, we show that
the mean is not microergodic: no estimator can ever be consistent for this
parameter and provide a lower bound for the variance of its MLE. For covariance
parameters, we give a general sufficient condition ensuring microergodicity.
This condition suggests that some parameters may not be estimated at the same
rate as others. We show that, indeed, maximum likelihood estimators of the
autocorrelation parameter converge at a slower rate than that of generally
microergodic parameters. We showed this theoretically in a symmetric tree
asymptotic framework and through simulations on a large real tree comprising
4507 mammal species.
