Max-Linear Competing Factor Models and Applications

在各个领域的应用中，极端风险分析发挥着重要的科学和社会作用。2007-2008年的金融危机证明了这一重要性，在此期间，几乎所有资产类别的同时下跌，使投资者几乎没有选择。这类事件可以用一个称为渐近相关性的指标来描述，渐近相关性是一个概率概念，用于描述两个风险变量同时出错。另一方面，虽然高频金融交易数据的极端性对经济影响巨大，但数据的基本结构至今被忽视。该研究项目的主要目标是找到一种统计方法来表征此类极端风险。本课题所研究的非线性竞争因子模型和非线性时间序列模型将弥补理论研究与实践之间的差距。新方法和统计工具的传播将使人们更好地了解同时下降和极端的共同运动。在多变量背景下，众所周知，非线性相关，非对称相关和渐近相关共存于金融时间序列，社会网络研究，气候研究，图像处理和许多其他应用领域。该项目的一个主要目标是同时对观测建模做出重要的方法和理论贡献，同时拥抱不同的变量依赖特征。该项目包括两个主要的分项目。第一个子计划提出一个最大线性竞争因子模型，它可以包含不同的变量依赖特征，但仍然具有简单的因子结构形式。第二个子项目建立了一个新的多变量时间序列模型（Copula结构M4过程），适用于高频日内收益的多变量最大值。使用这些模型，在一个典型的股票市场的资产的同时下降的概率进行评估。这两个子项目的整合提供了一个全面的框架，了解变量如何相互依赖，在高维和时间观测数据。

Throughout applications in diverse fields, analysis of extreme risks plays an important scientific and societal role. This importance is exemplified by the 2007-2008 financial crisis, during which the concurrent decline of almost every asset category gave investors few options. These kinds of events can be described by an index termed asymptotic dependence, a probability concept that has been used to depict two risk variables concurrently going wrong. On the other hand, although the extremes of high-frequency financial transaction data have a huge economic impact, the basic structure of the data has been ignored up to now. The primary goal of this research project is to find a statistical method for characterizing such extreme risks. The nonlinear competing factor model and nonlinear time series model under investigation in this project will bridge the gap between theoretical research and practice. The dissemination of new methodologies and statistical tools will lead to a better understanding of concurrent decline and extreme co-movement. In the multivariate context, it is well-known that nonlinear dependence, asymmetric dependence, and asymptotic dependence co-exist in financial time series, social network studies, climate studies, image processing, and many other application areas. A major goal of this project is to make significant methodological and theoretical contributions to modeling observations simultaneously, while embracing the different variable dependence features. The project consists of two main sub-projects. The first sub-project proposes a max-linear competing factor model that can incorporate the different variable dependence features but still possesses a simple form of factor structure. The second sub-project builds a new family of multivariate time series models (Copula Structured M4 Processes) suitable for multivariate maxima of high frequency intra-day returns. Using these models, the probability of the concurrent decline of assets in a typical stock market will be evaluated. The integration of the two sub-projects provides a comprehensive framework for understanding how variables depend on each other in high dimensional and temporal observational data.