Equilibrium in Multivariate Nonstationary Time Series

非平稳时间序列系统经常出现在经济学、地震学、神经科学和物理学中，在这些领域，平稳通常是均衡的同义词。这样的系统通常是多维的，它们的建模、预测和控制具有巨大的社会和科学影响。在这类系统的预测和控制中，分离和识别平衡或平稳特征是非常重要的。这项研究项目旨在开发方法来提取表现出平衡感或平稳感的多变量非平稳过程的方面。它本质上是跨学科的，可以直接应用于经济学、地震学和神经科学数据的分析。一名研究生将参与这项研究。本研究的目的是将协整的概念和理论从经济学理论中的多元综合时间序列提升到概率统计中更一般的多元非平稳时间序列。尽管协整理论在过去的40年里在计量经济学中发挥了核心作用，在经济学中也有充分的动机，但它的要求是序列必须是积分的(单位根非平稳的)，并且满足向量自回归和移动平均模型。这个项目的目标是避免这样的限制，而专注于一般的多变量非平稳时间序列。将开发三种不同的方法来计算协整向量和协整等级的相似度。第一种是与经典的Johansen方法一致的时间域法，它依赖于降阶回归和似然比检验。第二种方法是在谱域中，依靠投影寻踪的思想。它通过最小化衡量时变和恒定谱密度函数之间差异的投影指数来搜索候选线性组合的系数。第三种方法涉及时变协整设置，其中系数随时间分段恒定。它的成功实现依赖于非平稳过程变点检测问题的较好解决，本研究探索了一种新的解决方案。结果将对收集多变量时间序列数据的环境产生立竿见影的影响，例如在金融市场、流行病学、环境监测和全球变化方面。

Nonstationary time series systems appear routinely in economics, seismology, neuroscience, and physics, where stationarity is usually synonymous with equilibrium. Such systems are usually multidimensional, and their modeling, prediction, and control have tremendous social and scientific impacts. Isolating and identifying equilibrium or stationary features are of fundamental importance in prediction and control of such systems. This research project aims to develop methodologies for extracting aspects of multivariate nonstationary processes that display a sense of equilibrium or stationarity. It is interdisciplinary in nature and has immediate applications to the analysis of economics, seismology, and neuroscience data. A graduate student will be involved in the research. This research project aims to elevate the concept and theory of cointegration from multivariate integrated time series rooted in economics theory to the more general multivariate nonstationary time series setup in probability and statistics. In spite of its central role in econometrics in the last four decades and well-founded motivations in economics, the cointegration theory suffers from the requirements that the series be integrated (unit-root nonstationary) and satisfy a vector autoregressive and moving average model. The goal of this project is to avoid such restrictions and focus on general multivariate nonstationary time series. Three distinct methods for computing analogues of cointegrating vectors and the cointegrating rank will be developed. The first is a time-domain method in line with the classical (Johansen's) approach that relies on the reduced rank regression and likelihood ratio tests. The second method is in the spectral domain and relies on the idea of projection pursuit. It searches for coefficients of candidate linear combinations by minimizing a projection index measuring the discrepancy between time-varying and constant spectral density functions. The third method is concerned with a time-varying cointegration setup where the coefficients are piecewise constant over time. Its successful implementation rests on a good solution of the problem of change-point detection for nonstationary processes, and a novel solution is explored in this research. The results will have immediate impact in settings where multivariate time series data are collected, such as in financial markets, epidemiology, environmental monitoring, and global change.