  We develop a methodology for conducting inference on extreme quantiles of
unobserved individual heterogeneity (heterogeneous coefficients, heterogeneous
treatment effects, etc.) in a panel data or meta-analysis setting. Inference in
such settings 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 and an intermediate order theorem for noisy
estimates and appropriate rate and moment conditions. Both theorems are then
used to construct confidence intervals for extremal quantiles. The intervals
are simple to construct and require no optimization. Inference based on the
intermediate order theorem involves a novel self-normalized intermediate order
theorem. In simulations, our extremal confidence intervals have favorable
coverage properties in the tail. Our methodology is illustrated with an
application to firm productivity in denser and less dense areas.
