Extremal quantile regression: an overview

Abstract

Extremal quantile regression, i.e. quantile regression applied to the tails of the conditional distribution, counts with an increasing number of economic and financial applications such as value-at-risk, production frontiers, determinants of low infant birth weights, and auction models. This chapter provides an overview of recent developments in the theory and empirics of extremal quantile regression. The advances in the theory have relied on the use of extreme value approximations to the law of the Koenker and Bassett (1978) quantile regression estimator. Extreme value laws not only have been shown to provide more accurate approximations than Gaussian laws at the tails, but also have served as the basis to develop bias corrected estimators and inference methods using simulation and suitable variations of bootstrap and subsampling. The applicability of these methods is illustrated with two empirical examples on conditional value-at-risk and financial contagion.

Estimating selection models without an instrument with Stata

“…As is standard with extremal quantile regressions (see Chernozhukov et al, 2017), the rate of convergence is not the usual parametric root-n rate. Moreover, in this case, this rate depends on unknown features of the distribution of (D, Y * , X).…”

Factorisable Multitask Quantile Regression

“…As τ is close to zero or one, the 1/f τ increases and the bounds (3.8) and (3.9) in Theorem 3.2 become loose, suggesting the estimator may be inaccurate. Extreme quantile is often characterized through the low extremal order or extremal rank τ n (Chernozhukov, 1999(Chernozhukov, , 2005Chernozhukov, Fernández-Val, and Kaji, 2017). In particular, if nτ → c for some c ≥ 0 as τ → 0 as n → ∞ (respectively, n(1 − τ ) → c as τ → 1 for the right extreme quantile, by symmetry we only discuss the left extreme quantile), classical asymptotic analysis breaks down if c is small or equal to 0, which is called the "extreme" quantile.…”

Stress Testing German Industry Sectors: Results from a Vine Copula Based Quantile Regression

“…In the context of stress testing, the focus is on rare events arising from the tails of the distribution, which motivates the application of so-called extremal QR, or QR applied to the tails, (see, e.g., Chernozhukov (2005) of Chernozhukov et al (2017)). The quantile regression method is used to estimate parameters in accordance with the distribution of a dependent variable.…”