由於GARCH模型在多維度的情況下，所需估計的參數較多，因此本研究希望以降低維度來減少參數的估計。本文以九個國際股票市場的指數報酬資料為研究樣本，在考慮同時估計九個股市報酬的變異數下，利用兩個降維模型主成份分析(Principal Component Analysis, PCA)，以及Peña-Box模型來預測股市的波動度，並與單變量GARCH模型進行波動性預測能力之比較，判斷預測能力的好壞則是藉由方均根誤(RMSE)和風險值(VaR)來評斷之。實證結果顯示，主成份分析取五個主成份時其預測能力表現最佳，此外，也因為降維而達到簡化計算過程的效益。不過Peña-Box模型並不如預期，其波動度的預測能力並沒有比單變量GARCH模型來的好。

This thesis focuses on comparing three models to find a better way to estimate volatility where fewer parameters are involved and a more accurate volatility can be derived. The data analyzed in this research are the weekly returns of nine world stock market indices. Considering the volatilities of nine stock markets at the same time, we compare the forecasting ability of Principal Component Analysis and Peña-Box model with that of univariate GARCH model. The empirical results show that Principal Component Analysis with five components not only outperforms univariate GARCH model, but also simplifies the process of calculation by reducing dimension. However, the Peña-Box model is not as good as we expect, which is worthy of further study.

Contents
1 Introduction 1
2 Literature Review 4
2.1 A Brief Introduction of Volatility . . . . . . . . . . .. . . . 4
2.2 Volatility Forecasting Models . . . . . . . . . . . . . . . . . 5
2.2.1 The GARCH Family . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Application of GARCH Model . . . . . . . . . . . . . . . . . 7
2.3 Measuring Forecast Errors . . . . . . . . . . . . . . . . . . 10
2.4 Dimension Reduction . . . . . . . . . . . . . . . . . . . . . 12
3 Methodology 15
3.1 Univariate (G)ARCH Processes . . . . . . . . . . . . . . . . . 15
3.1.1 The ARCH Model . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.2 The GARCH Model . . . . . . . . . . . . . . .. . . . . . . . 16
3.2 Principal Component Analysis . . . . . . . . . . . . . . . . . 17
3.3 The Pe˜na-Box Model . . . . . . . . . . . . . . . . . . . . . 20
3.4 Forecasting Evaluation based on VaR and RMSE . . . . . . . . . 22
3.4.1 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.2 Root Mean Squared Error . . . . . . . . . . . . . . . . . . 25
4 Data and Empirical Results 27
4.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Analysis of the GARCH Model . . . . . . . . . . . . . . .. . . 28
4.3 Analysis of the PCA . . . . . . . . . . . . . . . . . . . .. . 29
4.4 Analysis of the Pe˜na-Box Model . . . . . . . . . . . . . . . 32
4.5 Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . 33
5 Conclusion 37

