Estimation of High Conditional Tail Risk Based on Expectile Regression

Abstract

Assessing conditional tail risk at very high or low levels is of great interest in numerous applications. Due to data sparsity in high tails, the widely used quantile regression method can suffer from high variability at the tails, especially for heavy-tailed distributions. As an alternative to quantile regression, expectile regression, which relies on the minimization of the asymmetric l2-norm and is more sensitive to the magnitudes of extreme losses than quantile regression, is considered. In this article, we develop a new estimation method for high conditional tail risk by first estimating the intermediate conditional expectiles in regression framework, and then estimating the underlying tail index via weighted combinations of the top order conditional expectiles. The resulting conditional tail index estimators are then used as the basis for extrapolating these intermediate conditional expectiles to high tails based on reasonable assumptions on tail behaviors. Finally, we use these high conditional tail expectiles to estimate alternative risk measures such as the Value at Risk (VaR) and Expected Shortfall (ES), both in high tails. The asymptotic properties of the proposed estimators are investigated. Simulation studies and real data analysis show that the proposed method outperforms alternative approaches.

A novel two‐step extrapolation‐insertion risk model based on the Expectile under the Pareto‐type distribution

“…The suggested choice of the k ‐value method based on this traditional Hill estimation is to select the intersection of the traditional Hill plot and the Expectile‐based Hill plot to determine the first stable component in the common stable point as the k ‐value. According to Hu et al [15], the final estimation of k is close to the estimation obtained by the quantile‐based traditional Hill method in most cases.…”

“…This section first introduces the method that Daouia et al [8] and Hu et al [15] estimate XES through asymptotic properties. In order to improve the existing methods and improve the prediction accuracy, a two‐step extrapolation‐insertion method is further introduced.…”

“…$$$ \overline{F}(y)=1-F(y)={y}^{-1/\gamma }l(y),\kern2em 0<\gamma <1 $$$, where $$$ l(y) $$$ is a slowly varying function, then $$$$ {\displaystyle \begin{array}{cc}& \frac{\mathrm{XE}{\mathrm{S}}_{\tau }}{\mathrm{QE}{\mathrm{S}}_{\tau }}\sim \frac{\xi_{\tau }}{q_{\tau }}\sim \frac{\mathbbm{E}\left[Y \mid Y>{\xi}_{\tau}\right]}{\mathbbm{E}\left[Y \mid Y>{q}_{\tau}\right]},\frac{XE{S}_{\tau }}{\xi_{\tau }}\\ {}& \kern1em \sim \frac{1}{1-\gamma}\sim \frac{\mathbbm{E}\left[Y \mid Y>{\xi}_{\tau}\right]}{\xi_{\tau }},\kern1em \tau \to 1.\end{array}} $$$$ Given the asymptotic properties of the above, the XES estimation obtained by Hu et al . [15] is …”

“…Expectile is particularly sensitive to tail behavior because it considers the amount of loss both above and at a given probability level, while VaR is only sensitive to the tail behavior at a given probability level. In terms of interpretability, the τ th quantile indicates the point where 100 τ % of Y have values less than this number, while the τ th Expectile specifies the position of $$$ {\xi}_{\tau } $$$ such that the average distance from the data below $$$ {\xi}_{\tau } $$$ to $$$ {\xi}_{\tau } $$$ itself is 100 τ % of the average distance between $$$ {\xi}_{\tau } $$$ and all the data [15]. The inference on Expectile is not governed by the estimation of the density function and it only requires that the second moment of Y be bounded (Corollary 4 in Holzmann and Klar [14]).…”

Expectile regression forest: A new nonparametric expectile regression model

“…The superiority of the linear expectile regression model has drawn attention in research. Many scholars have applied it to financial risk measurement, for example Happersberger et al (2020), Xu et al (2020), Hu et al (2021), Yao et al (2021) and Giacometti et al (2021).…”