  High-dimensional matrix-variate time series data are becoming widely
available in many scientific fields, such as economics, biology, and
meteorology. To achieve significant dimension reduction while preserving the
intrinsic matrix structure and temporal dynamics in such data, Wang et al.
(2017) proposed a matrix factor model that is shown to provide effective
analysis. In this paper, we establish a general framework for incorporating
domain or prior knowledge in the matrix factor model through linear
constraints. The proposed framework is shown to be useful in achieving
parsimonious parameterization, facilitating interpretation of the latent matrix
factor, and identifying specific factors of interest. Fully utilizing the
prior-knowledge-induced constraints results in more efficient and accurate
modeling, inference, dimension reduction as well as a clear and better
interpretation of the results. In this paper, constrained, multi-term, and
partially constrained factor models for matrix-variate time series are
developed, with efficient estimation procedures and their asymptotic
properties. We show that the convergence rates of the constrained factor
loading matrices are much faster than those of the conventional matrix factor
analysis under many situations. Simulation studies are carried out to
demonstrate the finite-sample performance of the proposed method and its
associated asymptotic properties. We illustrate the proposed model with three
applications, where the constrained matrix-factor models outperform their
unconstrained counterparts in the power of variance explanation under the
out-of-sample 10-fold cross-validation setting.
