ABSTRACT
This chapter reviews the theory of extremal quantile regression. It shows that each of the sequences produces different asymptotic approximation to the distribution of the quantile regression estimators. The chapter also reviews models for marginal and conditional extreme quantiles. It describes estimation and inference methods for extreme quantile models. It presents two empirical applications of extremal quantile regression to conditional VaR and financial contagion. The chapter provides typical modeling assumptions in extremal quantile regression. It discusses the estimation and inference methods for extremal quantile regression. Very low birthweights are connected with subsequent health problems and therefore extremal quantile regression can help identify factors to improve adult health outcomes. Zhang employed extremal quantile regression methods to estimate tail quantile treatment effects under a selection on observables assumption. The work of Portnoy and Koenker and Gutenbrunner et al. implicitly contained some results on extending the normal approximations to intermediate order regression quantiles in location models.
