Max-Linear Competing Factor Models and Applications

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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.