High-Dimensional Time Series, Common Factors, and Nonstationarity

在这个现代信息时代，随着计算能力的增加，访问和分析前所未有的大小和复杂性的数据已经变得司空见惯。在许多重要的统计应用中，变量或参数的数量现在与观测值的数量一样大，甚至比观测值的数量大得多。在这种情况下的推断被公认为是当代统计学的一个重要挑战，并已成为近年来积极研究的焦点。位于剑桥的牛顿研究所在2008年1月至6月期间开展了一项关于复杂高维数据的统计理论和方法的大规模研究计划。在此背景下，本项目致力于对各种实际问题所产生的超高维时间序列进行分析的理论和方法研究。例如，在投资组合优化和风险管理中，所涉及的资产数量通常为数百或数千。所谓面板数据，是指由p个长度为n的时间序列组成的数据，通常p大于或远大于n，对于这些大规模的多时间序列，降维是分析成功的关键。在这个项目中，我们提出了一个创新的因素建模技术，这是统计通用和计算有效。特别是，我们将在几个相互关联的领域进行研究，包括：（i）用非平稳因素对高维时间序列进行建模，（ii）基于因素建立高维波动率动态，包括使用高频数据的高维每日波动率;以及（iii）识别曲线时间序列的有限维度。从（i）的结果将是有用的建模和预测面板的时间序列所产生的经济学，商业，市场营销，社会学，生物学和生态学等。（ii）直接解决的重要问题，在现代金融，如资产定价，投资组合分配和风险管理。曲线时间序列分析（iii）将在例如环境研究（年度天气记录图表、年度污染图表）、金融（每日波动曲线、收益率曲线）、营销（销售图表）中找到应用。高维数据分析显然是当今统计学（包括生物统计学）和计量经济学（包括金融计量经济学）中最具活力的研究领域之一。该建议的新奇主要在于新的估计过程，该估计过程将估计潜在因素的问题（可能是非平稳的）转移到标准的特征分析中，因此适用于数千维的情况。在曲线时间序列框架中处理非平稳性的思想也是新的。

In this modern information age, with increasing computing power it has become commonplace to access and to analyze data of unprecedented size and complexity. In many important statistical applications, the number of variables or parameters is now as large as or even much larger than the number of observations. Inference under such a circumstance is generally acknowledged as an important challenge in contemporary statistics, and has been a focus point for active research lately. The Newton Institute in Cambridge has staged a large scale research programme on Statistical Theory and Methods for Complex, High-Dimensional Data in January -- June 2008. Against this background, the proposed project is devoted to the research on both theory and methodology for analyzing ultra-high dimensional time series which arise from various practical problems. For example, in portfolio optimization and risk management the number of assets concerned is typically in the order of hundreds or thousands. The so-called panel data, collected for various applications, consist of p time series of length n with, typically, p is larger or much larger than n.For analyzing those large scale multiple time series, dimension-reduction is a key for success. In this project we propose an innovative factor modelling technique which is statistically versatile and computationally effective. In particular, we will conduct the research in several interlocking areas including: (i) modelling high-dimensional time series with nonstationary factors, (ii) establishing high-dimensional volatility dynamics based on factors, including high-dimensional daily volatilities using high-frequency data; and (iii) identify finite dimensionality of curve time series. The results from (i) will be useful for modelling and forecasting panels of time series arising from economics, business, marketing, sociology, biology and ecology etc. (ii) addresses directly the important issues in modern finance such as asset pricing, portfolio allocation and risk management. Curve time series analysis (iii) will find applications in, for example, environment studies (annual weather record charts, annual pollution charts), finance (daily volatility curves, yield curves), marketing (sales charts). The freely-available softwares will be developed to implement the new methods.High-dimensional data analysis is clearly one of the most vibrant research areas in statistics (including biostatistics) and econometrics (including financial econometrics) these days. The novelty of this proposal lies mainly on the new estimation procedure which transfers the problem of estimating latent factors, which may be nonstationary, into a standard eigenanalysis, and therefore is applicable to the cases with the dimensionalities in the order of thousands. The idea of handling nonstationarity in the framework of curve time series is also new.