 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.
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.
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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.
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