  We develop a new methodology for forecasting matrix-valued time series with
historical matrix data and auxiliary vector time series data. We focus on a
time series of matrices defined on a static 2-D spatial grid and an auxiliary
time series of non-spatial vectors. The proposed model, Matrix AutoRegression
with Auxiliary Covariates (MARAC), contains an autoregressive component for the
historical matrix predictors and an additive component that maps the auxiliary
vector predictors to a matrix response via tensor-vector product. The
autoregressive component adopts a bi-linear transformation framework following
Chen et al. (2021), significantly reducing the number of parameters. The
auxiliary component posits that the tensor coefficient, which maps non-spatial
predictors to a spatial response, contains slices of spatially smooth matrix
coefficients that are discrete evaluations of smooth functions from a
Reproducible Kernel Hilbert Space (RKHS). We propose to estimate the model
parameters under a penalized maximum likelihood estimation framework coupled
with an alternating minimization algorithm. We establish the joint asymptotics
of the autoregressive and tensor parameters under fixed and high-dimensional
regimes. Extensive simulations and a geophysical application for forecasting
the global Total Electron Content (TEC) are conducted to validate the
performance of MARAC.
