On the Robustness of Shrinkage Predictors in Regression to Differences between Past and Future Data

SUMMARY Copas (1983) investigates the shrinkage associated with prediction in the usual linear multiple regression model and proposes using Stein-type predictors which anticipate this shrinkage. Assuming that the first two (population) moments of the regressor variables associated with individuals for which predictions are to be made coincide with the first two (sample) moments of those in the construction sample, preshrunk predictors give a uniformly lower prediction mean squared error than least squares. In this paper, we investigate the robustness of these shrinkage predictors to departures from this distributional assumption. It is argued that minimaxity results are not sufficient on their own and that the applicability of shrinkage prediction should be assessed in the light of expressions involving the true regression parameters. Numerical illustrations suggest that Copas's (1983) shrinkage predictor is not only robust to small differences between the appropriate moment matrices, but can also be applicable in the case of large and systematic differences between the construction sample and future individuals.