  Many existing approaches for generating predictions in settings with
distribution shift model distribution shifts as adversarial or low-rank in
suitable representations. In various real-world settings, however, we might
expect shifts to arise through the superposition of many small and random
changes in the population and environment. Thus, we consider a class of random
distribution shift models that capture arbitrary changes in the underlying
covariate space, and dense, random shocks to the relationship between the
covariates and the outcomes. In this setting, we characterize the benefits and
drawbacks of several alternative prediction strategies: the standard approach
that directly predicts the long-term outcome of interest, the proxy approach
that directly predicts a shorter-term proxy outcome, and a hybrid approach that
utilizes both the long-term policy outcome and (shorter-term) proxy outcome(s).
We show that the hybrid approach is robust to the strength of the distribution
shift and the proxy relationship. We apply this method to datasets in two
high-impact domains: asylum-seeker assignment and early childhood education. In
both settings, we find that the proposed approach results in substantially
lower mean-squared error than current approaches.
