Variable selection of higher-order partially linear spatial autoregressive model with a diverging number of parameters

In this paper, we consider problem of variable selection in higher-order partially linear spatial autoregressive model with a diverging number of parameters. By combining series approximation method, two-stage least squares method and a class of non-convex penalty function, we propose a variable selection method to simultaneously select both significant spatial lags of the response variable and explanatory variables in the parametric component and estimate the corresponding nonzero parameters. Unlike existing variable selection methods for spatial autoregressive models, the proposed variable selection method can simultaneously select significant explanatory variables and spatial lags of the response variable. Under appropriate conditions, we establish rate of convergence of the penalized estimator of the parameter vector in the parametric component and uniform rate of convergence of the series estimator of the nonparametric component, and show that the proposed variable selection method enjoys the oracle property. That is, it can estimate the zero parameters as exact zero with probability approaching one, and estimate the nonzero parameters as efficiently as if the true model was known in advance. Simulation studies show that the proposed variable selection method is of satisfactory finite sample properties. Especially, when the sample size is moderate, the proposed variable selection method even works well in the case where the correlation among the explanatory variables in the parametric component is strong. An application of the proposed variable selection method to the Boston house price data serves as a practical illustration.