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博碩士論文 etd-0129113-213605 詳細資訊
Title page for etd-0129113-213605
論文名稱
Title
連續時間隨機波動模型的適合度檢定
Goodness-of-fit test for Continuous Time Stochastic Volatility Models
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
131
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2013-01-21
繳交日期
Date of Submission
2013-01-29
關鍵字
Keywords
Bickel-Rosenblatt檢定、訊干比、微結構噪音、累積波動、高頻資料、適合度檢定、隨機波動模型、拔靴法、經驗特徵函數、V統計量
high frequency data, goodness-of-fit test, empirical characteristic function, bootstrap, Bickel-Rosenblatt test, integrated volatility, microstructure noise, signal-to-noise ratio, stochastic volatility models, V-statistics
統計
Statistics
本論文已被瀏覽 5710 次,被下載 266
The thesis/dissertation has been browsed 5710 times, has been downloaded 266 times.
中文摘要
在建立隨機微分方程模型的問題上,如何對連續時間隨機過程的平穩分佈做適合度檢定扮
演著很重要的角色。在本論文的第一部分,我們使用離散取樣的觀測值,提出兩種檢定類型來
對連續時間隨機波動模型做適合度的檢定。第一類的檢定方法,其建構的方式是測量經驗特
徵函數以及欲檢定真實特徵函數的累積平方差距。在虛無假設下,第一種提出來的檢定統計
量,可以證明出其漸進分佈服從一些中心化常態隨機變數相乘後的加權和。第二種的檢定方法
是Bickel-Rosenblatt檢定,其建構的方式是測量觀測值的無母數核密度函數估計值與真實核密
度函數的累積平方差距。在虛無假設下,可以證明此Bickel-Rosenblatt的檢定統計量為漸進常
態分佈。同時,我們使用Bickel-Rosenblatt的檢定來發展含有連結(copula)的多維隨機波動模
型。我們推導了此模型的漸進虛無分佈以及給一些二維情況下的例子。此兩種檢定統計量都會
實施模擬研究與實證分析。
本論文的第二部分,我們考慮了有市場微結構噪音的隨機波動模型的統計推論,此類的模
型通常用於建立高頻財務的資料。在高頻財務資料的分析上,累積波動的估計是一個非常重要
的問題。在此,我們考慮了由Lin (2007) 所提出的一個最小變異數不偏的累積波動估計值。此
最小變異數不偏估計值是由樣本自我共變異數的線性組合所組成,而且此估計量是在所有不偏
估計量的群組中,最小化有限樣本的變異數。此最小變異數不偏估計值的變異數以Op(n−1/4)
的速率收斂。在特殊的常數波動情況下,此最小變異數不偏估計值可達到最大概似估計值的效
率。一個遞迴的演算法亦被發展來計算出此最小變異數不偏估計值的最佳權重。我們分別對微
結構噪音的變異數以及四階累積動差(quarticity)提出了改進的估計值,其結果有助於累積波
動的估計。模擬結果顯示我們提出的估計值,比之在有限樣本下最先進的方法,可達到較高的
效率。最後,將實行一個實證分析來做為說明。我們同時考慮有微結構噪音的隨機波動模型的
適合度檢定。模型參數的動差估計值與一個基於特徵函數的適合度檢定亦被提出來。此提出的
適合度檢定將給予型一誤差與檢定力的模擬結果。
Abstract
A goodness-of-fit test for stationary distributions of continuous time stochastic processes
plays an important role in building up stochastic differential equation (SDE) models. In
the first part of this dissertation, we propose two types of goodness-of-fit tests for continuous
time stochastic volatility models (SVMs) based on discretely sampled observations.
The first type of test is constructed by measuring deviations between the empirical and
true characteristic functions obtained from the hypothesized stochastic volatility model.
It is shown that under the null, the first proposed test statistics asymptotically follow a
weighted sum of products of centered normal random variables. The second type of test
is the Bickel-Rosenblatt test which is constructed by measuring integrated squared deviations
between the nonparametric kernel density estimate from the observations and a parametric
fit of the density. It is shown that under the null hypothesis, the Bickel-Rosenblatt
test statistic is asymptotically normal. We also developed the Bickel-Rosenblatt test for
the multivariate SVMs with a copula link. Its asymptotic null distribution is derived and
bivariate examples are given. Simulation studies and real data analysis are conducted for
both proposed tests.
In the second part of this thesis, we consider inference for the SVMs with market
microstructure noises which are often used to model high frequency financial data. Estimation
of the integrated volatility is an important problem for high frequency financial
data analysis. We consider the minimum variance unbiased estimator (MVUE) of the
integrated volatility proposed by Lin (2007). The MVUE minimizes the finite sample
variance in the class of unbiased estimators which are linear combinations of the sample
autocovariance functions. The variance of the MVUE converges at a rate of Op(n−1/4). In
particular, the MVUE achieves the maximum likelihood estimator efficiency for the constant
volatility case. A recursive algorithm is developed to compute the optimal weights
of the MVUE. Improved estimators of the microstructure noise variance and the quarticity
are also proposed to facilitate the estimation procedure. Simulation results show our
proposed estimator attains higher efficiency than state-of-the-art methods for the finite
samples. Finally, a real data analysis is conducted for illustration. We also consider the
goodness-of-fit test for the SVMs with microstructure noises. Moment estimators of the
model parameters are proposed. A goodness-of-fit test based on the characteristic function
is proposed. Simulation results of sizes and powers of the proposed test are given.
目次 Table of Contents
論文審定書
誌謝
中文摘要
Abstract ii
1 Introduction 1
2 The Characteristic Function Based Test 5
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Test Statistic Based on the Full Sample . . . . . . . . . . . . . . . . 9
2.2 The Average Subsample Test . . . . . . . . . . . . . . . . . . . . . 9
2.3 Mixing Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Asymptotic Null Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Asymptotic Distribution of hat{T}n . . . . . . . . . . . . . . . . . . . . . 14
3.2 Asymptotic Distribution of hat{T}(j)n1 . . . . . . . . . . . . . . . . . . . . 21
4 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1 Bootstrap Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 Real Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 The Bickel-Rosenblatt Test 43
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 The Test Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Univariate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Asymptotic Null Distribution . . . . . . . . . . . . . . . . . . . . . 49
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Multivariate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1 Asymptotic Null Distribution . . . . . . . . . . . . . . . . . . . . . 58
4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Simulation and Empirical Studies . . . . . . . . . . . . . . . . . . . . . . . 62
5.1 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Real Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Concluding Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Minimum Variance Unbiased Estimator of Integrated Volatility in SVM 69
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2 Minimum Variance Unbiased Estimator . . . . . . . . . . . . . . . . . . . . 73
3 A Recursive Algorithm for Solving SL(θ) . . . . . . . . . . . . . . . . . . 81
4 Estimation of σ"^2and the quarticity Q . . . . . . . . . . . . . . . . . . . . . 85
5 Simulation and Empirical Studies . . . . . . . . . . . . . . . . . . . . . . . 88
5.1 Comparison of the Four Estimators . . . . . . . . . . . . . . . . . . 89
5.2 Empirical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Goodness-of-fit Test for the SVM with Microstructure Noise 93
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2 Microstructure Noise Distribution . . . . . . . . . . . . . . . . . . . . . . . 94
3 Parameter Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4 Goodness-of-fit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Bibliography 99
A Figures 105
B Tables 113
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