• Step 4: Repeat steps 2 and 3 B times obtaining B bootstrap replicates ($\hat{y}{T+h|T}^{*(1)}, \dots, \hat{y}{T+h|T}^{(B)}$ of $y_{T+h}$ and $(\hat{s}_{T+h|T}^{(1)}, \dots, \hat{s}{T+h|T}^{*(B)})$ of $s{T+h}$.
Denote by $\hat{G}y^*(x) = # (y{T+h|T}^* \le x) / B$, the empirical bootstrap distribution function of the B bootstrap replicates $(y_{T+h|T}^{(1)}, \dots, y_{T+h|T}^{(B)})$. Using $\hat{G}_y^*(x)$, one can compute the VaR of returns as the required quantile of $\hat{G}_y(x)$ and also 100(1 - $\delta$)% forecast intervals as follows
where $Q_x^* = \hat{G}_x^{*-1}$. Bootstrap forecast intervals of future volatilities can be obtained in a similar way. This procedure works well when the series is not contaminated by outliers, as shown in Pascual et al. (2006), however, when the series is contaminated by atypical observations the performance is poor, even using robust estimators of the parameters.
Figure 1 shows the results for volatilities using the QMLn estimator and some robust estimators, as BM of Müller and Yohai (2008), Bs of Carnero et al. (2012) and BVT of Boudt et al. (2013). Using any estimator the result when the series is not contaminated shows good performance, and when the series is contaminated in the middle the robust estimator improves the performance of the algorithm. However, when the series is contaminated near to the end of the series the algorithm has not good performance, even, when robust estimators were used. As a consequence of the high error coverage below, the returns estimate coverage is
Figure 1: Estimated coverage and coverages above and below of the 95% bootstrap forecast interval for volatilities using the PRR algorithm with QMLn, BM, Bs and BVT estimators. Sample size $T = 1000$. Based on 500 Monte Carlo replications.
overestimated, because the length of intervals in larger than necessary.
4. Robust bootstrap forecast intervals
As we have seen in the previous section, bootstrap forecast densities of returns and volatilities may be distorted when the data contains outliers even if the bootstrap forecast densities are based on robust estimators. These difficulties could be attributed to the important negative effect of bootstrap replicates where the proportion