A geographically neural network weighted regression with spatial autoregressive model: model design, estimation, and variable selection

Geographically weighted regression model with a spatially autoregressive term of the response variable (abbreviated to GWR-SAR model) is a powerful tool to tackle spatial nonstationarity in spatial autocorrelation and regression relationships. Generally, its estimation uses the weighted least squares. However, these weights are usually determined by the predetermined function in simple forms, which may compromise estimation accuracy for complex spatial relationships. The variable selection is important for the GWR-SAR model to decide whether the SAR term should be included to account for spatial autocorrelation and whether some of its coefficients should be identified as zero for constructing a more appropriate model. Taking these two issues into consideration, we design a data-driven model–the geographically neural network weighted regression with spatial autoregressive (GNNWR-SAR) model. Our model combines the spatial two-stage least squares and neural networks to adaptively capture complex spatial relationships for better estimation, and uses the adaptive Lasso to select variables for model specification. A simulation study demonstrates that our proposed GNNWR-SAR model outperforms the GWR-SAR model in fit, estimation, identifying the pre-designed zero coefficients for the data generated with multicollinearity, and handling spatial autocorrelation. The Boston housing price analysis justifies the applicability of our proposed GNNWR-SAR model.