Inference for conditional value-at-risk of a predictive regression

Yi He, Yanxi Hou, Liang Peng and Haipeng Shen

The Annals of Statistics, Forthcoming. AI Percentile: 98%

Conditional value-at-risk is a popular risk measure in risk management. We study the inference problem of conditional value-at-risk under a linear predictive regression model. We derive the asymptotic distribution of the least squares estimator for the conditional value-at-risk. Our results relax the model assumptions made in Chun et al. (2012) and correct their mistake in the asymptotic variance expression. We show that the asymptotic variance depends on the quantile density function of the unobserved error and whether the model has a predictor with infinite variance, which makes it challenging to actually quantify the uncertainty of the conditional risk measure. To make the inference feasible, we then propose a smooth empirical likelihood based method for constructing a confidence interval for the conditional value-at-risk based on either independent errors or GARCH errors. Our approach not only bypasses the challenge of directly estimating the asymptotic variance but also does not need to know whether there exists an infinite variance predictor in the predictive model. Furthermore, we apply the same idea to the quantile regression method, which allows infinite variance predictors and generalizes the parameter estimation in Whang (2006) to conditional value-at-risk in the supplementary material. We demonstrate the finite sample performance of the derived confidence intervals through numerical studies before applying them to real data.


Statistical inference for a relative risk measure

Yi He, Yanxi Hou, Liang Peng and Jiliang Sheng

Journal of Business & Economic Statistics, 37:2, 301-311, 2019. AI Percentile: 98%

For monitoring systemic risk from regulators’ point of view, this article proposes a relative risk measure, which is sensitive to the market comovement. The asymptotic normality of a nonparametric estimator and its smoothed version is established when the observations are independent. To effectively construct an interval without complicated asymptotic variance estimation, a jackknife empirical likelihood inference procedure based on the smoothed nonparametric estimation is provided with a Wilks type of result in case of independent observations. When data follow from AR-GARCH models, the relative risk measure with respect to the errors becomes useful and so we propose a corresponding nonparametric estimator. A simulation study and real-life data analysis show that the proposed relative risk measure is useful in monitoring systemic risk.

Paper R codes

Estimation of extreme depth-based quantile regions

Yi He and John H.J. Einmahl

Journal of the Royal Statistical Society - Series B , 79:449-461, 2017. AI Percentile: 99%

Consider the extreme quantile region induced by the half‐space depth function HD of the form

$$ \mathcal{Q}=\{x\in\mathbb{R}^d:HD(x,P)\leq \beta\}, $$

such that

$$ P\mathcal{Q}=p $$

for a given, very small p>0. Since this involves extrapolation outside the data cloud, this region can hardly be estimated through a fully non‐parametric procedure. Using extreme value theory we construct a natural semiparametric estimator of this quantile region and prove a refined consistency result. A simulation study clearly demonstrates the good performance of our estimator. We use the procedure for risk management by applying it to stock market returns.

Paper MATLAB codes