本論文著重在隱含波動指標不對稱性與預測誤差之實證研究，共包含三個部份，分別為「VIX與S&P 500指數之關係：門檻與不對稱性效果」、「隱含波動指標預測誤差之財務意涵」與「隱含波動指標與真實波動之關係：預測誤差與資訊內涵」，在此將三部份的內容簡述如下。
第一部分研究使用了TAR模型以檢查VIX門檻效果，以及應用ARJI模型檢測S&P 500股票指數報酬對VIX變動的不對稱性。實證結果證明了具有高與低恐慌係數的不對稱效果、VIX上升與下降係數的不對稱效果及高與低恐慌變異數係數的不對稱效果。特別是係數的不對稱性效果，描述了當在高恐慌區間VIX指標傾向於下降時，極端強烈的復甦與市場谷底之現象發生。另外，本研究證明了跳躍強度與VIX指標在不同的恐慌區間有相似的不對稱性。
第二部份本文應用了ARJI模型併入了預測誤差，以檢測台灣隱含波動的變動與相關決定因子(特別是預測誤差)之關係。實證結果證明了隱含波動的變動，顯著地受到當期報酬、落後報酬、隱含波動的落後變動、真實波動的同期(日)變動與落後預測誤差的影響。特別是在全觀察期的極端落後預測誤差與金融風暴期間的落後預測誤差，對隱含波動的當期變動有著非常不同的影響效果。
第三部份本文採用了正交檢定，以檢驗在臺灣是否預測誤差與過去的跳躍特徵之資訊內涵有關?實證結果發現，假如模型不納入預測誤差將會導致錯誤地拒絕正交，以致於因為預測誤差包含了相關的資訊內涵，造成TVIX對未來真實波動的預測不是很有效率，反之亦然。當然，本研究亦證明了金融危機期間由於具有異常的資訊內涵，造成落後的預測誤差對TVIX的當期變動只有很小的影響效果，因此這裡隱含了TVIX在金融危機期間具有不好的預測能力。

This dissertation focuses on asymmetry and forecast errors in the implied volatility index and it contains three parts. The first part is titled “Relationships between the VIX and the S&P 500 Index: Threshold and asymmetric effects”, the second part is named “The financial implications of forecast errors in the implied volatility index”, and the last one is “Relationships between the implied volatility index and realized volatility: Forecast errors and informational content”. A brief introduction of these three parts can be described as follow: 
The first part employs a TAR model to examine the VIX threshold effect and applies the ARJI model to investigate the various asymmetric effects in S&P 500 returns on changes in the VIX. The empirical results provide evidence of a high-low coefficient asymmetric effect, a rising-falling coefficient asymmetric effect and a high-low variance coefficient asymmetric effect. In particular, the coefficient asymmetric effects describe the phenomenon of extremely strong rallies and market bottoms in the high-fear regime when the VIX tends to fall. In addition, this study demonstrates that the jump intensity and the VIX have similar asymmetric effects in the different fear regimes.
The second part also applies the ARJI models that incorporate forecast errors to investigate the relationships between the changes in the implied volatility and the relevant determinant factors (especially forecast errors) in Taiwan. The empirical results provide evidence that the changes in the implied volatility are significantly affected by the contemporaneous returns, the lagged returns, the lagged changes in the implied volatility, the contemporaneous daily changes in the realized volatility and the lagged forecast errors. In particular, the extreme lagged forecast errors during the whole sample period and the lagged forecast errors during the financial crisis period have very various influences on the current changes in the implied volatility. 
The final part adopts the orthogonality tests to examine whether the forecast errors are related to the informational content about past jump characteristics in Taiwan. The empirical results demonstrate that, if not taking forecast errors into account will lead to wrongly reject orthogonality so that the TVIX is not an efficient forecast for the future realized volatility due to forecast errors containing the relevant informational content, and vice versa. Of course, this study also demonstrates that the lagged forecast errors during the financial crisis period only have small influences on the current changes in the TVIX owing to abnormally informational content. This implies that the TVIX possesses the poor predictive ability during the financial crisis period.

