金融波动分析的小波和频域方法研究
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摘要
论文以金融波动分析的小波和频域方法为研究内容，共分为八章，主要研究工作和创新之处在第二——第七章中。
第二章研究金融波动的频域分析方法。基于平稳和非平稳分整自回归滑动平均模型提出了具有非平稳性和长记忆性序列的主频率的估计方法，同时提出了包括模型定阶在内的基于极小化残差平方和的近似极大似然参数估计算法。并用该方法对中国股市收益序列进行了分析。
第三章研究金融波动的时间——尺度分析方法。提出了用基于MODWT系数的小波方差分析金融波动的长记忆性、用小波互相关系数和小波交叉互相关函数分析金融波动在不同尺度下相关性的方法。并用该方法对中国股市收益波动的长记忆性和相关性进行了分析。
第四章对LMSV过程DWT系数的特性进行分析。对于同一尺度下的DWT系数，分别进行了相关性分析和谱分析；对于不同尺度下DWT系数进行了相关性分析。结果表明，LMSV过程同一尺度下DWT系数是近似不相关的，不同尺度下DWT系数也是近似不相关性。
第五章研究基于小波变换的LMSV模型变结构分析方法。根据LMSV过程同一尺度下DWT系数的近似不相关性，在基于累积平方和的正态独立变量同方差性检验方法的基础上，提出了应用DWT系数累积平方和的LMSV模型单一变结构点的DWT-CUSUMSQ-MODWT检测与定位法，以及多个变结构点的DWT-ICUSUMSQ-MODWT检测与定位法，并用该方法对中国股市收益进行了波动变结构分析。
第六章研究基于小波变换的LMSV模型的估计方法。根据LMSV过程同一尺度下和不同尺度下DWT系数的近似不相关性，提出了基于小波变换的LMSV模型参数的伪极大似然估计方法以及潜在波动过程的估计方法；又提出了基于小波变换的LMSV模型波动长记忆参数的伪极大似然估计方法以及波动长记忆性的检验方法，并用该方法对上海股市收益进行了LMSV模型分析。
第七章研究基于小波变换的时变LMSV模型参数的估计方法。根据小波变换可将过程分解到不同的尺度上以及 LMSV过程同一尺度下和不同尺度下DWT系数的近似不相关性，提出了建立局部似然函数的方法，又根据DWT系数和MODWT系数之间的关系，将局部似然函数表示为模型参数和局部小波方差估计的形式，并用该方法对中国股市收益进行了时变LMSV模型参数的估计。
本论文的研究是国家自然科学基金资助项目“多变量时间序列的波动持续性及其在金融系统上的应用研究”（NO.70171001）的组成部分。
This dissertation studies the wavelet and frequency methods of financial volatility analysis. There are eight chapters in it. The main work and innovations are included in Chapter two to Chapter seven.
In Chapter two, the frequency domain method of financial volatility analysis is studied. The estimation method of the dominating frequency of time series which are non-stationary and have long memory is proposed on the basis of the invertible short and long memory autoregressive integrated moving average model. A algorithm including the determination of the model orders of approximate maximum likelihood parameters estimation based on minimizing the sum of the squared residuals is put forward. The method is used to the return series analysis of China stock markets.
In Chapter three, the time-scale method of financial volatility analysis is studied. Based on MODWT coefficients, a long memory analysis method using wavelet variance and a correlation analysis method using wavelet correlation and wavelet cross correlation for financial volatility are suggested. The methods are applied to the analysis of the properties of long memory and correlation of China stock markets volatility.
In Chapter four, the properties of DWT coefficients of LMSV process are analyzed. For DWT coefficients of the same scale, correlation analysis and spectral analysis are taken separately. For DWT coefficients of different scales, correlation analysis is taken. The result suggests that for both the same scale and different scales, DWT coefficients of LMSV process are approximately uncorrelated.
In Chapter five, the analysis method for structure change of LMSV model based on wavelet transformation is studied. According to the analysis results of DWT coefficients of LMSV process in the same scale, the method for detecting single structure change point using cumulative sums of squares of DWT coefficients and locating structure change point using MODWT coefficients is proposed as the DWT-CUSUMSQ-MODWT. The method for detecting and locating multiple structure change points of LMSV process is also put forward as DWT-ICUSUMSQ-MODWT. The methods suggested are proved to be effective and feasible by the structure change analysis of volatility of the return series of China stock markets.
In Chapter six, the estimation method of LMSV model based on wavelet transformation is studied. On the approximate uncorrelation property of DWT
coefficients of LMSV process in the same scale and different scales, first the quasi maximum likelihood estimation method of parameters and the estimation method of volatility process of LMSV model are presented. Then, the quasi maximum likelihood estimation method and the test method of long memory of volatility in LMSV model are proposed. The methods suggested are proved to be effective and feasible by the estimation of LMSV model of the return series of Shanghai stock market.
In Chapter seven, the parameters estimation method of time varying LMSV model based on wavelet transformation is studied. In terms of the fact that wavelet transformation can decompose a process to different scales and on the approximate uncorrelation property of DWT coefficients of LMSV process in the same scale and different scales, the local likelihood function of time varying LMSV model is set up. Then, according to the relationship between DWT and MODWT coefficients, the local likelihood function of time varying LMSV model is represented in the form of parameters of the model and estimator of local wavelet variance. The method suggested is applied to the parameters estimation of time varying LMSV model of return series of China stock markets.
