Variable selection of higher-order partially functional linear multiple varying coefficient spatial autoregressive model

In this article, we consider the variable selection problem in a higher-order functional linear partially varying coefficient model in which the explanatory variables include an infinite dimensional predictor procedure, treated as functional data, and a scalar variate. Our goal aims at proposing a variable selection method to simultaneously select both significant spatial lags of the response variable and explanatory variables in the non parametric component and estimate the corresponding non zero non parameters and functional components. Our proposed method is based on the analysis of B-spline basis, function principal component analysis, two-stage least squares method, and a group SCAD optimization algorithm or a group MCP optimization algorithm. Under appropriate conditions and appropriate tuning parameters, we establish the rate of convergence of the penalized estimator of the spatial lags of the response variable, also the uniform rate of convergence of the series estimator of the non parametric and functional components, and demonstrate that the proposed variable selection method enjoys the oracle property. That is, our proposed method can consistently remove the noise terms with probability tending to one and estimate the non zero components as efficiently as if the true model was known in advance. The performance of the proposed estimation procedure is examined in Monte Carlo simulation studies.