In this appendix, we provide additional simulation results on the finite sample performance of our suggested MS and MA procedures. The simulation setting for these results is described in the main text. For the normal distribution case, Tables 1 and 2 report the simulation results when the true spatial weights matrices are W 2 and W 3 , respectively. For the non-normal distribution case, Tables 3, 4 and 5 report the simulation results when the true spatial weights matrices are W 2 , W 3 and W 4 , respectively. The remaining tables, Tables 6-8, include the simulation results for the heteroskedastic case.

The basalt residual soil of high slope of BiWei highway in Guizhou province belongs to regional special soil. Comprehensive and systematic evaluation for engineering geologic characteristics of residual soil on basalt is necessary to the construction in this area and geological engineering investigation. Kaolinite is dominant in clay mineral of this soil, which has a higher clay particle content. The special material composition and structural morphology make it have the physical properties of high porosity, low density and high water content. But it has mechanical properties of higher strength. The soil has a low expansibility and weak water-resistance. Moreover, the soil is sensitive to environmental factors such as temperature and humidity. The mechanical properties become weak after soaking, leading to a larger collapse deformation and a heavy disintegration. Therefore, it is important to focus on the engineering disasters caused by rain water for example.

The matrix exponential spatial specification (MESS) is an alternative to the spatial autoregressive-type (SAR-type) specifications with several attractive properties. The spatial dependence in the MESS-type models is formulated through a matrix exponential term, and the estimation of these models may require the computation of the matrix exponential terms many times in an estimation procedure. In the literature, it is well documented that the computation of the matrix exponential terms can pose challenges in terms of reliability, stability, accuracy, and efficiency. We propose a matrix-vector products approach based on the truncation of Taylor series expansion of the matrix exponential terms for the fast estimation of MESStype models. We show how to efficiently implement this approach for a first-order MESS model, and provide extensive simulation evidence for its computational advantage over the default method utilized by a popular statistical software.

This web appendix provides the following: (A) some useful lemmas that will be used in the proofs of theorems below, (B) proofs of Lemmas 2.1, 3.1 and 3.2 in the main paper, (C) proofs of Theorems 3.1-3.3 in the main paper, (D) estimation of submodels MESDPS(1,0,0), MESDPS(0,1,0), MESDPS(1,1,0), MESDPS(1,0,1) and MESDPS(0,1,1), and (E) some more comprehensive Monte Carlo simulation results.

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