Empirical Bayes When Estimation Precision Predicts Parameters

Empirical Bayes methods usually maintain a prior independence assumption: The unknown parameters of interest are independent from the known standard errors of the estimates. This assumption is often theoretically questionable and empirically rejected. This paper instead models the conditional distribution of the parameter given the standard errors as a flexibly parametrized family of distributions, leading to a family of methods that we call CLOSE. This paper establishes that (i) CLOSE is rate-optimal for squared error Bayes regret, (ii) squared error regret control is sufficient for an important class of economic decision problems, and (iii) CLOSE is worst-case robust when our assumption on the conditional distribution is misspecified. Empirically, using CLOSE leads to sizable gains for selecting high-mobility Census tracts. Census tracts selected by CLOSE are substantially more mobile on average than those selected by the standard shrinkage method.