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5. Conclusion and Suggestions
Within this study, we demonstrate the important roles of VIX futures and risk-neutral skewness for future volatility forecasting during high and low market volatility state. It is well documented in the literature that the implied volatility extracted from VIX futures and the risk-neutral skewness have the predictive ability regarding future volatility.
Using S&P 500 index, S&P 500 index option prices and VIX futures prices, this study finds two important results. First, in the in-sample analysis, the VIX futures has a significant effect on forecasting future volatility, especially in the short-term forecast horizon (i.e., daily and weekly horizons). More specifically, the VIX futures has more significant effect on the high volatile market in the weekly and monthly regressions, while the VIX futures has more significant effect on the low volatile market in the daily regression. Second, in the out-of-sample analysis, the VIX futures improves significantly forecasting ability in the daily, weekly, and monthly regressions.
In addition, similar to the findings of Byun and Kim (2013), the risk-neutral skewness has more significant effect on the high volatile market in the monthly regression, while risk-neutral skewness has more significant effect on the low volatile market in the daily and weekly regressions. We also find the volatility forecasting model that take the information of risk-neutral skewness and VIX futures into account improves the forecasting ability in the in-sample analysis. However, for the out-of-sample analysis, the volatility forecasting model is valid only in the daily regression.
For the comparison between MRS-HAR models and HAR models, we find that the MRS-HAR models outperform the corresponding HAR models in the weekly and monthly regressions. However, the MRS-HAR models are outperformed by HAR models in the daily regression. As we mentioned in Section 4.4, it is probably caused by the parameter estimates instability.
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Finally, this study proposes some suggestions for further research. First, based on the proposition of Byun and Kim (2013), it is possible that the VIX of VIX (VVIX) derived from VIX option market prices exists the information content for future volatility forecasting. Second, regarding to the parameter estimates instability of Markov switching models, Book and Pick (2014) suggest the optimal weights for Markov regime-switching models. Last, relative to the unobserved state of Markov regime-regime-switching models, threshold models, which are alternative nonlinear models, use a threshold variable determining the regimes based on observable past level of volatility. These extensions are left for future research.
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The derivation of S from CBOE website is as follows:
3
Forward index price derived from index option prices.
First strike below .
: The midpoint of the bid-ask spread for each option with strike . : Risk-free interest rate to maturity.
: The time to maturity expressed as a fraction of a
1( )
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Appendix B
Table 7: RMSE, MAE and QLIKE for HAR Models
The table reports the 1-day- (daily), 5-days- (weekly), and 22-days-ahead (monthly) forecast errors of different HAR models during the whole sample period from 3 January 2006 to 31 October 2012. Three loss functions are adopted, namely RMSE, MAE and QLIKE. Note that RMSE is the square root of MSE.
Daily Weekly Monthly
RMSE MAE QLIKE RMSE MAE QLIKE RMSE MAE QLIKE model 0 6.134 4.239 3.655 4.819 3.160 3.673 4.973 3.137 3.746 model 1 5.850 4.067 3.657 4.662 3.109 3.676 4.943 3.148 3.748 model 2 5.837 4.018 3.634 4.641 3.075 3.649 5.033 3.218 3.715 model 3 5.808 4.012 3.628 4.618 3.117 3.656 4.806 3.040 3.732 model 4 5.817 3.958 3.590 4.623 3.112 3.617 4.937 3.174 3.690
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Table 8: Out-of-sample Forecasting Performance of MRS-HAR Models
The table shows the comparison of out-of-sample forecasting performance for 1-day- (daily), 5-days- (weekly), and 22-days-ahead (monthly) forecasts of different HAR models during the whole sample period from 3 January 2006 to 31 October 2012. In panel A, the t-statistic of the Diebold-Mariano test based on different loss function is reported. In panel B, for each forecast horizon, the first column is the t-statistic of the Weighted Likelihood Ratio test and the second column is the corresponded p-value are reported. Note that *, ** and *** denote Significant at 10%, 5% and 1%, respectively.
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