Research On Nonparametric Estimation Method Of Covariance Matrix In Heteroscedastic Model

An important assumption in the linear regression model is that each random error term satisfies the same variance.However,in solving practical problems,especially in regression cases with cross-sectional data,it is difficult to guarantee the homogeneity of variance for the random error term,which is called heteroscedasticity.If the ordinary least squares method is still used to estimate the parameters,the accuracy of the model parameter estimation and test results will be reduced,and the fitting and prediction results of the model will also produce a certain deviation.When there is heteroscedasticity in the model,the generalized least squares(GLS)method is often used for parameter estimation.This method is suitable for the case where the covariance matrix of the random error term is known.In most cases,the error covariance matrix is unknown,so it needs to be estimated first.In the case of unknown heteroscedastic structure,nonparametric estimation method can be used as a better choice.Because the fixed window width selected by the non-parametric estimation method KNW is separated from the actual data,the estimation has a large error.In this regard,this paper combines the idea of adaptive N-W kernel regression estimation,introduces variable window width when selecting window width,and proposes AKNW method to improve the accuracy of error variance estimation.HCCMEs proposed by scholars at home and abroad can reduce the impact of heteroscedasticity on model testing and prediction.However,because HCCMEs estimators are calculated based on the residuals obtained by OLS,and the estimators generated by OLS under heteroscedasticity are no longer valid,the effectiveness of HCCMEs estimators is reduced.Because EWLS has better estimation effect than OLS,many scholars derived WHCCMEs from EWLS residuals based on the construction method of HCCMEs,and it is better in model test.In this paper,four nonparametric methods are combined with EWLS,and based on this,the existing WHCCMEs are extended.The work of this paper is as follows :(1)Aiming at the problem that the fixed window width selected in the KNW estimation method is not combined with the sample observation value and is separated from the actual data,which leads to a large estimation error,a new estimation-AKNW is proposed.Experiments show that this method not only inherits the advantages of kernel regression,but also improves the accuracy of estimation by using variable window width,and can better deal with the problem of variance estimation when the independent variable changes in a large range.(2)The EWLS estimates based on different nonparametric methods are compared and analyzed in terms of regression coefficient estimation and model prediction,and the existing WHCCMEs are extended.Two new weighted estimators are proposed,namely WHC4 m and WHC5 m,which are HC4 m and HC5 m with weighted terms in HCCMEs.Through numerical simulation and example analysis,it is proved that the new weighted estimator has better performance than HC4 m and HC5 m.