In many empirical studies, researchers seek to estimate causal relationships using instrumental variables. When few valid instruments are available, researchers typically estimate models that impose a linear relationship between the dependent variable and the endogenous variable, even when the true model is likely to be non-linear. In the presence of non-linearity, ordinary least squares (OLS) and instrumental variable (IV) estimators identify different weighted averages of the underlying marginal causal effects, so the traditional Hausman test (applied to mis-specified linear models) is uninformative about endogeneity. We build on this insight to develop a new exogeneity test that is robust to non-linearity in the endogenous regressor. This test compares the IV estimator applied to the mis-specified linear model with an appropriately weighted average of all marginal effects estimated from the correctly specified non-linear model using OLS. Notably, our test works well even when only a single valid instrument is available, and the true model cannot be estimated using IV methods. We re-visit three recent empirical examples to show how the test can be used to shed new light on the role of non-linearity
