分位数回归理论及其应用
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摘要
分位数回归是给定回归变量X,估计响应变量Y条件分位数的一个基本方法。它不仅可以度量回归变量在分布中心的影响,而且还可以度量在分布上尾和下尾的影响,因此较之经典的最小二乘回归具有独特的优势。本文主要对分位数回归的理论、Copula分位数回归、极端分位数以及分位数回归在各个领域的应用进行了深入研究。论文的主要工作如下:
1.论文介绍了极值的基本理论,为以后的各章提供了理论基础。并选取logistic分布,应用二元超阈值模型和二元点过程模型度量沪深股市收益率的尾部相关性。结果表明:沪深股市收益率在尾部具有很强的相关性,并且这两种模型都不失为一种很好的建模方法。
2.论文构建线性条件分位数回归模型,分析澳大利亚西部Fremantle港地区在1897-1989年间年最高海平面高度与时间及年平均南方涛动指数之间的线性变化趋势,并与经典的最小二乘回归拟合进行比较。结果表明:在不同分位数下年最高海平面高度与时间及南方涛动指数之间所呈现的线性趋势是不同的,分位数回归比经典的最小二乘回归能够提供更多的信息,因此对于我们进行预测和防范具有十分重要的意义。
3.论文研究了Copula分位数回归,推导出几种常见Copula的分位数曲线,并应用模拟研究的方法说明分位数回归估计方法的精确性。在此基础之上,选取Clayton Copula,应用Copula非线性分位数回归模型度量沪深股市收益率在不同分位数下的风险相关性,并与由极值理论方法得到的结果进行比较。结果表明:在不同分位数下沪深股市具有不同的相关关系,比普通的回归方法能更全面的描述不同区域的风险相关关系,而极值理论方法侧重于极端情况下尾部指标的估计。
4.论文通过研究极端分位数的估计方法及渐近性质,把极端分位数所具有的行为特征应用到VaR的研究中,建立上海股市收益率的条件分位数模型,描述其在极端分位数下的变化趋势。并选取适当的尾部模型,在此基础之上应用外推法预测非常极端分位数下的条件VaR,并与直接由分位数回归模型预测的结果进行比较。结果表明:两种方法得到的结果变化趋势都是一致的,由外推法预测的结果相对小一些。
Quantile regression is a basic tool for estimating conditional quantiles of a response variable Y given a vector of regressors X. It can be used to measure the effect of regressors not only in the center of a distribution, but also in the upper and lower tails. So it has much more advantages than the classical least square regression. The theory of quantile regression, Copula quantile regression, extremal quantiles and applications of quantile regression in many fields are discussed in this paper. The main achievements of this work are listed as follows:
1. The basic extreme value theory is introduced, which is the basis of other chapters. We choose logistic distribution and use bivariate excess threshold model and bivariate point process model to measure the tail dependence of Shanghai and Shenzhen Stock market. The results show that the return rates of Shanghai and Shenzhen Stock markets have strong tail dependence and the two models are excellent for application.
2. The linear trend of the annual maximum sea level at Fremantle Port, Western Australia, related with time and Southern Oscillation index during 1897-1989 is analyzed by linear conditional quantile regression model. And the result is compared with that of the classical least square regression. The results show that, under different quantiles, the linear trend of the annual maximum sea level related with time and Southern Oscillation Index is different, and quantile regression can provide much more information than the classical least square regression. So it is of great significance for prediction and prevention.
3. The theory of Copula quantile regression is studied and the quantile curves of several common Copulas are obtained. The accuracy of quantile regression estimation is shown by simulation research. We choose clayton Copula and use Copula nonlinear conditional quantile regression model to measure the tail area risk dependence in Shanghai and Shenzhen stock markets. And then the result of this approach is compared with the tail dependence measure by extreme value method. The results show that Shanghai and Shenzhen stock markets have different risk dependence under different quantiles and extreme value theory method only focuses on the estimation of tail dependence.
4. By studying the estimation method and asymptotic behaviors of extremal quantiles, we apply its behaviors to the research of VaR. The conditional quantile regression model of return rates of Shanghai stock market is established, which describes the trend of rates under extremal quantiles. Conditional VaR in very extreme quantiles is predicated by using extrapolation methods under the proper tail model. Comparison with the prediction of the ordinary quantile regression model is also given. The results show that the tendencies of the two predictions are similar and the value estimated by the extrapolation methods is relatively small.
