Working Paper Series CONDITIONAL EXTREMES AND NEAR-EXTREMES

This paper develops a theory of high and low (extremal) quantile regression: the linear models, estimation, and inference. In particular, the models coherently com-bine the convenient, flexible linearity with the extreme-value-theoretic restrictions on tails and the general heteroscedasticity forms. Within these models, the limit laws for extremal quantile regression statistics are obtained under the rank conditions (exper-iments) constructed to reflect the extremal or rare nature of tail events. An inference framework is discussed. The results apply to cross-section (and possibly dependent) data. The applications, ranging from the analysis of babies ' very low birthweights, (5, s) models, tail analysis in heteroscedastic regression models, outlier-robust infer-ence in auction models, and decision-making under extreme uncertainty, provide th