PMSE performance of the Stein-rule and positive-part Stein-rule estimators in a regression model with or without proxy variables

Consider a linear regression model with some relevant regressors are unobservable. In such a situation, we estimate the model by using the proxy variables as regressors or by simply omitting the relevant regressors. In this paper, we derive the explicit formula of the predictive mean squared error (PMSE) of the Stein-rule (SR) estimator and the positive-part Stein-rule (PSR) estimator for the regression coefficients when the proxy variables are used. We examine the effect of using the proxy variables on the risk performances of the SR and PSR estimators. It is shown analytically that the PSR estimator dominates the SR estimator even when the proxy variables are used. Also, our numerical results show that using the proxy variables is preferable to omitting the relevant regressors.