Lagrange multiplier test for spatial autoregressive model with latent variables

The focus of this research is to develop a Lagrange multiplier (LM) test of spatial dependence for the spatial autoregressive model (SAR) with latent variables (LVs). It was arranged by the standard SAR, where the independent variables were replaced by factor scores of the exogenous latent variables from a measurement model (in structural equation modeling) as well as their dependent variables. As a result, an error distribution of the SAR-LVs should have a different distribution from the standard SAR. Therefore, this LM test for the SAR-LVs is based on the new distribution. The estimation of the latent variables used a weighted least squares (WLS) method. The estimation of the SAR-LVs parameter used a two-stage least squares (2SLS) method. The SAR-LVs model was applied to the model with a positive and negative spatial autoregressive coefficient to illustrate how it was interpreted.