Abstract

This article proposes a new approach to modeling high-dimensional time series by treating a p-dimensional time series as a nonsingular linear transformation of certain common factors and idiosyncratic components. Unlike the approximate factor models, we assume that the factors capture all the nontrivial dynamics of the data, but the cross-sectional dependence may be explained by both the factors and the idiosyncratic components. Under the proposed model, (a) the factor process is dynamically dependent and the idiosyncratic component is a white noise process, and (b) the largest eigenvalues of the covariance matrix of the idiosyncratic components may diverge to infinity as the dimension p increases. We propose a white noise testing procedure for high-dimensional time series to determine the number of white noise components and, hence, the number of common factors, and introduce a projected principal component analysis (PCA) to eliminate the diverging effect of the idiosyncratic noises. Asymptotic properties of the proposed method are established for both fixed p and diverging p as the sample size n increases to infinity. We use both simulated data and real examples to assess the performance of the proposed method. We also compare our method with two commonly used methods in the literature concerning the forecastability of the extracted factors and find that the proposed approach not only provides interpretable results, but also performs well in out-of-sample forecasting. Supplementary materials for this article are available online.

摘要

本文提出了一种建模高维时间序列的新方法，将p维时间序列视为某些公因子和特殊成分的非奇异线性变换。与近似因子模型不同，我们假设因子捕获了数据的所有重要动态，但横截面依赖性可能由因子和特殊成分来解释。在所提出的模型下，（a）因子过程是动态相关的，并且异质成分是白噪声过程，并且（b）异质成分的协方差矩阵的最大特征值可能随着维度p发散到无穷大增加。我们提出了一种针对高维时间序列的白噪声测试程序，以确定白噪声分量的数量，从而确定公因子的数量，并引入投影主成分分析（PCA）来消除异质噪声的发散效应. 对于固定p和发散p作为样本大小n，建立所提出方法的渐近性质增加到无穷大。我们使用模拟数据和真实示例来评估所提出方法的性能。我们还将我们的方法与文献中关于提取因子的可预测性的两种常用方法进行了比较，发现所提出的方法不仅提供了可解释的结果，而且在样本外预测中表现良好。本文的补充材料可在线获取。