GMM estimation and variable selection of semiparametric model with increasing dimension and high-order spatial dependence

To address various forms of spatial dependence and the heterogeneous effects of the impacts of some regressors, this paper concentrates on the generalized method of moments (GMM) estimation and variable selection of higher-order spatial autoregressive (SAR) model with semi-varying coefficients and diverging number of parameters. With the varying coefficient functions being approximated by basis functions, the GMM estimation procedure is firstly proposed and then, a novel and convenient smooth-threshold GMM procedure is constructed for variable selection based on the smooth-threshold estimating equations. Under some regularity conditions, the asymptotic properties of the proposed estimation and variable selection methods are established. In particular, the asymptotic normality of the parametric estimator is derived via a novel way based on some fundamental operations on block matrix. Compared to the existing estimation methods of semiparametric SAR models, our proposed series-based GMM procedure can simultaneously enjoy the merits of lower computing cost, higher estimation accuracy or higher applicability, especially in the case of heteroscedasticity. Extensive numerical simulations are conducted to confirm the theories and to demonstrate the advantages of the proposed method, in finite sample performance. Two real data analysis are further followed for application.