Inference on Extreme Quantiles of Unobserved Individual Heterogeneity

Abstract: We develop a methodology for conducting inference on extreme quantiles of unobserved individual heterogeneity (e.g., heterogeneous coefficients, treatment effects) in panel data and meta-analysis settings. Inference is challenging in such settings: only noisy estimates of heterogeneity are available, and central limit approximations perform poorly in the tails. We derive a necessary and sufficient condition under which noisy estimates are informative about extreme quantiles, along with sufficient rate and moment conditions. Under these conditions, we establish an extreme value theorem and an intermediate order theorem for noisy estimates. These results yield simple optimization-free confidence intervals for extreme quantiles. Simulations show that our confidence intervals have favorable coverage and that the rate conditions matter for the validity of inference. We illustrate the method with an application to firm productivity differences between denser and less dense areas.