  Tensor-valued data are becoming increasingly available in economics and this
calls for suitable econometric tools. We propose a new dynamic linear model for
tensor-valued response variables and covariates that encompasses some
well-known econometric models as special cases. Our contribution is manifold.
First, we define a tensor autoregressive process (ART), study its properties
and derive the associated impulse response function. Second, we exploit the
PARAFAC low-rank decomposition for providing a parsimonious parametrization and
to incorporate sparsity effects. We also contribute to inference methods for
tensors by developing a Bayesian framework which allows for including
extra-sample information and for introducing shrinking effects. We apply the
ART model to time-varying multilayer networks of international trade and
capital stock and study the propagation of shocks across countries, over time
and between layers.
