Empirical Likelihood For Spatial Autoregressive Models With Spatial Autoregressive Disturbances

The data appear in all aspects of life in the form of time and space,such as the number of asymptomatic infections,housing prices,and species in a certain place at a certain time.The research on spatial panel data has developed into spatial Econometrics,and has been widely used in many fields such as epidemiology,regional economics,and biological science.The spatial crosssectional data model has been at the core of spatial Econometrics,and its estimation methods can be extended to the spatial panel data model.Therefore,it is of great practical value and theoretical significance to study its statistical inference.Since Owen proposed the empirical likelihood method,it has attracted the attention of many scholars with its effectiveness of parametric method and the stability of the nonparametric method.The confidence interval constructed by empirical likelihood method has the advantages of domain retention,transformation invariance,and the shape of the confidence domain is determined by the data itself,which makes the method widely applied in various statistical models.However,the estimation equation obtained by the quasi likelihood method for the spatial cross-sectional data model is a linear-quadratic form of error,which poses a great challenge to the application of the empirical likelihood method for spatial models.To address this problem,Qin introduces a martingale sequence to transform the linearquadratic form into a linear form to solve the problem,according to the characteristics of the estimation equation obtained by the quasi likelihood method,and this approach of transformation also opens a way to use empirical likelihood methods to other spatial models.However,the empirical likelihood method has many limitations in its own theory,and it often has other trouble in several practical applications,such as the fact that the empirical likelihood ratio function may have no solution when the data size is small,and the traditional empirical likelihood fails when the data dimension is diverging.Therefore,innovative empirical likelihood methods of the spatial section data model studied in this paper have certain theoretical value and expands the statistical inference method of the spatial data model.The main research work of this paper can be summarized as follows:1.When sample size is small,the usual profile empirical likelihood ratio statistic may have no numerical solution,as the convex hull of the sample points may not contain the zero vector 0as its interior point.The adjusted empirical likelihood(AEL)method is very useful to solve this problem.This paper studies the adjustment empirical likelihood inference for the parameters in spatial autoregressive models with spatial autoregressive disturbances(SARSAR models).Firstly,construct the adjusted empirical likelihood ratio statistics,and prove that the limit distribution of the statistics is chi-square distribution,which has the same degree of freedom as the asymptotic distribution of the unadjusted empirical likelihood ratio statistics.Secondly,the theoretical distribution is used to construct the confidence region of the model parameters,and numerical experiments are designed to compare with the unadjusted empirical likelihood method.The simulation shows that when the sample size is small,the coverage probabilities of the confidence region by our recommended method is closer to the confidence level than the unadjusted method,and the calculation speed is faster,without the complex programs such as Bartlett correction and Bootstrap method.2.When the sample dimension is large,the traditional empirical likelihood ratio statistic becomes invalid,as the larger sample theory is not applicable in such case.The innovative empirical likelihood method performs well to solve this problem of dimension explosion.This paper studies the new block empirical likelihood inference for the parameters in SARSAR model.Firstly,construct the new block empirical likelihood ratio statistics,and prove that the limit distribution of the new statistics is chi-square distribution as well as the traditional methods.The degree of freedom of the new distribution is independent of the data dimension and is much less than the degree of freedom under the traditional method.Secondly,the theoretical distribution is used to construct the confidence region of the model parameters,and numerical experiments are designed to compare with the traditional empirical likelihood method.The simulation shows that when the dimension is divergent,the coverage probabilities of the confidence region by our new method is very closer to the nominal confidence level than the traditional method,and the calculation speed is much faster,without presupposition that the model is sparse or that the dimension must be less than the sample size.