Lectures in Econometric Theory

5.1. Stein’s theorem and the regression model. It was pointed out in Chapter 2, section 2.2, that if no a priori knowledge is specified concerning β, the criterion of minimization of a matrix-valued meansquare error is ill posed. Nevertheless, there are some cases in which the choice of a particular scalar-valued definition of mean-square error makes it possible to obtain estimators with lower mean-square error than the Gauss-Markoff estimator, for all β. In such cases, the Gauss-Markoff estimator is “inadmissible” in the sense of Wald (1950). A situation of this kind was first discovered by Stein (1956). Let us now consider Stein’s formulation. In the model