This paper proposes empirical Bayes shrinkage methods. Compared to common shrinkage methods, we do not assume that the unknown parameters are independent from the known standard errors. This prior independence assumption is both theoretically tenuous and often empirically rejected. We instead model the conditional distribution of the parameter given the standard errors as a location-scale family. This assumption leads to a family of methods that we call CLOSE. We establish that (i) CLOSE is rate-optimal for squared error Bayes regret up to logarithmic factors, (ii) squared error regret control is sufficient for a class of economic decision problems, and (iii) CLOSE is worst-case robust. We illustrate our method with an empirical application to the Opportunity Atlas and Creating Moves to Opportunity (Chetty et al., 2018; Bergman et al., 2019). For the decision problem of selecting high mobility Census tracts in Bergman et al. (2019), CLOSE selects Census tracts that are more economically mobile than the standard shrinkage method. This estimated gain is larger than the gain of using the standard method relative to selecting tracts uniformly at random.

翻译：本文提出了经验贝叶斯收缩方法。与常见的收缩方法相比，我们不假设未知参数与已知标准误差之间相互独立。这一先验独立性假设在理论上薄弱且常被实证拒绝。相反，我们将给定标准误差条件下的参数条件分布建模为位置-尺度族。该假设衍生出一系列我们称之为CLOSE的方法。我们证明：(i) CLOSE在均方误差贝叶斯遗憾上达到对数因子的率最优性；(ii) 均方误差遗憾控制足以应对一类经济决策问题；(iii) CLOSE具备最坏情况鲁棒性。我们通过实证应用展示该方法：基于“机会地图集”与“创造机会迁移”数据（Chetty等人，2018；Bergman等人，2019）。在Bergman等人（2019）中筛选高流动性人口普查区的决策问题中，CLOSE选取的人口普查区经济流动性高于标准收缩方法。这一估计增益超过标准方法相较于随机均匀选取区域的增益。