References
[1] Akgiray, V., 1989, ”Conditional Heteroskedasticity in Time Series of Stock Returns:
Evidence and Forecasts,” Journal of Business, 62, 55-80.
[2] Andersen, T., and Bollerslev, T., 1998, ”Answering the Skeptics: Yes, Standard Volatility
Models Do Provide Accurate Forecasts,” International Economic Review, 39 (4),
885-905.
[3] Andersen, T. G.; Bollerslev, T.; and Lange S., 1999, ”Forecasting Financial Market
Volatility: Sample Frequency vis-`a-vis Forecast Horizon,” Journal of Empirical Finance,
6 (5), 457-477.
[4] Blair, B.; Poon, S.H.; and Taylor, S.J., 2001, ”Forecasting S&P 100 Volatility: The
Incremental Information Content of Implied Volatilities and High Frequency Index
Returns,” Journal of Econometrics, 105, 5-26.
[5] Bollerslev, T., 1986, ”Generalized Autoregressive Conditional Heteroskedasticity,”
Journal of Econometrics, 31, 307-327.
[6] Bollerslev, T.; Engle, R.; and Woodridge, J., 1988, ”A Capital Asset Pircing Model
with Time Varying Covariances,” Journal of Political Economy, 96, 116-131.
[7] Bollerslev, T., 1990, ”Modeling the Coherence in Short-run Nominal Exchange Rates:
A Multivariate Generalized ARCH Model,” Review of Economics and Statistics, 72,
498-505.
[8] Brillinger, D. R., 1981, ”Time Series : Data Analysis and Theory,” San Francisco:
Holden-Day.
[9] Chauvet, M. and Potter, S., 2000, ”Coincident and Leading Indicators of the Stock
Marke”, Journal of Empirical Finance, 7, 87-111.
[10] Davidian, M., and Carroll, R. J., 1987, ”Variance Function Estimation,” Journal of
American Statistic Associate, 82, 1079-1091.
[11] Dillon, W. and Goldstein, M., 1984, ”Multivariate Analysis-method and Application.”
New York: Jone Wiley & Sons, Inc.
[12] Dunis, C. L.; Laws, J.; and Chauvin, S., 2000, ”The Use of Market Data and
Model Combination to Improve Forecast Accuracy,” working paper, Liverpool Business
School.
[13] Engle, R. F., 1982, ”Autoregressive Conditional Heteroskedasticity with Estimates of
the Variance of U.K. Inflation,” Econometirca, 50, 987-1008.

[14] Engle, R.F. and Bollerslev, T., 1986, ”Modelling the Persistence of Conditional Variances,”
Econometric Review, 5, 1-50.
[15] Engle, R. F. and Kroner, K. F., 1995, ”Multivariate Simultaneous Generalized ARCH,”
Econometric Theory, 11, 122-150.
[16] Engle, R., 2002, ”Dynamic Conditional Correlation: A Simple Class of Multivariate
Generalized Autoregressive Conditional Heteroskedasticity Models,” Journal of Business
and Economic Statistics, Vol. 20, No 3, 339-350.
[17] Figlewski, S., 1997, ”Forecasting Volatility,” NYU, Salomon Center, 6:1, 1-88.
[18] Forbes, K.J. and Rigobon, R., 2002, ”No Contagion, Only Interdependence: Measuring
Stock Market Comovements,” The Journal of Finance, Vol. LVII, 5, 2223-2261.
[19] Gau, Yin-Feng and Hsieh, Jia-Ho, 2002, ”Value at Risk and Multivariate GARCH
Models,” Taiwan Economic Review, Vol. 30, No 3, 273-312.
[20] Gau, Yin-Feng and Kuo, Hsien-Chung, 2004, ”Application of the DCC Model in the
International Stock Markets,” National Chi-Nan University.
[21] Geweke, J., 1977, ”The Dynamic Factor Analysis of Economic Time Series Models,”
Amsterdam: North-Holland, 365-383.
[22] Geweke, J. F. and Singleton, K. J., 1981, ”Maximum Likelihood Confirmatory Analysis
of Economic Time Series,” International Economic Review, 22, 37-54.
[23] Glosten, L.R.; Jagannathan, R.; and Runkle, D.E., 1993, ”On the Relation between
the Volatility of the Nominal Excess Return on Stocks,” Journal of Finance, 1779-1801.
[24] Hamilton, J. D., 1989, ”A New Approach to the Economic Analysis of Nonstationary
Time Series and the Business Cycle,” Econometrica, 57, 357-384.
[25] Hannan, E. J., 1981, ”Estimating the Dimension of a Linear System,” Journal of
Multivariate Analysis, 11, 459-473.
[26] Heynen, R.C. and Kat, H.M., 1994, ”Volatility Prediction: A Comparison of Stochastic
Volatility, GARCH(1,1) and EGARCH(1,1) Models,” Journal of Derivatives , 50-65.
[27] Hotelling, H., 1933, ”Analysis of a Complex of Statistical Variables into Principal
Components,” Journal of Education Psychology, 24, 417-441.
[28] Hu, Y. P. and Chou, R. J., 2004, ” On the Pe˜na-Box model,” Journal of Time Series
Analysis, 25, 811-830.

