High-dimensional time series often exhibit hierarchical structures represented by tensors, while statistical methodologies that can effectively exploit the structural information remain limited. We propose a supervised factor modeling framework that accommodates general hierarchical structures by extracting low-dimensional features sequentially in the mode orders that respect the hierarchical structure. Our method can select a small collection of such orders to allow for impurities in the hierarchical structures, yielding interpretable loading matrices that preserve the hierarchical relationships. A practical estimation procedure is proposed, with a hyperparameter selection scheme that identifies a parsimonious set of action orders and interim ranks, thereby revealing the possibly latent hierarchical structures. Theoretically, non-asymptotic error bounds are derived for the proposed estimators in both regression and autoregressive settings. An application to the IPIP-NEO-120 personality panel illustrates superior forecasting performance and clearer structural interpretation compared with existing methods based on tensor decompositions and hierarchical factor analysis.

翻译：高维时间序列常以张量形式呈现层级结构，然而能够有效利用这种结构信息的统计方法仍较为有限。我们提出一种监督因子建模框架，通过沿尊重层级结构的模式阶次依次提取低维特征，可适应一般性的层级结构。该方法能选择少量此类阶次以允许层级结构中的杂质存在，从而生成保留层级关系的可解释载荷矩阵。我们提出实用的估计程序，并设计超参数选择方案以识别简洁的作用阶次与中间秩集合，进而揭示可能存在的潜在层级结构。在理论层面，我们推导了回归与自回归场景下所提估计量的非渐近误差界。在IPIP-NEO-120人格面板数据上的应用表明，相比基于张量分解与层级因子分析的现有方法，本方法展现出更优的预测性能与更清晰的结构可解释性。