Bayesian Instrumental Variable Estimation Of Linear Measurement Error Model

Regression model with measurement error in predictive variables is one of the hot issues in statistics,which makes up for the deficiency of traditional regression model considering the predictive variables and has greater practical significance.The commonly used parameter estimation methods of this model are based on instrumental variables,using two-stage least squares estimation and other methods.In this paper,we study the problem of parameter estimation for measurement error model by combining Bayes method and instrumental variable.We calculate the posterior distribution of the parameter for different priors with known and unknown variance parameter respectively,and obtain the Bayes estimator(BE)of the parameters under the square loss.However,complex multiple integrals are involved in the calculation of BE,and it is difficult to obtain the explicit expression of estimator.Therefore,we use the linear Bayes method to obtain the expression of the linear Bayes estimator(LBE)for different priors and prove that the LBE is superior to the two-stage least squares estimator(TSLS)under the mean square error matrix criterion(MSEM).The numerical results show that the LBE is close to the BE,and both of them are better than the TSLS.Meanwhile,the LBE is very close to the real parameter value regardless of variance parameter is known or not.And with the increase of the sample size,the LBE gradually approaches to the BE.Finally,it is verified numerically that the LBE with instrumental variables is closer to the real parameter value than that without instrumental variables.