Inference on Extreme Quantiles of Unobserved Individual Heterogeneity

We develop a methodology for conducting inference on extreme quantiles of unobserved individual heterogeneity (heterogeneous coefficients, heterogeneous treatment effects, and other unobserved heterogeneity) in a panel data or meta-analysis setting. Examples of interest include productivity of most and least productive firms or prediction intervals for study-specific treatment effects in meta-analysis. Inference in such a setting is challenging. Only noisy estimates of unobserved heterogeneity are available, and approximations based on the central limit theorem work poorly for extreme quantiles. For this situation, under weak assumptions we derive an extreme value theorem for noisy estimates and appropriate rate and moment conditions. In addition, we develop a theory for intermediate order statistics. Both extreme and intermediate order theorems are then used to construct confidence intervals for extremal quantiles. The limiting distribution is non-pivotal, and we show consistency of both subsampling and simulating from the limit distribution. Furthermore, we provide a novel self-normalized intermediate order theorem. In a Monte Carlo exercise, we show that the resulting extremal confidence intervals have favorable coverage properties in the tail.