  The matrix exponential spatial models exhibit similarities to the
conventional spatial autoregressive model in spatial econometrics but offer
analytical, computational, and interpretive advantages. This paper provides a
comprehensive review of the literature on the estimation, inference, and model
selection approaches for the cross-sectional matrix exponential spatial models.
We discuss summary measures for the marginal effects of regressors and detail
the matrix-vector product method for efficient estimation. Our aim is not only
to summarize the main findings from the spatial econometric literature but also
to make them more accessible to applied researchers. Additionally, we
contribute to the literature by introducing some new results. We propose an
M-estimation approach for models with heteroskedastic error terms and
demonstrate that the resulting M-estimator is consistent and has an asymptotic
normal distribution. We also consider some new results for model selection
exercises. In a Monte Carlo study, we examine the finite sample properties of
various estimators from the literature alongside the M-estimator.
