Variable selection and estimation for high-dimensional spatial autoregressive models

Spatial regression models are important tools for many scientific disciplines including economics, business, and social science. In this article, we investigate postmodel selection estimators that apply least squares estimation to the model selected by penalized estimation in high-dimensional regression models with spatial autoregressive errors. We show that by separating the model selection and estimation process, the postmodel selection estimator performs at least as well as the simultaneous variable selection and estimation method in terms of the rate of convergence. Moreover, under perfect model selection, the ?₂ rate of convergence is the oracle rate of $\sqrt{s/n}$, compared with the convergence rate of $\text{?√?}\sqrt{s?\log ?p/n}$ in the general case. Here, n is the sample size and p, s are the model dimension and number of significant covariates, respectively. We further provide the convergence rate of the estimation error in the form of $\sup$ norm, and ideally the rate can reach as fast as $\text{?√?}\sqrt{\log ?s/n}$.