Estimating and Testing Non-Linear Models Using Instrumental Variables

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. ∗We thank Josh Angrist, David Card, Pedro Carneiro, Jim Heckman, Guido Imbens, the editor, two anonymous referees and seminar participants at the 2008 UM/MSU/UWO Summer Labor Conference, UCSD, and Stanford for insightful suggestions. We are also grateful to Matias Cattaneo and Javier Cano Urbina, who provided excellent research assistance and insightful substantive comments, as well as Martijn van Hasselt and Youngki Shin for their many comments and suggestions.