[29] Hu, Y. P.; Lin, L.; and Kao, J.W., 2004, ”Inter-market Linkage and Leading Indicators
of International Stock Markets,” National Chi-Nan University.
[30] Hu, Y. P.; Wang, C.C., 2004, ”Common Factors of International Interest Rates,”
National Chi-Nan University.
[31] Jorion, P., 2000, ”Value at Risk,” 2nd edition, McGraw-Hill.
[32] Kupiec, P. H., 1995, ”Techniques for Verifying the Accuracy of Risk Measurement
Models,” Journal of Derivatives, Vol. 3, 73-84.
[33] Liu, Y. A. and Pan, M. S., 1997, ”Mean and Volatility Spillover Effects in the U.S.
and Pacific-Basin Stock Markets,” Multinational Finance Journal, 1, 47-62.
[34] Longin, F. and Solnik, M.B., 1995, ”Is the Correlation on International Equity Returns
Constant: 1960-1990,” Journal of international Money and Finance, 14, 3-23.
[35] Masih, A. M. M. and Masih, R., 1997, ”Dynamic Linkage and the Propagation Mechanism
Driving Major International Stock Markets: An Analysis of the Pre-and-postcrash
Eras,” The Quarterly Review of Economics and Finance, 37, 859-885.
[36] Meric, I. and Meric, G., 1995, ”The Co-Movement Patterns of the World’s Nine Leading
Stock Markets,” American Business Review, 13(1), 106-112.
[37] Nasar, S., 1992, ”For Fed, a New Set of Tea Leaves,” New York Times.
[38] Nelson, D., 1991, ”Conditional Heteroskedasticity in Asset Returns: A New Approach,”
Econometrica, 59, 349-370.
[39] Pearson, K., 1901, ”On Lines and Planes of Closest Fit to Systems of Points in Space,”
Phil. Mag., 6 (2), 559-572.
[40] Pe˜na, D. and Box, G. E. P., 1987, ”Identifying a Simplifying Structure in Time Series,”
Journal of the American Statistical Association, 82, 836-844.
[41] Poon, S.H. and Granger, C.W.J., 2003, ”Forecasting Volatility in Financial Markets:
a Review,” Journal of Economic Literature, Vol. XLI, 478-539.
[42] Priestley, M.B., 1981, ”Spectrat Analysis and Time Series,” New York: Academic
Press, Vol.2.
[43] Quenouille, M. H., 1957, ”The Analysis of Multiple Time Series,” London: Charles W.
Griffin.
[44] Rao, T. S., 1975, ”An Innovation Approach to the Reduction of the Dimensions in a
Multivariate Stochastic System,” International Journal of Control, 21, 673-680.

[45] Reinsel, G. C., 1983, ”Some Results on Multivariate Autoregressive Index Models,”
Biometrika, 70, 145-156.
[46] Roll, R., 1988, ”The International Crash of October 1987,” Financial Analysts Journal,
44, 19-37.
[47] Sargent, T.J. and Sims, C.A., 1977, ”Business Cycle Modelling without Pretending to
Have too much a Priori Economic Theory,” New Methods in Business Cycle Research:
proceedings From a Conference, MN: Federal Reserve Bank of Minneapolis, 45-110.
[48] Taylor, S. J., 1987, ”Forecasting of the Volatility of Currency Exchange Rates,” International
Journal of Forecast, 3, 159-170.
[49] West, K. D. and Cho, D., 1995, ”The Predictive Ability of Several Models of Exchange
Rate Volatility,” Journal of Econometrics, 69 (2), 367-391.
[50] West, K. D.; Edison, H. J.; and Cho, D., 1993, ”A Utility Based Comparison of Some
Methods of Exchange Rate Volatility,” Journal of International Economic, 35, 23-45.
[51] Zakoian, J. M., 1994, ”Threshold heteroskedastic models,” Journal of Economic Dynamics
and Control, 18, 931-955.