<p>The semiparametric spatial autoregressive (SAR) models have received more and more attention due to its flexibility, compared to the parametric ones. However, existing literatures on estimation and inference of semiparametric SAR models were built on some pre-specified model frameworks, which shall suffer the risk of model mis-specification, because rarely can the analysts have a priori knowledge of the relationship between response variable and covariates in practice. To this end, this paper develops a double-regularized procedure for model detection and variable selection of semiparametric additive SAR model, based on the generalized method of moments and spline approximation. Under some regularity conditions, we establish asymptotic properties of the resulting estimators, including the convergence rate of functional estimators, asymptotic normality of parametric estimators as well as consistency of model selection. An efficient algorithm is provided for computation and the selection of tuning parameters is discussed. Large amounts of numerical simulations are conducted to demonstrate the finite sample performance of the proposed method. An empirical dataset is analyzed for further application.</p>