1.论文介绍了极值的基本理论,为以后的各章提供了理论基础。并选取logistic分布,应用二元超阈值模型和二元点过程模型度量沪深股市收益率的尾部相关性。结果表明:沪深股市收益率在尾部具有很强的相关性,并且这两种模型都不失为一种很好的建模方法。
2.论文构建线性条件分位数回归模型,分析澳大利亚西部Fremantle港地区在1897-1989年间年最高海平面高度与时间及年平均南方涛动指数之间的线性变化趋势,并与经典的最小二乘回归拟合进行比较。结果表明:在不同分位数下年最高海平面高度与时间及南方涛动指数之间所呈现的线性趋势是不同的,分位数回归比经典的最小二乘回归能够提供更多的信息,因此对于我们进行预测和防范具有十分重要的意义。
3.论文研究了Copula分位数回归,推导出几种常见Copula的分位数曲线,并应用模拟研究的方法说明分位数回归估计方法的精确性。在此基础之上,选取Clayton Copula,应用Copula非线性分位数回归模型度量沪深股市收益率在不同分位数下的风险相关性,并与由极值理论方法得到的结果进行比较。结果表明:在不同分位数下沪深股市具有不同的相关关系,比普通的回归方法能更全面的描述不同区域的风险相关关系,而极值理论方法侧重于极端情况下尾部指标的估计。
4.论文通过研究极端分位数的估计方法及渐近性质,把极端分位数所具有的行为特征应用到VaR的研究中,建立上海股市收益率的条件分位数模型,描述其在极端分位数下的变化趋势。并选取适当的尾部模型,在此基础之上应用外推法预测非常极端分位数下的条件VaR,并与直接由分位数回归模型预测的结果进行比较。结果表明:两种方法得到的结果变化趋势都是一致的,由外推法预测的结果相对小一些。
Quantile regression is a basic tool for estimating conditional quantiles of a response variable Y given a vector of regressors X. It can be used to measure the effect of regressors not only in the center of a distribution, but also in the upper and lower tails. So it has much more advantages than the classical least square regression. The theory of quantile regression, Copula quantile regression, extremal quantiles and applications of quantile regression in many fields are discussed in this paper. The main achievements of this work are listed as follows:
1. The basic extreme value theory is introduced, which is the basis of other chapters. We choose logistic distribution and use bivariate excess threshold model and bivariate point process model to measure the tail dependence of Shanghai and Shenzhen Stock market. The results show that the return rates of Shanghai and Shenzhen Stock markets have strong tail dependence and the two models are excellent for application.
2. The linear trend of the annual maximum sea level at Fremantle Port, Western Australia, related with time and Southern Oscillation index during 1897-1989 is analyzed by linear conditional quantile regression model. And the result is compared with that of the classical least square regression. The results show that, under different quantiles, the linear trend of the annual maximum sea level related with time and Southern Oscillation Index is different, and quantile regression can provide much more information than the classical least square regression. So it is of great significance for prediction and prevention.
3. The theory of Copula quantile regression is studied and the quantile curves of several common Copulas are obtained. The accuracy of quantile regression estimation is shown by simulation research. We choose clayton Copula and use Copula nonlinear conditional quantile regression model to measure the tail area risk dependence in Shanghai and Shenzhen stock markets. And then the result of this approach is compared with the tail dependence measure by extreme value method. The results show that Shanghai and Shenzhen stock markets have different risk dependence under different quantiles and extreme value theory method only focuses on the estimation of tail dependence.
4. By studying the estimation method and asymptotic behaviors of extremal quantiles, we apply its behaviors to the research of VaR. The conditional quantile regression model of return rates of Shanghai stock market is established, which describes the trend of rates under extremal quantiles. Conditional VaR in very extreme quantiles is predicated by using extrapolation methods under the proper tail model. Comparison with the prediction of the ordinary quantile regression model is also given. The results show that the tendencies of the two predictions are similar and the value estimated by the extrapolation methods is relatively small.
引文
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