TABLE OF CONTENTS                                                                                            
                                                                    Page
ACKNOWLEDGEMENTS                                              i
ABSTRACT IN CHINESE                                              ii
ABSTRACT IN ENGLISH                                              iv
LIST OF TABLES                                                     ix
LIST OF FIGURES                                                    x
1. Introduction	1
1.1 Motivations	1
1.2 Objectives	4
1.3 Flow Chart	6
2. Literature Review	7
2.1 Literature on the relationships between the changes in the volatility index
   and the corresponding stock market returns	7
2.2 Literature on the relationships between the implied volatility and the
realized volatility	9
2.3 Literature on informational content and forecast efficiency	12
3. Data Resources	14
3.1 Data resources in the Part I of this dissertation	14
3.2 Data resources in the Part II and Part III of this dissertation	14
4. Methodology	16
4.1 TAR model and ARJI model in the Part I of this dissertation	16
4.1.1 Threshold autoregressive model	16
4.1.2 ARJI model with the VIX threshold effect	17
4.2 ARJI model in the Part II of this dissertation	21
4.3 Orthogonality of forecast errors in the Part III of this dissertation	26
5. Empirical Results and Analysis	29
5.1 Empirical analysis of Part I - Relationships between the VIX and the 
S&P 500 Index: Threshold and asymmetric effects	29
5.1.1 Descriptive statistics	29
5.1.2 Estimation for TAR model and asymmetric effects in series	30
5.1.3 Estimation of the ARJI model with the VIX threshold effect	31
  5.2 Empirical Analysis of Part II - The financial implications of forecast
errors in the implied volatility index	38
5.2.1 Descriptive statistics	38
5.2.2 Comparison of models and determination of the superior model	41
5.2.3 Discussion of the financial implications in the superior model	45
5.2.4 Discussion of the financial implications of extreme forecast 
errors during the whole sample period or of forecast errors
during the financial crisis period	48
5.3 Empirical analysis of Part III - Relationships between the implied
   volatility index and realized volatility: Forecast errors and 
informational content	53
5.3.1 Re-examining the superior model by orthogonality tests of 
forecast errors	53
5.3.2 Discussion of the financial implications in forecast error
 during the financial crisis period by orthogonality tests	56
6. Conclusions	61
6.1 Relationships between the VIX and the S&P 500 Index: Threshold
and asymmetric effects	61
6.2 The financial implications of forecast errors in the implied volatility index	62
6.3 Relationships between the implied volatility index and realized volatility:
Forecast errors and informational content	64
Bibiography	65

LIST OF TABLES
Page
Table 5.1.1 Descriptive statistics	29
Table 5.1.2 Tests for the threshold effect in the VIX	31
Table 5.1.3 Empirical results and tests of the ARJI-M and ARJI-MV models	32
Table 5.1.4 Coefficient asymmetric effect for return and volatility	35
Table 5.1.5 Average and asymmetric effects for the jump intensity and the VIX	38
Table 5.2.1 Descriptive statistics	40
Table 5.2.2 Linear models	43
Table 5.2.3 ARJI model	44
Table 5.2.4 Linear (ARJI) models with extreme forecast errors or with forecast
 errors during the event period	52
Table 5.3.1 ARJI models during the whole sample period	55
Table 5.3.2 Orthogonality tests by extracting the jump intensities in ARJI model
 without forecast errors during the whole sample period	56
Table 5.3.3 Orthogonality tests by extracting the jump intensities in ARJI model
 with forecast errors during the whole sample period	56
Table 5.3.4 Orthogonality tests by extracting the jump intensities in ARJI model
 with forecast errors during the financial crisis period	59
Table 5.3.5 ARJI models with forecast errors during the whole of the sample
 period or during the financial crisis period	60
 
LIST OF FIGURES
Page
Figure 5.1.1 Daily S&P 500 Index and VIX	30
Figure 5.1.2 Daily Jump Intensity for S&P500 Returns and the VIX	37
Figure 5.2.1 TVIX and the 22-day moving average realized volatility	40
Figure 5.2.2 Forecast errors over the research period	41

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