Reduced-rank regressions are powerful tools used to identify co-movements within economic time series. However, this task becomes challenging when we observe matrix-valued time series, where each dimension may have a different co-movement structure. We propose reduced-rank regressions with a tensor structure for the coefficient matrix to provide new insights into co-movements within and between the dimensions of matrix-valued time series. Moreover, we relate the co-movement structures to two commonly used reduced-rank models, namely the serial correlation common feature and the index model. Two empirical applications involving U.S.\ states and economic indicators for the Eurozone and North American countries illustrate how our new tools identify co-movements.

翻译：降秩回归是识别经济时间序列中共变动的有力工具。然而，当我们观测矩阵值时间序列时，这一任务变得具有挑战性，因为每个维度可能具有不同的共变动结构。我们提出了具有张量结构的降秩回归方法，通过构建系数矩阵的张量结构，为矩阵值时间序列维度内部及维度间的共变动提供新的分析视角。此外，我们将共变动结构与两种常用的降秩模型——序列相关共同特征模型和指数模型——建立理论联系。通过美国各州经济数据以及欧元区与北美国家经济指标的两个实证应用，我们展示了新工具如何有效识别共变动模式。