Exponential Squared Loss Based Estimation For Partially Linear Panel Data Model With Fixed Effects And Application

Panel data is a form of data presented in two dimensions,point-in-time and individual,and has outstanding advantages over cross-sectional data and time series data.The partially linear panel data model has the advantages of panel data and retains the advantages of parametric and non-parametric models,avoiding the problems of "restrictive" of parametric models and "dimensional disaster" of non-parametric models.It has been widely used in the field of statistics and economics.However,in the study of practical problems,panel data models usually assume that the model errors obey normal distribution,which is difficult to be satisfied by the actual data,and most of the estimations obtained by using traditional methods are not robust.Therefore,this paper introduces an exponential squared loss estimation method to achieve robust estimation of partially linear panel data model with fixed effects for parametric and nonparametric components even when there are outliers such as spikes,thick tails,and outliers in the data.In this paper,a robust estimation method is proposed for a partially linear panel data model with fixed effects.The method is first based on auxiliary linear regression to eliminate fixed effects,and then uses a B-spline function to approximate the unknown nonparametric part and combines the projection matrix with an exponential squared loss function to obtain exponential squared loss estimations of the parameters and non-parameters.The asymptotic properties of the parametric estimations and the convergence rate of the nonparametric estimations are demonstrated under some regularity conditions.Some simulation studies illustrate that the proposed method is more robust than the semiparametric least squares dummy variable estimator.The simulation results show that the estimation method proposed in this paper has good robustness.In terms of application,the model and estimation method proposed in this paper are used to analyze and forecast the carbon emission research.Using the panel data of Chinese provinces from 2014 to 2018 as the sample,10 explanatory variables with significant effects on carbon emissions are selected.Firstly,the principal component analysis method is used for dimensionality reduction,and then the obtained principal components are used to fit a partially linear panel data model with fixed effects.The exponential squared loss estimation method is used to estimate the parametric and nonparametric components of the model,followed by the prediction of carbon emissions of Chinese provinces in 2019.The results show that the proposed estimation method is more effective than the semiparametric least squares dummy variable estimator in predicting carbon emissions in 2019.