  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.
