Better Measurement or Larger Samples? Data Collection for Policy Learning with Unobserved Heterogeneity

Abstract: Empirical research shows that individuals' responses to treatments vary along latent characteristics, such as innate ability or motivation. Therefore, a policymaker seeking to maximize welfare may consider designing policies based on observed characteristics and estimated latent traits. I characterize how the estimates' precision affects the worst-case performance of policies, deriving rate-sharp regret bounds for assignment rules that include or exclude them, highlighting new trade-offs with the policy space complexity. I then study how a policymaker can solve such trade-offs by designing tailored data collections, and derive the minimax optimal collection plan. In an empirical application in development economics, I show that including a proxy for entrepreneurs' business skills in targeting cash transfers increases welfare by 5%, and halves the probability of generating welfare losses. Moreover, I estimate the optimal allocation of resources between improving the precision of the proxy via repeated measurements and increasing sample size.