The Dilution Problem in Observational Studies on the Relationship Between Immigration and Crime

This research note critically reevaluates one of the most common empirical strategies to study the effect of immigration on crime -- namely, regressing changes in aggregate crime rates on exogenous shifts in local immigrant shares. I show that, because immigrants typically comprise only a small fraction of the population, even large crime-rate differences between immigrants and natives are mechanically diluted. As a result, null findings from such designs are predetermined and reveal little to no information about immigrant-native crime differentials. I derive a closed-form expression for the minimum detectable gap - the smallest immigrant–native crime difference these regressions can identify - and then use Monte Carlo simulations, calibrated to real-world immigration and crime data, to demonstrate that conventional designs only achieve adequate statistical power with implausibly large crime differentials and extreme immigration shocks.