Most factor modelling research in vector or matrix-valued time series assume all factors are pervasive/strong and leave weaker factors and their corresponding series to the noise. Weaker factors can in fact be important to a group of observed variables, for instance a sector factor in a large portfolio of stocks may only affect particular sectors, but can be important both in interpretations and predictions for those stocks. While more recent factor modelling researches do consider ``local'' factors which are weak factors with sparse corresponding factor loadings, there are real data examples in the literature where factors are weak because of weak influence on most/all observed variables, so that the corresponding factor loadings are not sparse (non-local). As a first in the literature, we propose estimators of factor strengths for both local and non-local weak factors, and prove their consistency with rates of convergence spelt out for both vector and matrix-valued time series factor models. Factor strength has an important indication in what estimation procedure of factor models to follow, as well as the estimation accuracy of various estimators (Chen and Lam, 2024). Simulation results show that our estimators have good performance in recovering the true factor strengths, and an analysis on the NYC taxi traffic data indicates the existence of weak factors in the data which may not be localized.

翻译：大多数关于向量或矩阵值时间序列的因子建模研究假设所有因子都是普遍的/强的，并将较弱因子及其对应的序列归为噪声。然而，较弱的因子实际上可能对一组观测变量至关重要，例如，大型股票投资组合中的行业因子可能仅影响特定行业，但在这些股票的解释和预测中却具有重要意义。尽管近年来的因子建模研究确实考虑了“局部”因子——即对应因子载荷稀疏的弱因子，但文献中的真实数据实例表明，因子可能因对大多数/所有观测变量影响较弱而成为弱因子，此时对应的因子载荷并不稀疏（非局部因子）。作为文献中的首次尝试，我们提出了针对局部和非局部弱因子的因子强度估计量，并证明了其在向量和矩阵值时间序列因子模型中的相合性及收敛速率。因子强度对因子模型的估计方法选择以及各类估计量的估计精度具有重要指示意义（Chen and Lam, 2024）。模拟结果表明，我们的估计量在恢复真实因子强度方面表现良好，对纽约出租车交通数据的分析则揭示了数据中可能存在非局部化的弱因子。