The research is sponsored by National Natural Science Foundation of China: Persistence in Volatility of Multivariate Time Series and Its Application in Financial System(NO.70171001).
第二章研究金融波动的频域分析方法。基于平稳和非平稳分整自回归滑动平均模型提出了具有非平稳性和长记忆性序列的主频率的估计方法，同时提出了包括模型定阶在内的基于极小化残差平方和的近似极大似然参数估计算法。并用该方法对中国股市收益序列进行了分析。
第三章研究金融波动的时间——尺度分析方法。提出了用基于MODWT系数的小波方差分析金融波动的长记忆性、用小波互相关系数和小波交叉互相关函数分析金融波动在不同尺度下相关性的方法。并用该方法对中国股市收益波动的长记忆性和相关性进行了分析。
第四章对LMSV过程DWT系数的特性进行分析。对于同一尺度下的DWT系数，分别进行了相关性分析和谱分析；对于不同尺度下DWT系数进行了相关性分析。结果表明，LMSV过程同一尺度下DWT系数是近似不相关的，不同尺度下DWT系数也是近似不相关性。
第五章研究基于小波变换的LMSV模型变结构分析方法。根据LMSV过程同一尺度下DWT系数的近似不相关性，在基于累积平方和的正态独立变量同方差性检验方法的基础上，提出了应用DWT系数累积平方和的LMSV模型单一变结构点的DWT-CUSUMSQ-MODWT检测与定位法，以及多个变结构点的DWT-ICUSUMSQ-MODWT检测与定位法，并用该方法对中国股市收益进行了波动变结构分析。
第六章研究基于小波变换的LMSV模型的估计方法。根据LMSV过程同一尺度下和不同尺度下DWT系数的近似不相关性，提出了基于小波变换的LMSV模型参数的伪极大似然估计方法以及潜在波动过程的估计方法；又提出了基于小波变换的LMSV模型波动长记忆参数的伪极大似然估计方法以及波动长记忆性的检验方法，并用该方法对上海股市收益进行了LMSV模型分析。
第七章研究基于小波变换的时变LMSV模型参数的估计方法。根据小波变换可将过程分解到不同的尺度上以及 LMSV过程同一尺度下和不同尺度下DWT系数的近似不相关性，提出了建立局部似然函数的方法，又根据DWT系数和MODWT系数之间的关系，将局部似然函数表示为模型参数和局部小波方差估计的形式，并用该方法对中国股市收益进行了时变LMSV模型参数的估计。
本论文的研究是国家自然科学基金资助项目“多变量时间序列的波动持续性及其在金融系统上的应用研究”（NO.70171001）的组成部分。
This dissertation studies the wavelet and frequency methods of financial volatility analysis. There are eight chapters in it. The main work and innovations are included in Chapter two to Chapter seven.
In Chapter two, the frequency domain method of financial volatility analysis is studied. The estimation method of the dominating frequency of time series which are non-stationary and have long memory is proposed on the basis of the invertible short and long memory autoregressive integrated moving average model. A algorithm including the determination of the model orders of approximate maximum likelihood parameters estimation based on minimizing the sum of the squared residuals is put forward. The method is used to the return series analysis of China stock markets.
In Chapter three, the time-scale method of financial volatility analysis is studied. Based on MODWT coefficients, a long memory analysis method using wavelet variance and a correlation analysis method using wavelet correlation and wavelet cross correlation for financial volatility are suggested. The methods are applied to the analysis of the properties of long memory and correlation of China stock markets volatility.
In Chapter four, the properties of DWT coefficients of LMSV process are analyzed. For DWT coefficients of the same scale, correlation analysis and spectral analysis are taken separately. For DWT coefficients of different scales, correlation analysis is taken. The result suggests that for both the same scale and different scales, DWT coefficients of LMSV process are approximately uncorrelated.
In Chapter five, the analysis method for structure change of LMSV model based on wavelet transformation is studied. According to the analysis results of DWT coefficients of LMSV process in the same scale, the method for detecting single structure change point using cumulative sums of squares of DWT coefficients and locating structure change point using MODWT coefficients is proposed as the DWT-CUSUMSQ-MODWT. The method for detecting and locating multiple structure change points of LMSV process is also put forward as DWT-ICUSUMSQ-MODWT. The methods suggested are proved to be effective and feasible by the structure change analysis of volatility of the return series of China stock markets.
In Chapter six, the estimation method of LMSV model based on wavelet transformation is studied. On the approximate uncorrelation property of DWT
coefficients of LMSV process in the same scale and different scales, first the quasi maximum likelihood estimation method of parameters and the estimation method of volatility process of LMSV model are presented. Then, the quasi maximum likelihood estimation method and the test method of long memory of volatility in LMSV model are proposed. The methods suggested are proved to be effective and feasible by the estimation of LMSV model of the return series of Shanghai stock market.
In Chapter seven, the parameters estimation method of time varying LMSV model based on wavelet transformation is studied. In terms of the fact that wavelet transformation can decompose a process to different scales and on the approximate uncorrelation property of DWT coefficients of LMSV process in the same scale and different scales, the local likelihood function of time varying LMSV model is set up. Then, according to the relationship between DWT and MODWT coefficients, the local likelihood function of time varying LMSV model is represented in the form of parameters of the model and estimator of local wavelet variance. The method suggested is applied to the parameters estimation of time varying LMSV model of return series of China stock markets.
The research is sponsored by National Natural Science Foundation of China: Persistence in Volatility of Multivariate Time Series and Its Application in Financial System(NO.70171001).
引文
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