根據Corsi（2004）的異質性自我迴歸模型(HAR)及修正後HAR模型(HAR-GARCH)來探討2012年7月至2016四年半間VIX指數的波動性預測，並運用MAE、MSE統計損失函數來評估其預測績效；本研究同時加入已實現波動VIX期貨、已實現偏態、已實現峰態、已實現波動VIX指數、VIX波動率之風險溢酬等，探討是否影響VIX指數的波動性預測。
在研究期間中，透過HAR及HAR-GARCH模型能夠有效的預測樣本外VIX指數，迴歸結果發現已實現波動率VIX期貨、已實現偏態及VIX波動率之風險溢酬對於VIX指數具有資訊內涵，且確實能夠提升VIX指數的預測能力。預測績效結果顯示，MAE與MSE的損失函數其值較小，使用SPA Test檢測分析模型卻無達到統計之顯著性，代表外生變數之加入確實能夠增加預測績效但差異不大。此外，加入二階動差之HAR-GARCH模型在MAE檢定下，比基準HAR模型來的更有良好之預測效果，並使用實際商品VIX期貨、VIX ETPs來評估其實際績效。

This study uses HAR model of Corsi (2004) and HAR-GARCH model to predict out-of-sample VIX during July 2012 to December 2016. To explore the information content of additional variables, we use realized volatility of VIX futures, realized skewness of VIX futures, realized kurtosis of VIX futures, VIX volatility, realized volatility of VIX and risk premium of VIX volatility as the exogenous variables in HAR model and HAR-GARCH model to predict one-day-ahead and five-days-ahead VIX. The predicted performance is evaluated by mean absolute error (MAE) and mean squared error (MSE), and statistical significance is provided by superior predictive ability (SPA) test. 
The empirical results indicate that the HAR and HAR-GARCH model can predict out-of-sample VIX effectively, and incorporating realized volatility of VIX futures, realized skewness of VIX futures and risk premium of VIX volatility into HAR model and HAR-GARCH model can enhance the out-of-sample predictive performance, which implies that these variables contain information content in predicting VIX. However, the SPA test shows that the superior predictive performances are statistically significant. In addition, the predictive performance of HAR-GARCH model is better than HAR model, indicating that modeling time-varying variance of VIX is important for predict out-of-sample VIX. Furthermore, using actual commodities like VIX Futures, VIX ETPs to evaluate their actual performance.

目錄
第一章  緒論 1
第一節  研究背景與動機 1
第二節  研究目的 4
第三節  研究架構 5
第四節  研究流程圖 6
第二章  文獻探討 7
第一節  VIX指數介紹 7
第二節  波動性預測理論基礎 11
第三節  本章小結 23
第三章  研究方法 24
第一節  單根檢定 24
第二節  外生變數 27
第三節  迴歸模型 31
第四節  迴歸假設 33
第五節  模型預測能力比較 34
第四章  實證結果 38
第一節  敘述統計 38
第二節  單根檢定 45
第三節  迴規模型 46
第四節  模型預測能力比較 54
第五節  實際商品預測分析 64
第五章  結論 67
參考文獻 69
一、國內文獻 69
二、國外文獻 69

表目錄
【表2.1.1】2009年至2016年VXX及VXZ每年之日平均成交量(單位:股) 9
【表2.1.2】2011年至2016年VIXY及VIXM每年之日平均成交量(單位:股) 10
【表2.1.3】選擇權與期貨之差異 10
【表4.1.1】 每日原始資料基本敘述統計量 40
【表4.1.2】 外生變數之基本敘述統計 44
【表4.2.1】VIX指數、VIX期貨、VXX及VIXY之單根檢定 45
【表4.3.2】加入外生變數HAR模型之迴歸結果 48
【表4.3.3】相關係數及Q檢定 50
【表4.3.4】基準HAR-GARCH迴歸模型結果 50
【表4.3.5】相關係數及Q檢定 51
【表4.3.6】加入外生變數HAR-GARCH模型之迴歸結果 52
【表4.3.7】加入外生變數HAR-GARCH模型之迴歸結果 53
【表4.4.1】為MAE一日樣本平均數之檢定結果 56
【表4.4.2】為MSE一日樣本平均數之檢定結果 57
【表4.4.3】為SPA檢定之MAE一日顯著程度檢定結果 58
【表4.4.4】為SPA檢定之MSE一日顯著程度檢定結果 59
【表4.4.5】為MAE五日樣本平均數之檢定結果 60
【表4.4.6】為MSE五日樣本平均數之檢定結果 60
【表4.4.7】為SPA檢定之MAE五日顯著程度檢定結果 61
【表4.4.8】為SPA檢定之MSE五日顯著程度檢定結果 62
【表4.5.1】HAR模型之實際績效結果 64
【表4.5.2】HAR-GARCH模型之實際績效結果 65

