Estimating and Testing Models with Many Treatment Levels and Limited Instruments

Empirical researchers interested in the causal effect of the endogenous regressor often use instrumental variables. When few valid instruments are available, they typically estimate restricted specifications that impose uniform per unit treatment effects, even when these effects are likely to vary. We show that in these cases, ordinary least squares and instrumental variables estimators identify different weighted averages of all per unit effects, so the traditional Hausman test is uninformative about endogeneity. We develop a new exogeneity test that works even when the true model cannot be estimated using IV methods as long as a single valid instrument is available. We revisit three recent empirical examples to demonstrate the practical value of our test.