We study the dynamics of matrix-valued time series with observed network structures by proposing a matrix network autoregression model with row and column networks of the subjects. We incorporate covariate information and a low rank intercept matrix. We allow incomplete observations in the matrices and the missing mechanism can be covariate dependent. To estimate the model, a two-step estimation procedure is proposed. The first step aims to estimate the network autoregression coefficients, and the second step aims to estimate the regression parameters, which are matrices themselves. Theoretically, we first separately establish the asymptotic properties of the autoregression coefficients and the error bounds of the regression parameters. Subsequently, a bias reduction procedure is proposed to reduce the asymptotic bias and the theoretical property of the debiased estimator is studied. Lastly, we illustrate the usefulness of the proposed method through a number of numerical studies and an analysis of a Yelp data set.

翻译：我们通过提出一种矩阵网络自回归模型来研究具有观测网络结构的矩阵值时间序列的动态，该模型包含行和列网络中的主体。我们纳入协变量信息和一个低秩截距矩阵。我们允许矩阵中的观测不完整，且缺失机制可以依赖协变量。为了估计该模型，我们提出了一种两步估计方法。第一步旨在估计网络自回归系数，第二步旨在估计回归参数（这些参数本身是矩阵）。理论上，我们首先分别建立自回归系数的渐近性质和回归参数的误差界。随后，我们提出一种偏差缩减程序以减小渐近偏差，并研究去偏估计量的理论性质。最后，我们通过一系列数值研究和对Yelp数据集的分析来说明所提出方法的实用性。