臺灣博碩士論文加值系統

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

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