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Estimating and Testing Models with Many Treatment Levels and Limited Instruments

  • Lance Lochner
  • Enrico Moretti

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.

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File URL: http://www.nber.org/papers/w17039.pdf
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Paper provided by National Bureau of Economic Research, Inc in its series NBER Working Papers with number 17039.

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Date of creation: May 2011
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Handle: RePEc:nbr:nberwo:17039
Note: CH ED LS
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  1. Carneiro, Pedro & Heckman, James J. & Vytlacil, Edward, 2009. "Evaluating Marginal Policy Changes and the Average Effect of Treatment for Individuals at the Margin," IZA Discussion Papers 4324, Institute for the Study of Labor (IZA).
  2. James J. Heckman & Lance J. Lochner & Petra E. Todd, 2008. "Earnings Functions and Rates of Return," University of Western Ontario, CIBC Centre for Human Capital and Productivity Working Papers 20082, University of Western Ontario, CIBC Centre for Human Capital and Productivity.
  3. Yitzhaki, Shlomo, 1996. "On Using Linear Regressions in Welfare Economics," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(4), pages 478-86, October.
  4. Jaeger, David A & Page, Marianne E, 1996. "Degrees Matter: New Evidence on Sheepskin Effects in the Returns to Education," The Review of Economics and Statistics, MIT Press, vol. 78(4), pages 733-40, November.
  5. Mogstad, Magne & Wiswall, Matthew, 2010. "Linearity in Instrumental Variables Estimation: Problems and Solutions," IZA Discussion Papers 5216, Institute for the Study of Labor (IZA).
  6. Lance Lochner & Enrico Moretti, 2001. "The Effect of Education on Crime: Evidence from Prison Inmates, Arrests, and Self-Reports," NBER Working Papers 8605, National Bureau of Economic Research, Inc.
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