  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.
