We develop theoretical foundations for widely used falsification tests for instrumental variable (IV) designs. We characterize these tests as conditional independence tests between negative control variables - proxies for potential threats - and either the IV or the outcome. We find that conventional applications of these falsification tests would flag problems in exogenous IV designs, and propose simple solutions to avoid this. We also propose new falsification tests that incorporate new types of negative control variables or alternative statistical tests. Finally, we illustrate that under stronger assumptions, negative control variables can also be used for bias correction.
