High-Dimensional Time Series, Common Factors, and Nonstationarity

在这个现代信息时代，随着计算能力的提高，访问和分析前所未有的规模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，甚至远远大于观测值的数量。在这种情况下的推理被普遍认为是当代统计学的一个重要挑战，也是近年来活跃研究的焦点。剑桥牛顿研究所于2008年1月至6月开展了一项关于复杂高维数据的统计理论与方法的大规模研究项目。在此背景下，本课题致力于研究因各种实际问题而产生的超高维时间序列分析的理论和方法。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。所谓的面板数据，是为各种应用收集的，由p个长度为n的时间序列组成，通常p大于或远远大于n。对于分析这些大规模的多时间序列，降维是成功的关键。在这个项目中，我们提出了一种创新的因子建模技术，它具有统计上的通用性和计算上的有效性。特别是，我们将在几个相互关联的领域进行研究，包括：(i)用非平稳因素建模高维时间序列，（ii）基于因素建立高维波动动力学，包括使用高频数据的高维每日波动；(3)识别曲线时间序列的有限维数。(i)的结果将有助于建模和预测来自经济学、商业、市场营销、社会学、生物学和生态学等领域的时间序列面板。（ii）直接解决现代金融中的重要问题，如资产定价、投资组合配置和风险管理。曲线时间序列分析（iii）将在环境研究（年度天气记录图、年度污染图）、金融（每日波动曲线、收益率曲线）、市场营销（销售图）等领域得到应用。将开发免费软件来实现这些新方法。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。该方法的新颖之处主要在于将潜在因素的估计问题转化为标准特征分析，从而适用于数千维数的情况。在曲线时间序列的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.