Max-Linear Competing Factor Models and Applications

在不同领域的应用中，极端风险分析发挥着重要的科学和社会作用。这种重要性在2007-2008年的金融危机中得到了体现，在那次危机期间，几乎每一种资产类别的同时下跌，让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数来描述，这是一个概率概念，用来描述两个同时出错的风险变量。另一方面，尽管高频金融交易数据的极值对经济产生了巨大的影响，但数据的基本结构到目前为止一直被忽视。这项研究项目的主要目标是找到一种统计方法来表征这种极端风险。本项目正在研究的非线性竞争因素模型和非线性时间序列模型将在理论研究和实践之间架起一座桥梁。传播新的方法和统计工具将有助于更好地理解同时下降和极端共同变动。在多变量背景下，众所周知，在金融时间序列、社会网络研究、气候研究、图像处理等许多应用领域中，非线性相关性、非对称相关性和渐近相关性并存。该项目的一个主要目标是在采用不同的变量相关性特征的同时，为模拟观测做出重大的方法学和理论贡献。该项目由两个主要的子项目组成。第一个子项目提出了一个最大线性竞争因素模型，该模型可以包含不同的变量依赖特征，但仍然具有简单的因素结构形式。第二个子项目构建了一族新的多变量时间序列模型(Copula结构化M4过程)，适用于高频日内收益的多变量极大值。利用这些模型，我们将评估典型股票市场中资产同时下跌的概率。这两个子项目的结合为理解高维和时态观测数据中的变量如何相互依赖提供了一个全面的框架。

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.