Collaborative Research: Copulas, Tail Copulas, Garch and Extreme Values in Dependence Modelling and Risk Management

经济学、生存分析、医学、生物统计学、保险、金融等社会科学领域的多变量数据序列通常是非线性的、非正态分布的，具有非线性的协同运动和相互作用。然而，现有的多变量时间序列数据分析的计量经济学和统计学方法大多局限于多变量正态或条件正态框架。这个跨学科的合作研究项目将研究基于copula的半参数模型的估计、测试和经验应用，这些模型用于分析表现出非线性共运动、不对称、重尾、可能删节和/或其他非正态模式的多变量数据序列。这类模型将通过指定非参数（或半参数）的边际分布和指定捕获多变量参数（或半参数）之间依赖关系的联结函数，部分地解决“维数诅咒”问题。研究将涉及对经验似然法、筛极大似然法、自举法、极值理论、基于残差的加权经验过程理论等几种现代统计方法的新颖修正和应用。研究成果将包括几篇关于独立数据和截尾数据的基于copula的半参数多元模型的估计和测试的原创论文，以及非线性和可能的重尾时间序列数据的半参数动态模型的估计和测试。该项目将产生新的评估和测试软件，可以从作者的网页上免费下载。本项目的研究成果将对基于copulas、tail copulas和Garch的半参数多元模型估计和检验的计量经济学和统计学文献做出重要贡献，并将在经济学（如产业组织、收入不平等、卫生经济学）、保险、生物统计学、医学等非线性依赖重要的领域发挥重要作用。该项目将开发新的方法，并应用现代统计方法来探索多元序列之间的非线性依赖关系，并通过经济、保险、金融、统计和其他科学的实证应用来理解人类和社会相互作用的复杂动态。由于金融和保险从业者已经将非线性时间序列模型与copula和/或尾copula结合起来，启发式地模拟多元期权定价，投资组合风险价值，相关违约和信用风险以及不同序列之间的时变非对称非线性协同运动，本项目的结果将指导应用研究人员进行统计可靠的经济政策评估，金融预测和风险管理。主要的教育计划是培养经济学、统计学、金融学的博士研究生，使其成为相关学科的研究人员，并为经济学、管理学、统计学、数量金融学的学生开设新的课程。该奖项是2006财政年度数学科学优先领域数学社会和行为科学特别竞赛的一部分。

Multivariate data series in economics, survival analysis, medical science, biostatistics, insurance, finance, and other fields in social sciences are typically nonlinear, non-normally distributed, and have nonlinear co-movements and interactions. However, the existing econometrics and statistics methods in multivariate time series data analysis are largely confined to multivariate normal or conditional normal framework. This interdisciplinary collaborative research project will examine the estimation, testing, and empirical applications of broad classes of copula-based semiparametric models for analyzing multivariate data series that exhibit nonlinear co-movements, asymmetric, heavy-tailed, possibly censored and/or other non-normal patterns. These classes of models will partially solve the "curse-of-dimensionality" problem by specifying the marginal distributions nonparametrically (or semiparametrically) and specifying the copula functions that capture the dependence among the multivariate variables parametrically (or semiparametrically). The research will involve novel modifications and applications of several modern statistical methods such as empirical likelihood method, sieve maximum likelihood method, bootstrap method, extreme value theory, residual-based weighted empirical process theory, etc. Research outputs will include several original papers on estimation and testing of copula-based semiparametric multivariate models for independent data and for censored data, as well as estimation and testing of semiparametric dynamic models for nonlinear and possibly heavy-tailed time series data. The project will produce new estimation and testing software that can be freely downloaded from the authors' web pages. The results from this project will make significant contributions to econometrics and statistics literature on estimation and testing of semiparametric multivariate models based on copulas, tail copulas, and Garch with heavy tails, and will be very useful in economics (such as industrial organization, income inequality, health economics), insurance, biostatistics, medical science, and other fields where nonlinear dependence is important. This project will develop new methodologies and apply modern statistical methods to explore nonlinear dependence among multivariate series and to understand complicated dynamics of human and social interactions with empirical applications to economics, insurance, finance, statistics, and other sciences. Since practitioners in finance and insurance have been combining nonlinear time series models with copulas and/or tail copulas to heuristically model multivariate option pricing, portfolio Value-at-Risk, correlated default and credit risk, and the time-varying asymmetric nonlinear co-movements among different series, the results from this project will guide applied researchers to perform statistically reliable economic policy evaluations, financial forecasts, and risk managements. The main educational plans are to train PhD students in economics, statistics, and finance to become capable researchers in related topics, and to develop new courses for students in economics, management, statistics, and quantitative finance. This award was supported as part of the fiscal year 2006 Mathematical Sciences priority area special competition on Mathematical Social and Behavioral Sciences (MSBS).