Statistical Inference Of Partially Linear Varying Coefficient Spatial Autoregressive Models

Spatial data generally have spatial autocorrelation.If classical regression model is used to fit the data with spatial dependent structure,the resulting estimators will not be consistent.Spatial autoregressive models are regarded as the most powerful and useful tool to deal with the spatial dependent data.It is well known that the traditional spatial autoregressive model has the assumption that the dependent variable is effected by the predictors via a linear mode.When the model is misspecification,in other words,when there is a nonlinear relationship,the statistical inference based on simple linear model would give misleading results.Therefore,it is urgent to study the semi-parametric regression model with spatial structure.This paper studies partial linear varying coefficient spatial autoregressive models(PLVCSARM),which has the good interpretability of the linear component and flexibility of the nonparametric part,including classical linear regression model,nonparametric regression models and spatial autoregressive models as a special case.In addition,PLVCSARM has wide adaptability and can better solve social and economic problems.Therefore,statistical inference for PLVCSARM is extremely important for theoretical or empirical studies.The research of this model has important practical significance.To the best of our knowledge,most existing estimation procedures are limited to parametric spatial autoregressive models or partially linear spatial autoregressive models.To end this gap,in this paper,we develop statistical inference for PLVCSARM.We first consider approximating nonparametric components by polynomial spline method,and then use profile quasi-maximum likelihood estimation method to obtain the estimation of model parameters.The proposed estimation method is simple and easy to implement with existing statistical soft,and have some mild assumptions on distribution of errors.Under some mild condition,we investigate the asymptotic properties of the proposed estimation methods.It is found that the convergence rate of finite dimensional parameter vectors depends on some characteristics of the spatial weight matrix in the model.Further,in large samples,when the reciprocal of spatial weighting matrix order is uniformly bounded,the convergence rate of spatial parameter estimators is parametric convergence rate.When the reciprocal of spatial weighting matrix order tends to be infinite,the convergence rate of some spatial parameter estimators is slower.In addition,based on certain conditions,the suitable convergence rate of nonparametric components is given.Finally,the finite sample properties of the proposed estimation method are investigated via Monte Carlo simulation studies,and the developed methodology is illustrated by an analysis of the Boston housing price data.