Bootstrap Version of Rao–Blackwellization to Two-Step and Instrumental Variable Estimators

We focus on Arnold and Katti’s application of the Rao–Blackwell theorem to improve preliminary test estimators. The improvement uniformly reduces the mean squared error (MSE) by replacing the pre-test estimator with a suitable conditional expectation. This paper suggests a novel use of the bootstrap to compute the relevant conditional expectation numerically. We illustrate with two econometric estimators. First, a pre-test estimator of the scale elasticity (SCE) for US production of metals. Second, instrumental variables (IV) estimator of the marginal propensity to consume (MPC) of the Haavelmo model. We use relatively large simulation experiments to show MSE reductions in Rao–Blackwellized pre-test and IV estimators. These illustrations show that one can use our bootstrap version to improve these 2-step estimators, and perhaps others.