The profiled variable selection for high-dimensional spatial auto-regressive model

The spatial auto-regressive (SAR) model serves as a foundational tool for investigating spatial dependencies across various scientific fields. In this paper, we proposed a sequential variable selection approach named the profiled variable selection (PVS) procedure for the SAR model. The PVS procedure was designed to efficiently handle high-dimensional cases while maintaining high scalability, even as the number of potential covariates $p$ grows exponentially with the sample size $n$. However, existing penalization methods are inadequate in variable selection for the SAR model when $p>n$, due to the necessity for a consistent initial estimator. The selection consistency of the PVS procedure was established under mild conditions. Numerical simulations demonstrated the promising performance of the PVS procedure in variable selection for SAR models. Additionally, a real data analysis on housing prices across Chinese cities illustrated the practical utility of our method.