Estimating and Testing Models with Many Treatment Levels and Limited Instruments
Many empirical microeconomic studies estimate econometric models that assume a single finite-valued discrete endogenous regressor (for example: different levels of schooling), exogenous regressors that are additively separable and enter the equation linearly; and coefficients (including per-unit treatment effects) that are homogeneous in the population. Empirical researchers interested in the causal effect of the endogenous regressor often use instrumental variables. When few valid instruments are available, researchers typically estimate restricted specifications that impose uniform per-unit treatment effects, even when these effects are likely to vary depending on the treatment level. In these cases, ordinary least squares (OLS) and instrumental variables (IV) estimators identify different weighted averages of all per-unit effects, so the traditional Hausman test (based on the restricted specification) is uninformative about endogeneity. Addressing this concern, we develop a new exogeneity test that compares the IV estimate from the restricted model with an appropriately weighted average of all per-unit effects estimated from the more general model using OLS. Notably, our test works even when the true model cannot be estimated using IV methods as long as a single valid instrument is available (e.g. a single binary instrument). We re-visit three recent empirical examples that examine the role of educational attainment on various outcomes to demonstrate the practical value of our test.
Previously circulated as "Estimating and Testing Non-Linear Models Using Instrumental Variables." 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 their suggestions. We also thank Matias Cattaneo and Javier Cano Urbina for their excellent research assistance and comments, as well as Martijn van Hasselt and Youngki Shin for their many comments and suggestions. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.
Lance Lochner & Enrico Moretti, 2015. "Estimating and Testing Models with Many Treatment Levels and Limited Instruments," The Review of Economics and Statistics, MIT Press, vol. 2(97), pages 387-397, May. citation courtesy of