Robust variable selection with exponential squared loss for the spatial autoregressive model

Spatial dependent data frequently occur in spatial econometrics and endemiology. In this work, we propose a class of penalized robust regression estimators based on exponential squared loss with independent and identical distributed errors for general spatial autoregressive models. A penalized exponential squared loss with the adaptive lasso penalty is employed for simultaneous model selection and parameter estimation. Under mild conditions, we establish the asymptotic and oracle property of the proposed estimators The induced nonconvex nondifferentiable mathematical programming offer challenges for solving algorithms. We specially design a block coordinate descent (BCD) algorithm equipped with CCCP procedure for efficiently solving the subproblem. Moreover, we provide a convergence guarantee of the BCD algorithm. Every limit point of the iterated solutions is proved a stationary point. We also present a convergence speed of spatial weight ρk. Numerical studies illustrate that the proposed method is particularly robust and applicable when the outliers or intensive noise exist in the observations or the estimated spatial weight matrix is inaccurate. All the source code could be freely downloaded from https://github.com/Isaac-QiXing/SAR.