Value-at-Risk Forecasting Ability of Filtered Historical Simulation for Non-Normal GARCH Returns

As a hybrid methodology to estimate VaR, that combines the use of parametric modelling with the use of bootstrapping techniques, filtered historical simulation (FHS) should not be sensitive to the use of alternative distributions assumed in the filtering stage. However, recent studies (Kuester et al. 2006) have found that the distribution used in the filtering stage can influence the VaR estimates obtained in the context of this methodology. Using Extreme Value Theory (EVT) this paper explains that the VaR estimates for lower probabilities should not be sensitive to the distribution assumed in the filtering stage of the FHS method. However, for higher probabilities, the EVT results do not hold and therefore the use of alternative distributions might impact the VaR estimates. These theoretical results are tested using both simulated and real data. Three different realistic data generating processes were considered to generate several series of simulated returns. Additionally, three competing models, differing in the innovations assumption, were tested: a normal-GARCH, a t-GARCH and a skew-t-GARCH. Our backtesting results indicate that FHS can forecast VaR with accuracy for data which exhibits a high incidence of zeros, time-varying skewness, asymmetric effects to return shocks on volatility, as well as other stylized facts. Importantly, our results for the simulated data demonstrate that, for lower probabilities, the choice of the distribution assumed in the filtering stage has no impact on the performance of FHS as an accurate method to forecasting VaR. Additionally, 40 years of daily data on six well known active stock indices are used to empirically evaluate the FHS VaR estimates. Four competing GARCH-type specifications, combined with three different innovation assumptions (normal, Student-t and skew-Student t), are used to capture time series dynamics. Based on a sample of several VaR probabilities, the results of the dynamic quantile (DQ) tests clearly indicate that the use of asymmetric GARCH models (specifically GJR and GJR in Mean) generally improve the VaR forecasting performance of FHS. In addition, the choice of a skew-Student t distribution for the innovation process slightly improves the performance results of the GJR in Mean model. When different VaR probabilities are used, the choice of an appropriate model specification seems to be more important than the choice of a suitable distribution assumption. With respect to the lower VaR probability tested (1%), the results show that, as expected, the VaR estimate is very similar regardless of the GARCH model and distribution assumed.