Applications of James-Stein Shrinkage (I): Variance Reduction without Bias

In a linear regression model with homoscedastic Normal noise, I consider James-Stein type shrinkage in the estimation of nuisance parameters associated with control variables. For at least three control variables and exogenous treatment, I show that the standard least-squares estimator is dominated with respect to squared-error loss in the treatment effect even among unbiased estimators and even when the target parameter is low-dimensional. I construct the dominating estimator by a variant of James-Stein shrinkage in an appropriate high-dimensional Normal-means problem; it can be understood as an invariant generalized Bayes estimator with an uninformative (improper) Jeffreys prior in the target parameter.