| Spatial econometric model is one of the widely used econometric models,and it is also an important branch of econometrics.Its basic content is based on the spatial effect of variables in regional scientific model.Generally,the traditional econometric model is mainly linear model,and the model usually makes assumptions in advance.In practical application,there are not only simple linear relations,but also nonlinear relations.The single use of parametric model can not fully explain complex economic problems and economic phenomena,which has also become one of the main limitations of spatial econometric model.Therefore,in order to overcome the above problems,this paper introduces the spatial measurement single index variable coefficient model.The single index variable coefficient model of spatial measurement is a kind of semi parametric model with strong interpretability.And this kind of model can not only effectively avoid the problem of "Curse of dimension",but also better deal with the problem of spatial correlation,and fully reflect the interaction between variables and reflect the spatial characteristics.This paper constructs a single index variable coefficient model of spatial measurement with the function of dimensionality reduction-spatial lag single index variable coefficient model and spatial error single index variable coefficient model;At the same time,in order to solve the possible spatial correlation and spatial heterogeneity between cross-sectional data units,this paper combines the advantages of spatial econometric model and single index variable coefficient model,adopts the idea of geographic weighted regression model,and takes spatial location as an important factor to reflect the spatial heterogeneity of data,The spatial position of the observation object is introduced into the single index variable coefficient model to deal with this problem.For the above constructed model,its estimation method,large sample nature and small sample performance are systematically studied and analyzed,and the model is applied to the problem of environmental pollution.Specifically,the research contents and results of this paper can be summarized as follows:First,the estimation of single index model and partially linear single index model.In this paper,a local penalty spline estimation method based on coefficient of variation is constructed.The radial basis spline function is used to construct the decreasing function.Through spline approximation or local quadratic approximation method and Levenberg Marquardt algorithm.the parameter estimates of local penalty spline estimation based on ridge regression and lasso are obtained,Finally,the correctness of the method is verified by Monte Carlo simulation,and three parameter estimation cases of non penalty spline estimation,unif-orm penalty spline estimation and local penalty spline estimation are discussed.Second,the study of spatial lag/error single index variable coefficient.In this paper,a new spatial lag/error single index variable coefficient model is proposed.The model setting form is divided into parametric part and nonparametric part,which reflects the flexibility of the model.On this basis,the estimation method of the model is constructed by combining spline estimation method and maximum likelihood estimation method,the parameter estimates are obtained,and the consistency and asymptotic normality of the estimators are proved.Finally,the performance of the estimation method under limited samples is verified by Monte Carlo simulation,and the total number of different spatial objects,different spatial correlation coefficients Parameter estimation and graph fitting under different error term variance and different spatial weight matrix.Thirdly,the local penalty spline estimation of spatial lag/error single index variable coefficient model is studied.Based on the above research contents,for the spatial lag/error single index variable coefficient model,the local penalty spline estimation based on ridge regression and the local penalty spline estimation based on lasso are used to estimate the parameters in the model.Firstly,the model is estimated and reasoned;Secondly,the effectiveness and correctness of the estimation method are verified by statistical simulation.The parameter estimation of uniform penalty spline estimation and local penalty spline estimation under different error term variance and different spatial weight are analyzed.Finally,when the error term variance is small and the data is dense,the uniform penalty spline estimation is better than the local penalty spline estimation,but the local penalty spline still has excellent estimation results and fitting effect;On the contrary,when the error term variance is large and the data is scattered,the local penalty spline estimation is better than the uniform penalty spline estimation,which shows that the local penalty spline estimation has good stability.Fourth,in the empirical application,the theoretical methods are applied to the research on the spatial effects of environmental pollution and industrial structure.Through the selection of variables and the test of data,the multiple linear regression model,spatial lag model and spatial lag single index variable coefficient model are selected to analyze the environmental pollution problems in China.The final results show that foreign investment Industrial profit and industrial structure promote industrial pollution,while the level of science and technology can inhibit industrial pollution.It provides an effective empirical basis for China’s industrial development and environmental problems.This paper combines the characteristics of theoretical methods and practical applications.Theoretical methods are the improvement of model construction and estimation methods.Combined with the form of spatial econometric model and single index model,it fully solves the spatial correlation and spatial heterogeneity.It has certain value in both theoretical research and practical application.Estimation methods also have application value in the fields of statistics,economics and so on. |
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