Statistical Inference Of Semi-Functional Spatial Autoregressive Model Based On Local Linear Regression Method

Partially linear spatial autoregressive models have more advantages in explaining spatial effects than traditional spatial autoregressive(SAR)models.Therefore,these models are increasingly becoming an important part of the theoretical framework of spatial econometric models.In recent years,SAR models have been increasingly applied to fileds with spatial effects,in which some observations are functional data.In order to explain clearly the influence of functional variables in SAR models,this paper firstly introduces one functional variable into partially linear SAR models,which named semi-functional partially linear spatial autoregressive model(SFPLSAR),and the functional variable is explained in nonparametric form.The estimation methods of partially linear SAR models are mature,and the usual estimation method is a multi-step estimation method.Therefore,this paper based on quasi-maximum likelihood estimation(QMLE)method and local linear regression method,construct a two-step estimation method to estimate the nonparametric and parametric part.The QMLE method is a usual parameter estimation method in spatial econometric model estimation.It needs to make a normal assumption about the distribution of the data,involving the estimation of variance.However,the generalized moment estimation method(GMM)is another commonly used estimation method in the spatial econometric model.This method solves the problem of QMLE and does not require assumptions about the distribution.Since the determined spatial autoregressive parameters can not accurately represent the spatial autocorrelation relationship,this paper constructs a semi-functional partially linear varying coefficient spatial autoregressive(SFPLVCSAR)model with varying spatial autoregressive parameters.The three-step estimation method based on non-parametric generalized moment estimation method(NPGMM)is used to estimate the varying coefficient and non-parametric part of the model.For the above two parts,this paper gives the asymptotic behaviours of parameters and nonparametric part,and finally carries out numerical simulation to verify the theoretical results.The main structure of the paper is as follows:In chapter 1,we first give a brief introduction of the research background and significance of this article.Secondly,according to the domestic and foreign related literatures of SAR model,functional data and their parameters and non-parametric estimation methods are systematically introduced.Finally,we describe the main contents,difficulties and new points of the paper.In chapter 2,a systematic introduction to the methods used in this paper is given.Firstly,the local linear regression method is introduced.Secondly,the SAR models and the commonly used QMLE method and GMM method are briefly introduced.In chapter 3,based on the research of partially linear SAR model and QMLE method,firstly,the SFPLSAR model constructed in this chapter is introduced.Secondly,the local linear regression estimation method is used to estimate the unknown functional in the SFPLSAR model.Then,based on the preliminary estimation results of the functional,the QMLE method is used to estimate the parameter part,so as to obtain a more accurate estimation result of the nonparametric part;The asymptotic behaviours of the parameters and the nonparametric part and the proofs is given.Finally,the feasibility of the two-step estimation method based on QMLE method for SFPLSAR model estimation are proved by numerical simulation.The model and method of this chapter are applied to the study of Canadian meteorological data to analyze the spatial autocorrelation of rainfall.In chapter 4,based on the research of varying corfficient SAR model and NPGMM,we firstly introduce SFPLVCSAR model.Secondly,as with the first step of the two-step estimation method based on the QMLE method,a preliminary estimate of the non-parameter is first obtained based on the local linear regression estimation method;then the varying coefficient is estimated using the NPGMM method.And prove the asymptotic distribution of the varying coefficient and the convergence speed of the non-parametric part.Finally,numerical simulation is used to verify that the three-step estimation method based on NPGMM is applicable to the statistical inference of the SFPLVCSAR model.In chapter 5,we give a summary of the whole paper,and outline a future research plan.