Specification test for high-dimensional partially linear varying coefficient spatial autoregressive model

This paper considers a specification test method for the partially linear varying coefficient spatial autoregressive models with high-dimensional covariates. By combining the one-dimensional linear projection of the covariates with the residual marked empirical process, we propose a projection-based test method. This method is not only suitable for high-dimensional covariates, but also effectively avoids the subjective selection of parameters (such as bandwidth). Under mild conditions, the consistency of the proposed method is established. Furthermore, we show that the proposed test method can distinguish Pitman local alternatives converging to the null at the usual parametric rate. To improve the finite sample properties of the proposed test, we develop a wild bootstrap method to obtain the critical values or p-values, and show the validity of the proposed bootstrap method. We conduct Monte Carlo studies to investigate the finite sample performance of our new procedure, and find that the proposed test methods produce satisfactory results. Finally, the proposed test methods are illustrated by two real data examples.