Exponential Squared Loss-based Estimation And Application Of Instrumental Variable Models

In this paper,focusing on the linear regression models and partial linear regression models with endogenous covariates,the linear instrumental variable models and partial linear instrumental variable models are studied,and the exponential squared loss-based estimation for the two models is proposed.For the instrumental variable linear regression model,in order to overcome the deviation caused by endogenous variables,the instrumental variables are introduced to eliminate the endogeneity of covariates,and then the exponential squared loss-based estimator is obtained by constructing objective function.Under some regular conditions,the asymptotic properties of the proposed estimator are discussed.The simulation compares the advantages and disadvantages of four estimation methods: naive least square estimation,naive exponential squared loss-based estimation,least square estimation using instrumental variables and exponential squared loss-based estimation using instrumental variables.The simulation results show that the exponential squared loss-based estimation using instrumental variables proposed in this paper effectively eliminates the endogeneity of covariates,and it has good robustness.Further,a real data aiming to study the identical twins' economic returns to schooling is analyzed by the proposed method.For a partial linear instrumental variable model,the instrumental variables are first introduced,and then the non-parametric components in the model are parameterized by the B-spline basis function and then get eliminated by the projection technique.Afterwards,the exponential squared loss-based estimator is obtained for the parametric components by maximizing objective function.After that,constructing another objective function to obtain the estimator for the non-parametric components.Further,under some mild regulations,we establish some asymptotic properties of the resulting estimators.Also,the finite sample property is discussed by simulation and the simulation results show that the proposed method is non-biased and has outstanding robustness.At last,the method proposed is used to analyze the relationship between the economic level and the openness and market scale for the 31 provinces in China.