  Matrix-valued time series data are frequently observed in a broad range of
areas and have attracted great attention recently. In this work, we model
network effects for high dimensional matrix-valued time series data in a matrix
autoregression framework. To characterize the potential heterogeneity of the
subjects and handle the high dimensionality simultaneously, we assume that each
subject has a latent group label, which enables us to cluster the subject into
the corresponding row and column groups. We propose a group matrix network
autoregression (GMNAR) model, which assumes that the subjects in the same group
share the same set of model parameters. To estimate the model, we develop an
iterative algorithm. Theoretically, we show that the group-wise parameters and
group memberships can be consistently estimated when the group numbers are
correctly or possibly over-specified. An information criterion for group number
estimation is also provided to consistently select the group numbers. Lastly,
we implement the method on a Yelp dataset to illustrate the usefulness of the
method.
