Recursive-Design Residual Bootstrap for Semi-Strong GARCH-type Volatility Models and Conditional Value-at-Risk Estimation

This paper discusses two bootstrap algorithms to quantify the uncertainty around point estimates of volatility parameters and conditional Value-at-Risk estimates. In the first part of the paper, we elaborate on a recursive-design residual bootstrap for semi-strong GARCH-type volatility models. Additionally, we discuss the inconsistency of the fixed-design residual bootstrap in case of dependency in the innovations. The claims are substantiated by preliminary proofs and a simulation study. In the second part of the paper we discuss the model of León et al. (2005). The model allows for time-varying conditional skewness and kurtosis and is therefore, if used correctly, suited to model time-varying conditional quantiles. Afterwards, we propose a recursive-design residual bootstrap for conditional Value-at-Risk, where the innovations are allowed to have time-varying conditional higher moments. Again, the simulation study provides evidence that our presented bootstrap procedure can be used to quantify the uncertainty around the conditional Value-at-Risk estimates. Moreover, we show that the fixed-design residual bootstrap underestimates the uncertainty around those point estimates. As a result, we observe that the average coverage rates falls short of their nominal value.

Key words: Residual bootstrap; GARCH; Value-at-Risk; SE residuals