Title: Unbiased Shrinkage Estimation,
Abstract: Shrinkage estimation usually reduces variance at the cost of bias. But when
we care only about some parameters of a model, I show that we can reduce
variance without incurring bias if we have additional information about the
distribution of covariates. In a linear regression model with homoscedastic
Normal noise, I consider shrinkage estimation of the nuisance parameters
associated with control variables. For at least three control variables and
exogenous treatment, I establish 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 a
high-dimensional Normal-means problem. It can be interpreted as an invariant
generalized Bayes estimator with an uninformative (improper) Jeffreys prior in
the target parameter.