Bayesian-INLA Estimation Of Two Types Of Nonparametric Spatial Error Model

The current dominant paradigm in spatial econometrics is still a parametric one,These models use different linear assumptions to model the spatial autocor-relation and spatial spillover effects.But in reality,the relationship between the variables is complicated,in most case,it would present nonlinear characteristics.Only use the linear assumption is unable to explore the true relationship between variables.Therefore,introducing nonparametric models to considering potential nonlinearities is necessary.Since Lesage used Bayesian method to test the parameters of the spatial econo-metric model in 1997,the practice of treating the spatial relationship as a latent random effect with spatial structure is still the focus of current research.However,under the Bayesian framework,Bayesian inference generally obtain joint posterior distribution of the parameters and the statistical information such as the mean,variance,and confidence interval of the marginal distribution by using the Monte Carlo(MCMC)sampling.The biggest limitation of this kind of method is that when dealing with complex models,especially when the samples are highly cor-related,the model needs to be divided into blocks,hyperparameterized,or more complicated sampling methods are used,but even so,there are still problems with convergence speed and calculation time.In addition,although the Matlab MCMC tool can change the default a priori value,it still takes time to consider whether it is feasible for non-default settings.integrated nested laplace approximation method(INLA)proposed by Rue et al.In 2009 provides a fast and effective solution to the posterior marginal inference of the Bayesian layered model,Due to the Gaussian Markov of the latent Gaussian model,the random field has sparse precision matrix,which make integrated nested laplace approximation method can greatly reduce the calculation time under the premise of ensuring accuracy.Both bayesian estimation of nonparametric spatial econometric models and the estimation of spatial econometric models using the Laplace approximation method are rarely studied,Therefore,this paper combines nonparametric spline methods and Bayesian-INLA to estimate the nonparametric spatial error model.This has theoretical and practical significance for the expansion of spatial econometric models and the application of widened Bayesian estimation methods.The work done in this article has the following aspects:First,for the widely existing nonlinear relationships between spatial data vari-ables,this paper uses nonparametric spline methods to deal with nonparametric terms,and introduces one-dimension,two-dimension nonparametric terms into the spatial error model to effectively explore the true response relationship between spatial variables.Secondly,under the Bayesian framework,the INLA method is used instead of MCMC sampling to approximate bayesian posterior marginal distribution of the nonparametric spatial error model.The latent Gaussian Markov random field is con-structed to maintain the sparseness of the accuracy matrix to ensure the calculation speed of INLA.Finally,on the basis of theoretical research,the estimation method of this two types of modeles is applied in empirical analysis.