A simple diagnostic to investigate instrument validity and heterogeneous effects when using a single instrument

Many studies that use instrumental variables are based on a first stage linear in the instrument. Using only linear first stages may miss important information about effect heterogeneity and instrument validity. Analyzing fifteen studies using linear first stages, we find ten with significant nonlinearities. Six of these ten have statistically different second stage estimates. Additional analysis is necessary when results are sensitive to first stage choice. We provide a framework to reconcile these differences by determining those patterns of heterogeneity that are consistent with instrument validity. If these patterns violate economic reasoning, then the validity of the instrument is questioned.