4. Empirical Analysis
4.2 In-sample Performance
Figure 2: Time series of daily VIX, VIX futures, Residual of VIX and Spread.
Figure 2 exhibits three time series plots from 3 January 2006 to 31 October 2012. First is the time series of daily VIX and VIX futures; Second is the time series of the difference between daily realized volatility and VIX (RV-VIX) (left scale) and Residual of VIX (right scale); Third is the time series of RV-VIX (left scale) and Spread (right scale). Gray area denotes the Financial Crisis from November 2007 to June 2009 and Spread denotes the difference of VIX futures and VIX. (Here, we roughly regard the Financial Crisis as the periods of high volatility state.)
4.2 In-sample Performance
To investigate if there exists the information content of Residual of VIX and risk-neutral skewness for future volatility forecasting, the following regression models with Markov regime-switching are adopted:
2006 2007 2008 2009 2010 2011 2012
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2006 2007 2008 2009 2010 2011 2012
-0.015 -0.01 -0.005 0 0.005
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time horizons. In the low volatility state, the size and the significance of βSK decrease with the forecast horizon, and the coefficient is negative and highly significant. However, in the high volatility state, the size and the significance of βSK increase with the forecast horizon, and the coefficient is negative and highly significant except the daily forecast horizon. The coefficient of βSK is larger in the high volatility state than low volatility state for all forecast horizons. These results are consistent with Byun and Kim (2013), which suggest that the risk-neutral skewness has more explanatory power in the short-term regression than the long-short-term regression and is more important, especially in the high volatility state.The estimates of βIM verify the effectiveness of Residual of VIX in different time horizons. In the low volatility state, the size βIM decreases with the forecast horizon, but the coefficient is significant for all forecast horizons. On the other hand, in the high volatility state, the size and the significance of βIM increase with the forecast horizon, and the coefficient is negative and high significant for all forecast horizons. In addition, the estimates of βIM is significant in the high volatility state at monthly regression, as compared with the estimates of βIV. These results imply that VIX futures somewhat has information content for future volatility forecasting, especially in the high volatility state.
Regarding the average log likelihood, the MRS-HAR-RV-IV-IM model outperforms the MRS-HAR-RV-IV model, stressing the relevance of Residual of VIX as a predictor.
10 To clearly report the coefficient estimates of each regression, the realized volatility is scaled by 100 times the square root of 252 of the original realized volatility and VIX is scaled by the square root of 252/365 of the original VIX.
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Besides, the MRS-HAR-RV-IV-IM-SK model outperforms the others. Last, in the each model, both of the probability of staying in the regime 1 and regime 2 increase with the forecast horizon, however, the magnitude is larger in the regime 2 than regime 1. This implies that it is not easy to change regimes, especially in the monthly time horizon and high volatility state. On the other hand, the expected duration of regime 1 and regime 2 generally decrease from Model 0 to Model 4 under different forecast horizon.
Figure 3 plots daily realized volatility and estimated smoothed probability of regime 2 (i.e., Pr(St =2 |YT)) from 3 January 2006 to 31 October 2012. All the models provide similar regime estimates, detecting the high volatility from November 2007 to June 2009.
Apart from the MRS-HAR-RV model, the other models detect the high volatility on 27 April 2006.
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Table 2: In-sample Performance Result for Future Volatility of MRS-HAR Models
The table presents the estimation result of risk-neutral skewness and Residual of VIX for future volatility. The sample period is from 3 January 2006 to 31 October 2012, for a total 1704 daily observations. The specification for M0:RV, = β + β RV , + β RV , + β RV , + ε, ; M1:RV, = β + β RV , + β RV , + β RV , + β VIX + ε, ; M2:RV, = β + β RV ,+ β RV , + β RV , + β VIX + β SKEW + ε, ; M3: RV, = β + β RV , + β RV , + β RV , + β VIX + β reVIX + ε, ; M4: RV, = β + β RV ,+ β RV , + β RV , + β VIX + β reVIX + β SKEW + ε, .reVIXt represents the Residual of VIX which is the residuals obtained from regressing VIX futures on VIX. For each model, the left column is the estimated coefficient in the regime 1 and the right column is in the regime 2. Pii indicates the probability of staying in regime I and all the Pii are significant at 1%. Duration indicates the expected duration of each regime which is calculated by 1/ (1-Pii) and the unit is days. Last, Log likelihoodfor each model is reported in last row. Note that the parentheses is t-statistics and *, ** and *** denote Significant at 10%, 5% and 1%,
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(Continued from previous page.) Panel B : Weekly forecast horizon
Model M0 M1 M2 M3 M4
β 2.752*** 8.526*** 0.611*** 3.775*** 4.369** 46.323*** 0.424* 4.348*** 4.200** 49.990***
(17.089) (7.439) (3.232) (2.794) (2.485) (4.054) (1.916) (3.046) (2.382) (4.706) β 0.087*** 0.304*** 0.023 0.215*** 0.026 0.201*** 0.020 0.144** 0.020 0.127**
(4.188) (5.276) (1.369) (3.648) (1.540) (3.511) (1.161) (2.530) (1.179) (2.298) β 0.275*** 0.315*** 0.182*** 0.398*** 0.175*** 0.346*** 0.164*** 0.232*** 0.162*** 0.183**
(9.721) (3.195) (7.708) (4.505) (7.275) (3.922) (6.800) (2.652) (6.733) (2.129) β 0.361*** 0.195** 0.146*** -0.200** 0.138*** -0.142 0.154*** -0.103 0.147*** -0.047 (15.740) (2.533) (6.081) (-2.040) (5.621) (-1.462) (6.395) (-1.120) (6.010) (-0.504) β 0.444*** 0.575*** 0.449*** 0.568*** 0.466*** 0.676*** 0.467*** 0.666***
(16.290) (4.471) (16.227) (4.555) (15.703) (5.441) (15.715) (5.535)
β -0.104* -1.127*** -0.098* -1.143***
(-1.932) (-5.337) (-1.915) (-5.857)
β -0.031** -0.358*** -0.031** -0.384***
(-2.150) (-3.753) (-2.151) (-4.352)
σ 4.866 86.753 4.027 78.218 3.979 75.229 3.976 70.870 3.850 66.960 (26.228) (14.398) (26.281) (14.473) (26.235) (14.605) (26.254) (14.626) (26.137) (14.900) P"" 0.976 0.909 0.970 0.887 0.970 0.890 0.969 0.886 0.968 0.889 Duration 40.82 10.96 33.11 8.88 33.25 9.11 32.14 8.80 31.30 8.99
Log. Lik. -4454.256 -4336.161 -4326.934 -4320.424 -4308.799
(Continued to next page.)
