Some Impossibility Results for Inference With Cluster Dependence with Large Clusters.

There have appeared in the literature various approaches of inference for models with cluster dependence with few clusters, where the dependence structure within each cluster is unknown. These approaches are all different from the "standard" approach of asymptotic inference based on an asymptotically pivotal test. One may wonder whether it is possible to develop a standard asymptotic inference in this situation. To answer this question, we focus on a Gaussian experiment, and present a necessary and sufficient condition for the cluster structure that the long run variance is consistently estimable. Our result implies that when there is at least one large cluster, the long run variance is not consistently estimable, and hence, the standard approach of inference based on an asymptotically pivotal test is not possible. This impossibility result extends to other models that contain a Gaussian experiment as a special case. As a second focus, we investigate the consistent discrimination of the common mean from a sample with cluster dependence. We show that when the observations consist of large clusters, it is necessary for the consistent discrimination of the mean that the sample has at least two large clusters. This means that if one does not know the dependence structure at all, it is not possible to consistently discriminate the mean.