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

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 a sufficient condition for a collection plan to be minimax optimal. 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.