Identifying the structure of high-dimensional time series via eigen-analysiss

Cross-sectional structures and temporal tendency are important features of high-dimensional time series. Based on eigen-analysis on sample covariance matrices, we propose a novel approach to identifying four popular structures of high-dimensional time series, which are grouped in terms of factor structures and stationarity. The proposed three-step method includes:a ratio statistic of empirical eigenvalues;a projected Augmented Dickey-Fuller Test;a new unit-root test based on the largest empirical eigenvalues.
a ratio statistic of empirical eigenvalues; a projected Augmented Dickey-Fuller Test; a new unit-root test based on the largest empirical eigenvalues. We develop asymptotic properties for these three statistics to ensure the feasibility of the whole identifying procedure. Finite sample performances are illustrated via various simulations. We also analyze U.S. mortality data, U.S. house prices and income, and U.S. sectoral employment, all of which possess cross–sectional dependence and non-stationary temporal dependence. It is worth mentioning that we also contribute to statistical justification for the benchmark paper by Lee and Carter (1992) in mortality forecasting.

截面结构与时序趋势是高维时间序列的重要特征。基于样本协方差矩阵的特征分析，本文提出一种新颖的高维时间序列四类典型结构识别方法，该四类结构按因子结构与平稳性进行分类。所提出的三步法包含：经验特征值比率统计量、投影增广迪基-富勒检验（Augmented Dickey-Fuller Test）、以及基于最大经验特征值的新型单位根检验。经验特征值比率统计量、投影增广迪基-富勒检验（Augmented Dickey-Fuller Test）、以及基于最大经验特征值的新型单位根检验。本文推导了这三类统计量的渐近性质，以确保整体识别流程的可行性。通过多组仿真实验验证了该方法的有限样本表现。本文还分析了美国死亡率数据、美国房价与收入数据以及美国行业就业数据，上述数据集均存在截面相依性与非平稳时序相依性。值得一提的是，本文还为死亡率预测领域的经典基准文献——Lee与Carter（1992）的研究提供了统计合理性佐证。