Cross-Sectional Matrix Exponential Spatial Models: A Comprehensive Review and Some New Results

In this paper, we provide a comprehensive review of the literature on estimation, inference, and model selection approaches for cross-sectional matrix exponential spatial models. We first discuss the properties of the matrix exponential specification in modeling cross-sectional dependence in comparison to the spatial autoregressive specification. We then provide a survey of the existing estimation and inference methods for cross-sectional matrix exponential spatial models. We carefully discuss summary measures for the marginal effects of regressors, detail the matrix–vector product method for efficient computation of matrix exponential terms, and then explore model selection approaches. 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 presenting several new results. We propose an M-estimation approach for models with heteroskedastic error terms and demonstrate that the resulting M-estimator is consistent and asymptotically normally distributed. Moreover, we provide additional results for model selection exercises. Finally, in a Monte Carlo study, we evaluate the finite sample properties of various estimators from the literature alongside the M-estimator.