圖目錄
【圖1.4.1】研究流程圖 6
【圖2.1.1】VXX和VXZ之成交量對照圖 8
【圖2.1.2】VIXY和VIXM之成交量對照圖 9
【圖4.1.1】VIX指數時間序列資料趨勢圖 41
【圖4.1.2】VX時間序列資料趨勢圖 41
【圖4.1.3】VXX時間序列資料趨勢圖 41
【圖4.1.4】VIXY時間序列資料趨勢圖 41
【圖4.1.5】VVIX時間序列資料趨勢圖 42
【圖4.1.6】S&P 500指數時間序列資料趨勢圖 42
【圖4.1.7】SKEW指數時間序列資料趨勢圖 42

一、國內文獻
鄒紹輝 (2006)，「隱含波動率之模型及預測：以台灣市場為例」國立中央大學統計研究所碩士學位論文
于曉蕾 (2009)，「基於HAR模型對中國股票市場已實現波動率的研究」吉林大學碩士學位論文
二、國外文獻
Abadir, K., & Talmain, G. (2002). Aggregation, persistence and volatility in a macro model. The Review of Economic Studies, 69(4), 749-779.
Adebiyi, A. A., Adewumi, A. O., & Ayo, C. K. (2014). Comparison of ARIMA and artificial neural networks models for stock price prediction. Journal of Applied Mathematics, 2014.
Akhtar, S., Ansari, V. A., & Ansari, S. A. (2017). Day-of-the-Week Effect in Fear Gauge: Evidence from India. IUP Journal of Applied Finance, 23(1), 44.
Amihud, Y. (2002). Illiquidity and stock returns: cross-section and time-series effects. Journal of financial markets, 5(1), 31-56.
Andersen, T. G., & Bollerslev, T. (1997). Heterogeneous Information Arrivals and Return Volatility Dynamics: Uncovering the Long‐Run in High Frequency Returns. The journal of Finance, 52(3), 975-1005.
Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2001). The distribution of realized exchange rate volatility. Journal of the American statistical association, 96(453), 42-55.
Andersen, T. G., Bollerslev, T., & Diebold, F. X. (2007). Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility. The Review of Economics and Statistics, 89(4), 701-720.
Ariyo, A. A., Adewumi, A. O., & Ayo, C. K. (2014, March). Stock price prediction using the ARIMA model. In Computer Modelling and Simulation (UKSim), 2014 UKSim-AMSS 16th International Conference on (pp. 106-112). IEEE.
Audrino, F., Huang, C., & Okhrin, O. (2016). Flexible HAR Model for Realized Volatility. University of St. Gallen.
Barndorff-Nielsen, O. E., & Shephard, N. (2005). Variation, jumps, market frictions and high frequency data in financial econometrics. Available at SSRN 751984.
Barone-Adesi, G., Engle, R. F., & Mancini, L. (2008). A GARCH option pricing model with filtered historical simulation. Review of Financial Studies, 21(3), 1223-1258.
Bhardwaj, G., & Swanson, N. R. (2006). An empirical investigation of the usefulness of ARFIMA models for predicting macroeconomic and financial time series. Journal of Econometrics, 131(1), 539-578.
Bollerslev, T., Kretschmer, U., Pigorsch, C., & Tauchen, G. (2009). A discrete-time model for daily S & P500 returns and realized variations: Jumps and leverage effects. Journal of Econometrics, 150(2), 151-166.
Box, G. E., & Jenkins, G. M. (1976). Time series analysis: forecasting and control, revised ed. Holden-Day.
Busch, T., Christensen, B. J., & Nielsen, M. Ø. (2011). The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets. Journal of Econometrics, 160(1), 48-57.
Byun, S. J., & Kim, J. S. (2013). The information content of risk-neutral skewness for volatility forecasting. Journal of Empirical Finance, 23, 142-161.
Byun, S. J., & Min, B. (2013). Conditional Volatility and the GARCH Option Pricing Model with Non‐Normal Innovations. Journal of Futures Markets, 33(1), 1-28.
Chan, W., Jha, R., & Kalimipalli, M. (2006). The economic value of trading with realized volatility in the S&P 500 index options market. Working paper.
Corsi, F. (2004). A simple long memory model of realized volatility. Available at SSRN 626064.
Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7(2), 174-196.
Corsi, F., & Reno, R. (2009). HAR volatility modelling with heterogeneous leverage and jumps. Available at SSRN 1316953.
Corsi, F., Audrino, F., & Renó, R. (2012). HAR modeling for realized volatility forecasting. Handbook of Volatility Models and Their Applications, 363 - 382
Deng, G., McCann, C., & Wang, O. (2012). Are VIX futures ETPs effective hedges?. The Journal of Index Investing, 3(3), 35-48.
Dennis, P., Mayhew, S., & Stivers, C. (2006). Stock returns, implied volatility innovations, and the asymmetric volatility phenomenon. Journal of Financial and Quantitative Analysis, 41(02), 381-406.
Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American statistical association, 74(366a), 427-431.
Evans, T., & McMillan, D. G. (2007). Volatility forecasts: the role of asymmetric and long-memory dynamics and regional evidence. Applied Financial Economics, 17(17), 1421-1430.
Fernandes, M., Medeiros, M. C., & Scharth, M. (2014). Modeling and predicting the CBOE market volatility index. Journal of Banking & Finance, 40, 1-10.
Fradkin, A. (2007). The Informational Content of Implied Volatility in Individual Stocks and the Market. Unpublished manuscript, Duke University.
Fuertes, A. M., Izzeldin, M., & Kalotychou, E. (2009). On forecasting daily stock volatility: The role of intraday information and market conditions. International Journal of Forecasting, 25(2), 259-281.
Han, H., Kutan, A. M., & Ryu, D. (2015). Modeling and predicting the market volatility index: The case of VKOSPI (No. 2015-7). Economics Discussion Papers.
Hansen, P. R. (2005). A test for superior predictive ability. Journal of Business & Economic Statistics, 23(4), 365-380.
Hansen, P. R., Huang, Z., & Shek, H. H. (2012). Realized garch: a joint model for returns and realized measures of volatility. Journal of Applied Econometrics, 27(6), 877-906.
Hao, J., & Zhang, J. E. (2013). GARCH option pricing models, the CBOE VIX, and variance risk premium. Journal of Financial Econometrics, 11(3), 556-580.
Hillebrand, E., & Medeiros, M. C. (2010). The benefits of bagging for forecast models of realized volatility. Econometric Reviews, 29(5-6), 571-593.
Jiang, I., Hung, J. C., & Wang, C. S. (2014). Volatility Forecasts: Do Volatility Estimators and Evaluation Methods Matter?. Journal of Futures Markets, 34(11), 1077-1094.
Kalimipalli, M., Chan, W. H., & Jha, R. (2009). The Economic Value of Using Realized Volatility in Forecasting Future Implied Volatility, The Journal of Financial Research, 32(3), 231-259.
Kambouroudis, D. S., & McMillan, D. G. (2016). Does VIX or volume improve GARCH volatility forecasts?. Applied Economics, 48(13), 1210-1228.
Kanniainen, J., Lin, B., & Yang, H. (2014). Estimating and using GARCH models with VIX data for option valuation. Journal of Banking & Finance, 43, 200-211.
Karolyi, G. A., Lee, K. H., & Van Dijk, M. A. (2012). Understanding commonality in liquidity around the world. Journal of Financial Economics, 105(1), 82-112.
Khan, M. A. I. (2011). Financial volatility forecasting by nonlinear support vector machine heterogeneous autoregressive model: evidence from Nikkei 225 stock index. International Journal of Economics and Finance, 3(4), 138.
Li, H., & Hong, Y. (2011). Financial volatility forecasting with range-based autoregressive volatility model. Finance Research Letters, 8(2), 69-76.
Liu, Q., Guo, S., & Qiao, G. (2015). VIX forecasting and variance risk premium: A new GARCH approach. The North American Journal of Economics and Finance, 34, 314-322.
Liu, L. Y., Patton, A. J., & Sheppard, K. (2015). Does anything beat 5-minute RV? A comparison of realized measures across multiple asset classes. Journal of Econometrics, 187(1), 293-311.
Marcos Vinicio & Pedro L. (2013). Modeling and Forecasting Realized Volatility: Evidence from Brazil. Brazilian Review of Econometrics, 31, 315-337.
McAleer, M., & Medeiros, M. C. (2008). A multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries. Journal of Econometrics, 147(1), 104-119.
Mincer, J. A., & Zarnowitz, V. (1969). The evaluation of economic forecasts. In Economic forecasts and expectations: Analysis of forecasting behavior and performance (pp. 3-46). NBER.
Park, Y. H. (2015). Volatility-of-volatility and tail risk hedging returns. Journal of Financial Markets, 26, 38-63
Phillips, P. C., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 335-346.
Said, S. E., & Dickey, D. A. (1984). Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika, 71(3), 599-607.
Tian, F., Yang, K., & Chen, L. (2017). Realized volatility forecasting of agricultural commodity futures using the HAR model with time-varying sparsity. International Journal of Forecasting, 33(1), 132-152.
White, H. (2000). A reality check for data snooping. Econometrica, 68(5), 1097-1126.