  We consider a high-dimensional model in which variables are observed over
time and space. The model consists of a spatio-temporal regression containing a
time lag and a spatial lag of the dependent variable. Unlike classical spatial
autoregressive models, we do not rely on a predetermined spatial interaction
matrix, but infer all spatial interactions from the data. Assuming sparsity, we
estimate the spatial and temporal dependence fully data-driven by penalizing a
set of Yule-Walker equations. This regularization can be left unstructured, but
we also propose customized shrinkage procedures when observations originate
from spatial grids (e.g. satellite images). Finite sample error bounds are
derived and estimation consistency is established in an asymptotic framework
wherein the sample size and the number of spatial units diverge jointly.
Exogenous variables can be included as well. A simulation exercise shows strong
finite sample performance compared to competing procedures. As an empirical
application, we model satellite measured NO2 concentrations in London. Our
approach delivers forecast improvements over a competitive benchmark and we
discover evidence for strong spatial interactions.
