Inference for partially linear additive higher-order spatial autoregressive model with spatial autoregressive error and unknown heteroskedasticity

This article extends spatial autoregressive model with spatial autoregressive disturbances (SARAR(1,1)) which is the most popular spatial econometric model to the case of an arbitrary finite number of nonparametric additive terms and spatial autoregressive models with spatial autoregressive disturbances of arbitrary finite order (SARAR(R,S)). We propose a sieve two-stage least squares (S2SLS) regression and generalized method of moments (GMM) procedure of the high-order spatial autoregressive parameters of the disturbance process. Under some sufficient conditions, we show that the proposed estimator for the finite dimensional parameter is n consistent and asymptotically normally distributed. We show that each proposed estimator for the additive terms is consistent and also asymptotically normally distributed at a rate slower than n. Consistent estimators for the asymptotic variances of the proposed estimators are provided. In addition, using asymptotic properties to make statistical inference for the parametric and additive is also considered. Monte Carlo evidence suggests that the estimation procedure performs reasonably well in small samples and the proposed approach has some practical value. The proposed method is applied to analyzing factors which affect haze pollution.