Inference for Conditional Value-at-Risk of a Predictive Regression

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 an infinite variance, which make it quite challenging to actually quantify the uncertainty of the conditional risk measure. To make the inference practically 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 quantile regression method, which allows infinite variance predictor and generalizes the parameter estimation in Whang (2006) to conditional value-at-risk. The finite sample performance of the derived confidence intervals is demonstrated through numerical studies before applying to real data.