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This paper focuses on a setting with observations having a cluster dependencestructure and presents two main impossibility results. First, we show that whenthere is only one large cluster, i.e., the researcher does not have anyknowledge on the dependence structure of the observations, it is not possibleto consistently discriminate the mean. When within-cluster observations satisfythe uniform central limit theorem, we also show that a sufficient condition forconsistent $\sqrt{n}$-discrimination of the mean is that we have at least twolarge clusters. This result shows some limitations for inference when we lackinformation on the dependence structure of observations. Our second resultprovides a necessary and sufficient condition for the cluster structure thatthe long run variance is consistently estimable. Our result implies that whenthere is at least one large cluster, the long run variance is not consistentlyestimable.

Some Impossibility Results for Inference With Cluster Dependence with Large Clusters