  Spatial econometric research typically relies on the assumption that the
spatial dependence structure is known in advance and is represented by a
deterministic spatial weights matrix. Contrary to classical approaches, we
investigate the estimation of sparse spatial dependence structures for regular
lattice data. In particular, an adaptive least absolute shrinkage and selection
operator (lasso) is used to select and estimate the individual connections of
the spatial weights matrix. To recover the spatial dependence structure, we
propose cross-sectional resampling, assuming that the random process is
exchangeable. The estimation procedure is based on a two-step approach to
circumvent simultaneity issues that typically arise from endogenous spatial
autoregressive dependencies. The two-step adaptive lasso approach with
cross-sectional resampling is verified using Monte Carlo simulations.
Eventually, we apply the procedure to model nitrogen dioxide ($\mathrm{NO_2}$)
concentrations and show that estimating the spatial dependence structure
contrary to using prespecified weights matrices improves the prediction
accuracy considerably.
