Gaussian Heteroskedastic Empirical Bayes without Independence

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.