Testing the Exclusion Restriction with a Single Instrument: Identification through Fourth-Order Cumulants under Latent Confounding

With a single instrument the exclusion restriction is untestable from second moments, and adding the instrument to the structural equation fails because the regressor is endogenous. When the endogeneity comes from a latent common cause with non-Gaussian components, the restriction becomes generically testable from the fourth-order cumulants of the first-stage and 2SLS residuals: a closed-form scalar $\theta$ is zero under exclusion and non-zero otherwise, while the direct effect itself is identified only up to two points. A companion statistic measures identification strength. Simulations show correct size under confounding, where the naive OLS test always rejects. In Card's (1995) data the test is uninformative, and says so.