臺灣博碩士論文加值系統

本研究試圖找出國際匯率及國際股市的波動因子，並証實降維後的波動因子模型比傳統波動模型有較佳的表現。本文以芝加哥商品交易所(CME)的兩檔外匯期貨做為國際匯率的代表，另外以摩根史坦利(MSCI)十筆國家指數做為國際股市的代表。降維的方法則採用主成份分析(Principal component Analysis)及修正後的條件不齊一(MCH)等法來進行，同時採用Cholesky解構法(Cholesky Decomposition)將共變數矩陣加以轉置以便估計多變量模型，最後採用兩種均方誤差(MSE)來進行模型預測好壞的比較。
國際匯率的實証結果顯示，以第一主成份所做的單變量因子模型表現比傳統的雙變量模型佳。而國際股市的實証結果顯示，由兩個主成份所構成的雙變量GARCH模型以及由Cholesky解構後所估的雙變量因子模型其表現能力均優於單變量GARCH模型；但三變量的因子模型不僅估計過程費時，其表現也不佳。至於MCH的因子模型表現比單變量GARCH稍差，可以嘗試先採用VAR模型估計以做進一步的討論。

This thesis tends to identify the volatility factor of foreign currency and world stock markets. We also testify whether factor volatility models by dimension reduction are well performing than traditional volatility models. Two currency futures traded in CME and ten MSCI developed market indices are adopted as foreign currency and world stock markets respectively. The methods of dimension reduction used in the research are principal com-ponent analysis (PCA) and modified conditionally heteroskedastic (MCH) factor model. In order to fit multivariate model, one of method we adopted is reparameterized the co-variance matrix by Cholesky decomposition. We use two loss functions (MSE1 and MSE2) to compare the forecast ability.
The result of foreign currency shows that univariate factor model, by using first principal component as a factor, outperforms original bivariate models. As for the results of world stock markets, two bivariate factor models, both bivariate GARCH model of first two principal component and bivariate model with Cholesky decomposition, also outper-form the univariate GARCH model generally. However, the trivariate factor model not only time consuming to fit, but also performs worse than univariate GARCH model. As for MCH factor model, it performs little worse than univariate GARCH model, suggesting that a VAR model could be adopted for further discussion.

1 INTRODUCTION 1
2 LITERATURE REVIEW 3
2.1 Introduction to Volatility 3
2.2 Volatility Models 4
2.2.2 Asymmetric GARCH Models 5
2.2.3 Multivariate GARCH Models 6
2.3 Common Factors on Volatility and Dimension Reduction 6
2.4 The Modified Conditionally Heteroskedastic Factor Model 8
2.5 Measuring Forecast Error 9
3. METHODOLOGY 11
3.1 Univariate Volatility Process 11
3.1.1 ARCH Model 11
3.1.2 GARCH Model 11
3.2 Multivariate Volatility Process 12
3.2.1 Bivariate GARCH Model 12
3.2.2 Cholesky Decomposition for Reparameterization 13
3.2.3 Higher Dimensional Volatility Model 15
3.3 Principal Component Analysis 16
3.4 Estimating the MCH Factor Model 18
3.5 Model Evaluation 19
3.5.1 Realized Variance for Daily Data 19
4. EMPIRICAL RESULTS: FOREIGN CURRENCY 21
4.1 Data Description 21
4.2 Univariate ARMA Model 22
4.3 Conducting Factor Model by Principal Component Analysis 22
4.3.1 Principal Component Analysis 22
4.3.2 Estimating Univariate ARCH Model 23
4.4 Estimating Bivariate Model 24
4.4.1 Bivariage GARCH Model 24
4.4.2 Bivariate Model by Cholesky Decomposition 25
4.5 Model Evaluation 26
5 EMPIRICAL RESULTS: WORLD STOCK MARKETS 27
5.1 Data Description 27
5.2 Univariate ARMA Model 28
5.3 Conducting Factor Model by Principal Component Analysis 28
5.3.1 Principal Component Analysis 28
5.3.2 Estimating Bivariate GARCH Model 29
5.3.3 Estimating Bivariate Model by Cholesky Decomposition 30
5.3.4 Estimating Trivariate Model by Cholesky Decomposition 31
5.4 Conducting the MCH Factor Model 32
5.5 Model Evaluation 33
6 CONCLUSION 35
REFERENCE 37


List of Tables
Table 4.1 Descriptive statistics of daily return series from CME currency futures 40
Table 4.2 Correlation matrix of 2 daily return series 40
Table 4.3 p-value of Q-statistics of 2 daily return series 40
Table 4.4 Coefficients of fitted AR models 40
Table 4.5 Eigenvalues of covariance matrix (PCA) 41
Table 4.6 Eigenvectors 41
Table 4.7 Coefficients of univariate ARCH model 41
Table 4.8 Ljung-Box Q-Statistics for residuals of univariate ARCH model 41
Table 4.9 Coefficients of bivariate GARCH model 42
Table 4.10 Ljung-Box Q-Statistics for residuals of bivariate GARCH model 42
Table 4.11 Coefficients of bivariate model with Cholesky Decomposition 43
Table 4.12 Ljung-Box Q-Statistics for residuals of bivariate model with Cholesky Decomposition 43
Table 4.13 Results of MSE 44
Table 4.14 Standardized MSE (Using ARCH_PCA1 as base) 44
Table 5.1 Descriptive statistics of 10 weekly return series from MSCI indices 45
Table 5.2 Correlation matrix of 10 weekly return series 45
Table 5.3 p-value of Q-statistics of 10 weekly return series 46
Table 5.4 Coefficients of fitted ARMA models 46
Table 5.5 Eigenvalues of covariance matrix (PCA) 47
Table 5.6 Eigenvectors 47
Table 5.7 Coefficients of bivariate GARCH model 48
Table 5.8 Ljung-Box Q-Statistics for residuals of bivariate GARCH model 48
Table 5.9 Coefficients of bivariate model with Cholesky Decomposition 49
Table 5.10 Ljung-Box Q-Statistics for residuals of bivariate model with Cholesky Decomposition 49
Table 5.11 Coefficients of trivariate model with Cholesky Decomposition 50
Table 5.12 Ljung-Box Q-Statistics for residuals of trivariate model with Cholesky Decomposition 51
Table 5.13 The transformed matrix 52
Table 5.14 Coefficients of bivariate model with Cholesky Decomposition (MCH) 53
Table 5.15 Ljung-Box Q-Statistics for residuals of bivariate model with Cholesky Decomposition (MCH) 53
Table 5.16 Coefficients of univariate GARCH models 54
Table 5.17 Results of MSE 55
Table 5.18 Standardized MSE (Using uni_GARCH as base) 56

List of Figures
Figure 4.1 Movement of daily return series from CME currency futures 57
Figure 5.1 Movement of 10 weekly return series from MSCI indices 58

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