Matrix-valued time series data are widely available in various applications, attracting increasing attention in the literature. However, while nonlinearity has been recognized, the literature has so far neglected a deeper and more intricate level of nonlinearity, namely the {\it row-level} nonlinear dynamics and the {\it column-level} nonlinear dynamics, which are often observed in economic and financial data. In this paper, we propose a novel two-way threshold matrix autoregression (TWTMAR) model. This model is designed to effectively characterize the threshold structure in both rows and columns of matrix-valued time series. Unlike existing models that consider a single threshold variable or assume a uniform structure change across the matrix, the TWTMAR model allows for distinct threshold effects for rows and columns using two threshold variables. This approach achieves greater dimension reduction and yields better interpretation compared to existing methods. Moreover, we propose a parameter estimation procedure leveraging the intrinsic matrix structure and investigate the asymptotic properties. The efficacy and flexibility of the model are demonstrated through both simulation studies and an empirical analysis of the Fama-French Portfolio dataset.

翻译：矩阵值时间序列数据在各领域应用中广泛存在，日益受到学界关注。然而，尽管非线性特征已得到认识，现有文献仍忽略了更深层次且更复杂的非线性动态——即常出现于经济金融数据中的行级非线性动态与列级非线性动态。本文提出一种新颖的双向阈值矩阵自回归模型。该模型旨在有效刻画矩阵值时间序列在行与列两个维度上的阈值结构。与现有仅考虑单一阈值变量或假设矩阵整体结构一致变化的模型不同，本模型通过两个阈值变量分别捕捉行与列的差异化阈值效应。相较于现有方法，该建模策略实现了更强的降维能力并具有更优的可解释性。此外，我们提出一种基于矩阵内在结构的参数估计方法，并研究了其渐近性质。通过模拟研究及对Fama-French投资组合数据集的实证分析，验证了模型的有效性与灵活性。