In this paper, we focus on the variable selection techniques for a class of semiparametric spatial regression models which allow one to study the effects of explanatory variables in the presence of the spatial information. The spatial smoothing problem in the nonparametric part is tackled by means of bivariate splines over triangulation, which is able to deal efficiently with data distributed over irregularly shaped regions. In addition, we develop a unified procedure for variable selection to identify significant covariates under a double penalization framework, and we show that the penalized estimators enjoy the oracle property. The proposed method can simultaneously identify non-zero spatially distributed covariates and solve the problem of leakage across complex domains of the functional spatial component. To estimate the standard deviations of the proposed estimators for the coefficients, a sandwich formula is developed as well. In the end, Monte Carlo simulation examples and a real data example are provided to illustrate the proposed methodology. All technical proofs are given in the supplementary materials.
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