Estimation For Several Classes Of Functional Spatial Lag Quantile Regression Models

The classical spatial econometric model mainly studies the quantitative relationship between spatial data based on full consideration of spatial correlation.Traditional spatial data is usually presented in the form of vector or matrix by discrete observations in a certain order.With the rapid development of data collection and storage technology,a continuous type of data began to appear in spatial data sets.This kind of data is obtained through high-frequency observation.Because the observation interval is too dense,these observations can be regarded as a continuous change process,and then show a curvilinear function feature.Academic circles call this data functional data.In fact,functional data is the discrete realization of functions in the sample interval.The function data has the feature of infinite dimension,which makes it contain more data information than the traditional discrete data,and this feature of infinite dimension makes the traditional statistical analysis tools built under the finite framework no longer applicable.Therefore,the emergence of functional data brings both opportunities and challenges to the statistical analysis of spatial data.Through combing the existing literature,we find that there are three problems in the existing research: first,the research on the integration of functional data and spatial econometric model is still in its infancy,and the research results in relevant fields are few;Second,most of the existing studies are based on the estimation of mean regression,and mean regression is too sensitive to the non-normal distribution data such as peak-thick tail,outlier,heteroscedasticity,which makes the estimation results not robust;Third,the models proposed by the existing research usually set that there is only linear relationship between variables.The structure of the model is too simple and idealized,and cannot well describe the complex relationship structure such as the interaction effect or nonlinear relationship between variables.In order to solve the above problems,for different types of spatial data samples including functional data,this thesis proposes several types of functional spatial lag quantile regression models,taking full account of various possible relationship structures between data.The specific research contents are as follows:(1)The estimation of functional linear spatial lag quantile regression model are studied.First,in the context of spatial correlation of data,functional data is introduced into the research scope of spatial econometrics.For the sake of robustness,this thesis takes quantile regression as a robust alternative to mean regression,and uses scalar data as response variables and functional data as covariates to establish a functional linear spatial lag quantile regression model.Secondly,the functional principal component analysis method is used to reduce the dimension of the functional covariates and the corresponding slope function,and the infinite dimensional data is converted into a finite dimensional matrix or vector.Then,the quantile regression estimation method of tool variables is designed for the model to solve the endogenous problem caused by spatial correlation and overcome the defect that mean regression is sensitive to error distribution and outliers.Thirdly,under the given regular conditions,the asymptotic normality of the parameter estimator and the optimal convergence rate of the function estimator are proved.Finally,numerical simulation and example application analysis show that the method proposed in this chapter can handle functional data and non-normal distribution data well.(2)The estimation of partial functional linear spatial lag quantile regression model are studied.First,in the context of mixed data composed of functional data and scalar data,not only the influence of functional covariates on response variables should be considered,but also the linear relationship between multiple scalar covariates and response variables should be fully considered.And for the need of robustness,this thesis selects quantile regression instead of mean regression,and builds partial functional linear space lag quantile regression model.Secondly,functional principal component analysis is used to reduce the dimensions of functional covariates and corresponding slope functions.At the same time,the quantile regression estimation method of instrumental variables is constructed to estimate the model.Thirdly,under certain regular conditions,the process of proving that the model parameter estimator obeys normal distribution and the slope function estimator achieves the best convergence rate is given.Fourth,through numerical simulation to compare the estimation effect of the method proposed in this chapter with the mean regression method.The simulation results show that the proposed method is superior to the mean regression method when dealing with data with different error distributions and outliers.In particular,when the variance of the random error distribution is infinite,the mean regression method fails and the method proposed in this chapter is still robust.Finally,the model is applied to an example of industrial carbon productivity impact factor analysis.The empirical results show that the method proposed in this chapter can well deal with mixed data with functional data and scalar data and data with non-normal distribution.In addition to the influence of functional data,the model also accurately depicts the linear relationship between other scalar covariates and response variables.