INFERENCE ON EXTREME QUANTILES OF UNOBSERVED INDIVIDUAL HETEROGENEITY
针对面板数据和元分析中未观测个体异质性的极端分位数，提出一种推断方法，推导了噪声估计对极端分位数有信息性的充要条件，并给出无需优化的置信区间。
We develop a methodology for conducting inference on extreme quantiles of unobserved individual heterogeneity (e.g., heterogeneous coefficients and 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 (CIs) for extreme quantiles. Simulations show that our CIs 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 across areas of varying population density. By analyzing the left tails of the productivity distributions, we find no evidence of stronger firm selection in more densely populated areas.