Cautions on Tail Index Regressions and a Comparative Study with Extremal Quantile Regression

Abstract

We re-visit tail the index regressions framework. For linear specifications, we find that the usual full rank condition can fail because conditioning on extreme outcomes causes regressors to degenerate to constants. Taking this into account, we provide additional regular conditions and establish its asymptotics in this irregular setup. For more general specifications, the conditional distribution of the covariates in the tails concentrates on the values at which the tail index is minimized. Such issue does not exist for the extremal quantile regression framework, where the tail index is assumed constant. Simulations support these findings. Using daily S&P 500 returns, we find that the extremal quantile regression framework appears more suitable than tail-index regression with respect to the tail rank condition.