(3)This thesis studies the estimation of the variable coefficient partial function quantile regression model with spatial lag.First of all,in the background of mixed data composed of functional data and scalar data,not only the influence of functional covariates on response variables,but also the influence of possible interaction between scalar covariates on response variables are considered.Based on the idea of robustness and the dimensionality reduction effect of "variable coefficient model",this thesis takes scalar data as the response variable,functional data and scalar data as two different covariates to establish a variable coefficient partial functional spatial lag quantile regression model.Secondly,functional principal component analysis is used to decompose functional covariates and corresponding slope functions to achieve the purpose of dimensionality reduction.At the same time,the B-spline is used to approximate the variable coefficient function,which is expressed as the product of the spline basis matrix and the coefficient vector to be estimated to realize the linearization of the nonparametric smooth function.Then,the quantile regression method of instrumental variables is constructed to solve the estimation problem of the model and the endogenous impact of spatial correlation.Thirdly,under the previously set regular conditions,the consistency and asymptotic normal distribution of parameter estimators are given,and the function estimators obtain the optimal convergence rate in the sense of nonparametric estimation minimax(minmax),and the corresponding demonstration process is provided.Fourth,the advantages and disadvantages of the proposed method and the mean regression method are compared by numerical simulation.The results of numerical simulation show that the estimation results of mean regression method vary greatly among different error distributions,and the simulation failure occurs when the error distribution is Cauchy distribution with large variance.The proposed method shows strong stability when dealing with different error distributions and outlier data.Finally,the proposed model and method are applied to an example of influencing factors analysis of industrial carbon productivity.The results of example analysis show that the model proposed in this chapter can well deal with data with functional data and non-normal distribution.The results show that the interaction effect between scalar covariates on response variables can be effectively captured,and the flexibility and interpretability of the model have been greatly improved.(4)In this thesis,we study the estimation of partial functional linear additive spatial lag quantile regression model.First of all,in the background of mixed,on the one hand,consider the influence of functional covariates on response variables,on the other hand,consider the multiple nonlinear relationships between scalar covariates and response variables.And based on the robustness of the estimation and the dimensionality reduction function of the "additive model",this thesis uses scalar data as the response variable,functional data and scalar data as two different covariates to establish a partial functional linear additive spatial lag quantile regression model.Secondly,by centralizing the B-spline basis function,the constrained quantile regression objective function optimization problem is transformed into an unconstrained optimization problem,solving the identification problem of additive models.At the same time,functional principal component analysis is used to process functional covariates and corresponding slope functions,transforming them into a product of a given matrix and the coefficient vector to be evaluated.Then,under the condition that the spatial effect violates the hypothesis of sample independence,the quantile regression method of instrumental variables is constructed to estimate the model to avoid biased estimators.Thirdly,according to some assumptions,the asymptotic normal distribution of the parameter estimator and the optimal convergence rate of the function estimator are derived and demonstrated in detail.Fourth,the effectiveness of the proposed method is verified by numerical simulation.The results of numerical simulation show that the proposed method has better stability when dealing with different random error distribution and outlier data than the mean regression method.Finally,the proposed models and methods are applied to the functional data of the level of openness,public participation in environmental regulation and the impact of urbanization on industrial carbon productivity.The analysis results show that the models and methods proposed in this chapter can well deal with data with functional and non-normal distribution,and can accurately describe multiple nonlinear relationships between scalar covariates and response variables.The practicability and interpretability of the model have been greatly improved.From the above research content,we can conclude that the innovation of this article lies in:(1)The thesis establishes two parameter functional spatial models that can not only describe the impact of functional data but also interpret spatial correlation.The thesis breaks the traditional practice of spatial statistical analysis that only studies data from a vector perspective,ignoring the continuity and complex structure between data.The model estimation method designed in this thesis not only solves the problem of dimensionality reduction and model endogeneity of functional data,but also overcomes the shortcomings of mean value regression that is sensitive to outliers and error distributions,ensuring the robustness of the estimation results.(2)This thesis constructs a variable coefficient model that can not only reflect the impact and spatial effects of functional data,but also describe the impact of interaction effects between scalar data.The thesis has changed the modeling method in which there is only a linear relationship between the preference setting variables of parametric models.This thesis presents a model estimation method that takes into account both processing functional data and estimating nonparametric functions.The assumption that random errors must obey a normal distribution is relaxed,which expands the applicability of the model and greatly improves the robustness of the model.(3)The thesis establishes an additive model that can not only reflect the impact and spatial dependence brought by functional data,but also capture multiple nonlinear effects brought by scalar data.The thesis solves the drawback of parametric models that mistakenly assume the nonlinear relationship between variables as a linear relationship.The model estimation method proposed in this thesis can not only process functional data but also meet the recognition conditions of additive models.The thesis transforms the constrained objective function optimization problem into an unconstrained optimization problem,greatly simplifying the estimation process of the model.The models and methods proposed in this thesis can effectively handle data sets with spatial correlation,functional characteristics,spikes,thick tails,outliers,skewed distributions,and other large data characteristics,as well as accurately describe complex relational structures such as linear and nonlinear relationships between data.Text research provides a set of statistical analysis tools with concise structure,effective estimation,and accurate prediction for modeling and analyzing spatial data in the big data era.