An extension of Kelejian's J-test for non-nested spatial models

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Comparing Implementations of Estimation Methods for Spatial Econometrics (2015, methodology): “1. Notation Here we adopt the notation ρLag for the spatial autoregressive parameter on the spatially lagged dependent variable y, and ρErr for the spatial autoregressive parameter on the spatially lagged residuals. In Ord (1975), ρ is used for both parameters, but subsequently two schools have deve”
Estimation and Selection of Spatial Weight Matrix in a Spatial Lag Model (2019, background): “Kelejian and Piras (2011) and Kelejian and Piras (2016) proposed J-tests to find from among a number of alternatives with differently specified spatial weight matrices.”
Functional-coefficient spatial autoregressive models with nonparametric spatial weights (2016, background): “Recently, Kelejian (2008) and Kelejian and Piras (2011) have proposed to use J-tests to select the proper interactive structure from several candidate interactive structures chosen by researchers.”
Spatial weights matrix selection and model averaging for spatial autoregressive models (2017, methodology): “We first investigate the performance of the J-test in Kelejian and Piras (2011).”