  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.
