A Residual Bootstrap for Conditional Value-at-Risk

This paper proposes a fixed-design residual bootstrap method for the two-step estimator of Francq and Zakoïan (2015) associated with the conditional Value-at-Risk. The bootstrap's consistency is proven under mild assumptions for a general class of volatility models and bootstrap intervals are constructed for the conditional Value-at-Risk to quantify the uncertainty induced by estimation. A large-scale simulation study is conducted revealing that the equal-tailed percentile interval based on the fixed-design residual bootstrap tends to fall short of its nominal value. In contrast, the reversed-tails interval based on the fixed-design residual bootstrap yields accurate coverage. In the simulation study we also consider the recursive-design bootstrap. It turns out that the recursive-design and the fixed-design bootstrap perform equally well in terms of average coverage. Yet in smaller samples the fixed-design scheme leads on average to shorter intervals. An empirical application illustrates the interval estimation using the fixed-design residual bootstrap.