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Estimating and Testing Models with Many Treatment Levels and Limited Instruments

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  • Lance Lochner
  • Enrico Moretti

Abstract

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.

Suggested Citation

  • Lance Lochner & Enrico Moretti, 2011. "Estimating and Testing Models with Many Treatment Levels and Limited Instruments," NBER Working Papers 17039, National Bureau of Economic Research, Inc.
  • Handle: RePEc:nbr:nberwo:17039
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    Cited by:

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    4. Shoya Ishimaru, 2021. "Empirical Decomposition of the IV-OLS Gap with Heterogeneous and Nonlinear Effects," Papers 2101.04346, arXiv.org, revised Jun 2022.
    5. Mario Fiorini & Katrien Stevens, 2021. "Scrutinizing the Monotonicity Assumption in IV and fuzzy RD designs," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 83(6), pages 1475-1526, December.
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    7. Tymon S{l}oczy'nski, 2018. "Interpreting OLS Estimands When Treatment Effects Are Heterogeneous: Smaller Groups Get Larger Weights," Papers 1810.01576, arXiv.org, revised May 2020.
    8. Tymon Sloczynski, 2018. "A General Weighted Average Representation of the Ordinary and Two-Stage Least Squares Estimands," Working Papers 125, Brandeis University, Department of Economics and International Businesss School.
    9. Tymon S{l}oczy'nski, 2020. "When Should We (Not) Interpret Linear IV Estimands as LATE?," Papers 2011.06695, arXiv.org, revised Sep 2022.
    10. Cygan-Rehm, Kamila & Wunder, Christoph, 2018. "Do working hours affect health? Evidence from statutory workweek regulations in Germany," Labour Economics, Elsevier, vol. 53(C), pages 162-171.
    11. Francesco Fasani & Tommaso Frattini & Luigi Minale, 2021. "Lift the Ban? Initial Employment Restrictions and Refugee Labour Market Outcomes," Journal of the European Economic Association, European Economic Association, vol. 19(5), pages 2803-2854.
    12. Firmin DOKO TCHATOKA & Jean-Marie DUFOUR, 2016. "Exogeneity Tests, Incomplete Models, Weak Identification and Non-Gaussian Distributions : Invariance and Finite-Sample Distributional Theory," Cahiers de recherche 14-2016, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    13. Katrine V. Løken & Magne Mogstad & Matthew Wiswall, 2012. "What Linear Estimators Miss: The Effects of Family Income on Child Outcomes," American Economic Journal: Applied Economics, American Economic Association, vol. 4(2), pages 1-35, April.
    14. Javier Cano-Urbina & Lance Lochner, 2019. "The Effect of Education and School Quality on Female Crime," Journal of Human Capital, University of Chicago Press, vol. 13(2), pages 188-235.
    15. Gaurab Aryal & Manudeep Bhuller & Fabian Lange, 2022. "Signaling and Employer Learning with Instruments," American Economic Review, American Economic Association, vol. 112(5), pages 1669-1702, May.
    16. Tymon Sloczynski, 2021. "When Should We (Not) Interpret Linear IV Estimands as LATE?," CESifo Working Paper Series 9064, CESifo.
    17. Firmin Doko Tchatoka & Jean-Marie Dufour, 2016. "Exogeneity tests, weak identification, incomplete models and non-Gaussian distributions: Invariance and finite-sample distributional theory," School of Economics Working Papers 2016-01, University of Adelaide, School of Economics.
    18. Kamila Cygam-Rehm & Christoph Wunder, 2018. "Do Working Hours Affect Health? Evidence from Statutory Workweek Regulations in Germany," SOEPpapers on Multidisciplinary Panel Data Research 967, DIW Berlin, The German Socio-Economic Panel (SOEP).
    19. Sergi Jiménez-Martín & Cristina Vilaplana Prieto, 2013. "Informal Care and intergenerational transfers in European Countries," Working Papers 2013-25, FEDEA.
    20. Kamila Cygan-Rehm & Christoph Wunder, 2018. "Do Working Hours Affect Health? Evidence from Statutory Workweek Regulations in Germany," CESifo Working Paper Series 7098, CESifo.
    21. Wossen, Tesfamicheal & Abay, Kibrom A. & Abdoulaye, Tahirou, 2022. "Misperceiving and misreporting input quality: Implications for input use and productivity," Journal of Development Economics, Elsevier, vol. 157(C).
    22. Doko Tchatoka, Firmin & Dufour, Jean-Marie, 2020. "Exogeneity tests, incomplete models, weak identification and non-Gaussian distributions: Invariance and finite-sample distributional theory," Journal of Econometrics, Elsevier, vol. 218(2), pages 390-418.
    23. Shoya Ishimaru, 2021. "What Do We Get from Two-Way Fixed Effects Regressions? Implications from Numerical Equivalence," Papers 2103.12374, arXiv.org, revised Oct 2022.

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    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • J0 - Labor and Demographic Economics - - General

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