GMM Estimation Of Semiparametric Spatial Regressive Models

In the study of real economic issues,a large number of economic variables usually have spatial correlation between adjacent regions(units),such as economic growth,industrial development,resource endowment,environmental pollution and so on,which often display regional clustering,etc..Moreover,they tend to have both linear and nonlinear relationships with explanatory variables.How to accurately describe the connection between them and grasp the inherent regularity by using spatial models is a research work with important theoretical significance and practical value.Semiparametric regression models are important in econometrics due to can effectively avoid misspecified models and more flexibility.However,multivariate semiparametric regression models often face "curse of dimensionality" when the di-mension of regressors is higher,that is,the estimation accuracy decreases rapidly as the dimension of the explanatory variable increases.To avoid the "curse of di-mensionality",statisticians further proposed various nonparametric/semiparametric regression models with "reduced-dimension property".Considering the possible co-existence of linear and nonlinear relationships,the nonparametric/semiparametric regression models are extended to partially linear nonparametric/semiparametric regression models.Among them,partially linear single-index models and partial-ly linear additive models are the two most concerned semiparametric models with"reduced-dimension property".For these two types of models,scholars have done a lot of research on their estimation methods and obtained plentiful and significant achievements.Based on the partially linear single-index models and partially linear additive models,this paper proposes partially linear single-index spatial regression models and partially linear additive spatial regression models that are suitable for studying spatial data,and attempts to construct estimation methods of the models,derive large sample properties and investigate the performance of estimators in finite sam-ple.The advantages of the proposed models are as follows:(1)They can avoid the risk of misspecified in parametric models.(2)They can overcome "curse of dimen-sionality" existing in nonparametric models.(3)They can investigate the linear and nonlinear effects of explanatory variables on response variables simultaneously.(4)They can observe the spatial effect of response variables in different regions(units).On the basis of systematic summary,review and analysis of the existing research literature,this paper has completed the following main research contents:(1)We present four kinds of spatial semiparametric models which include partially linear single-index spatial autoregression model,partially linear single-index model with autoregression disturbances,partially linear additive spatial autoregression model and partially linear additive model with autoregression disturbances.(2)The es-timators of unknown parameters and unknown functions are constructed according to the specific characteristics of each model.(3)The large sample properties of es-timators are proved by mathematical and statistical theoretical methods.(4)The Monte Carlo numerical simulation method is used to assess the small sample per-formance under different spatial correlation coefficients,different sample sizes and spatial weight matrices with different complexity.(5)The proposed methods are ap-plied to analyze the influence factors of Boston housing price and PM2.5 in China's provinces respectively.The main research results of this paper are summarized as follows:(1)Ac-cording to the characteristics of the four models,the GMM estimations of unknown parameters and unknown functions are obtained by combining the local linear esti-mation method and instrumental variables method.(2)The asymptotic normality of each estimator is proved under some regular assumptions.(3)The Monte Carlo simulation results show that:?The sample mean(MEAN)is closer to the true val-ue,sample standard deviation(SD)and mean square error(MSE)rapidly decrease as sample size increases.They show that the estimation effect of the parameter-s performs well;?The median(MEDIAN)and standard deviation(SD)of mean absolute deviation error of unknown function in partially linear single-index spatial regression model decrease as sample size increases.They indicate that the unknown function estimator performs well in finite samples;?The sample mean(MEAN)and sample standard deviation(SD)of root average square error of unknown function in partially linear additive spatial regression model decrease as sample size increases.They indicate that the unknown function estimator performs well in finite samples;?By selecting Case spatial weight matrix and Rook spatial weight matrix,it is found that the complexity of the spatial weight matrix has a great influence on the estimation effect of the spatial correlation coefficient,but a small influence on the estimation effect of other unknown parameters and unknown function.(4)According to the analysis of the influence factors of the housing price in Boston,we find that the RAD is positively correlated with the housing price,while the PTRATIO in the region is negatively correlated with the housing price.CRIM,NOX,RM,TAX and LSTAT have nonlinear effects on house prices.The analysis of influence factors of PM2.5 in China's provinces shows that:The PGDP,POP and ENS develop in the same direction as PM2.5.PM2.5 decreases with the increase of FOI?PCI?RAIN and FOC.Furthermore,these variables also have nonlinear influence on PM2.5.All results are basically consistent with the existing literature research results.The combination of theoretical argumentation,simulation investigation and real data analysis is the research feature of this paper.The research methods of this paper can be further extended to the study of similar models,and the estimation techniques can be applied to the study of realistic economic problems,which display theoretical significance and application value.