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(Continued from previous page.) Panel C : Monthly forecast horizon
Model M0 M1 M2 M3 M4
β 3.882*** 13.940*** 2.035*** 12.057*** 4.009*** 56.356*** 1.318*** 13.732*** 4.472*** 59.772***
(25.271) (16.005) (12.400) (10.198) (4.775) (6.237) (5.201) (10.943) (2.756) (6.930) β 0.064*** 0.147** -0.001 0.212*** 0.000 0.186*** 0.000 -0.055 0.012 0.106* (4.512) (2.504) (-0.142) (3.596) (-0.011) (3.244) (0.005) (-0.921) (0.833) (1.848) β 0.214*** 0.437*** 0.165*** 0.354*** 0.162*** 0.334*** 0.204*** 0.480*** 0.185*** 0.200***
(9.326) (5.149) (7.404) (4.363) (7.696) (4.326) (9.747) (8.049) (8.213) (2.605) β 0.355*** 0.083 0.076*** 0.221** 0.076*** 0.242*** 0.151*** 0.106 0.061*** 0.287***
(16.541) (1.331) (3.204) (2.539) (3.353) (2.830) (5.824) (1.273) (2.655) (3.493) β 0.440*** -0.107 0.434*** -0.089 0.396*** 0.163 0.397*** 0.051 (19.209) (-0.913) (18.757) (-0.772) (14.971) (1.259) (15.795) (0.421)
β -0.229*** -2.066*** -0.163*** -1.025***
(-5.140) (-8.284) (-3.315) (-5.684)
β -0.016** -0.373*** -0.017 -0.399***
(-2.393) (-4.975) (-1.295) (-5.567)
σ 4.529 75.738 2.832 78.162 2.833 73.566 4.417 60.170 2.809 68.567 (25.498) (14.585) (24.722) (16.177) (24.704) (16.219) (25.799) (14.040) (24.654) (16.189) P"" 0.988 0.962 0.983 0.959 0.983 0.959 0.987 0.954 0.984 0.961 Duration 83.95 26.49 58.47 24.10 58.55 24.29 76.89 21.86 61.27 25.53
Log. Lik. -4330.345 -4204.835 -4192.623 -4241.768 -4170.533
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Figure 3: Daily realized volatilityand estimated smoothed probability of #$(&'= (|*+).
Figure 3 plots the daily realized volatilityand estimated smoothed probability of Pr(-.= 2|01) from 3 January 2006 to 31 October 2012. The first panel is the S&P 500 daily realized volatility and from the second to the last panels are Pr2(-.= 2|01) for all Markov switching model discussed in Section 3.
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1 S&P 500 daily realized v olatility
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1 M0: Pr(St=2|Y )
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1 M1: Pr(St=2|Y )
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1 M2: Pr(St=2|Y )
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1 M3: Pr(St=2|Y )
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1 M4: Pr(St=2|Y )
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In Section 4.2, we report the results of the in-sample analysis with the time-varying forecasting performance of the information content of VIX futures and risk-neutral skewness based on the level of market volatile. In this Section, we use a re-estimate procedure to estimate the coefficient of the forecasting model. After that, we would use the loss functions to evaluate the accuracy of the forecasts.
We obtain the out-of-sample forecasts of future volatility from the estimates with an increasing window scheme. First of all, we use the first m=1200 observations to initialize the models. Then, we re-estimate the forecasting model at each day t conditional on all the observations available up to day t−1.11 That is, the first h-steps-ahead forecasts is based on ( ,y y1 2,...,ym), the second h-steps-ahead forecasts is based
Table 3 reports some classic loss functions of forecasting accuracy, namely RMSE13, MAE and QLIKE. It is evident that the MRS-HAR-RV-IV-IM model outperforms the MRS-HAR-RV-IV model from daily to monthly forecast horizon. In this case, VIX futures seems to provide incremental information to forecast future volatility. Similarly, the MRS-HAR-RV-SK model outperforms the MRS-HAR-RV-IV model at all forecast horizons except the RMSE criteria at daily horizon. Furthermore, the MRS-HAR-RV-IV-IM-SK model provides good forecasts at all forecast horizons. However, at monthly
11 We also specify the Residual of VIX with an increasing window scheme.
12 Since we use the first 1200 daily observations to estimate the parameter of the model, the sample period of out-of-sample analysis is from 13 October 2010 to 31 October 2012.
13 To clearly report the results, we adopt RMSE instead of MSE, and RMSE is the square root of MSE.