He, Y., Hou, Y., Peng, L. and Shen, H. (2020). Inference for Conditional Value-at-Risk of a Predictive Regression The Annals of Statistics, 48(6):3442–3464.

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 (Oper. Res. 60 (2012) 739–756) 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 (Econometric Theory 22 (2006) 173–205) 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.