IDEAS home Printed from https://ideas.repec.org/p/iza/izadps/dp11866.html
   My bibliography  Save this paper

A General Weighted Average Representation of the Ordinary and Two-Stage Least Squares Estimands

Author

Listed:
  • Sloczynski, Tymon

    (Brandeis University)

Abstract

It is standard practice in applied work to study the effect of a binary variable (“treatment”) on an outcome of interest using linear models with additive effects. In this paper I study the interpretation of the ordinary and two-stage least squares estimands in such models when treatment effects are in fact heterogeneous. I show that in both cases the coefficient on treatment is identical to a convex combination of two other parameters (different for OLS and 2SLS), which can be interpreted as the average treatment effects on the treated and controls under additional assumptions. Importantly, the OLS and 2SLS weights on these parameters are inversely related to the proportion of each group. The more units get treatment, the less weight is placed on the effect on the treated. What follows, the reliance on these implicit weights can have serious consequences for applied work. I illustrate some of these issues in four empirical applications from different fields of economics. I also develop a weighted least squares correction and simple diagnostic tools that applied researchers can use to avoid potential biases. In an important special case, my diagnostics only require the knowledge of the proportion of treated units.

Suggested Citation

  • Sloczynski, Tymon, 2018. "A General Weighted Average Representation of the Ordinary and Two-Stage Least Squares Estimands," IZA Discussion Papers 11866, Institute of Labor Economics (IZA).
  • Handle: RePEc:iza:izadps:dp11866
    as

    Download full text from publisher

    File URL: https://docs.iza.org/dp11866.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Monica Martinez-Bravo, 2014. "The Role of Local Officials in New Democracies: Evidence from Indonesia," American Economic Review, American Economic Association, vol. 104(4), pages 1244-1287, April.
    2. James J. Heckman, 1976. "The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for Such Models," NBER Chapters, in: Annals of Economic and Social Measurement, Volume 5, number 4, pages 475-492, National Bureau of Economic Research, Inc.
    3. Federico A. Bugni & Ivan A. Canay & Azeem M. Shaikh, 2018. "Inference Under Covariate-Adaptive Randomization," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(524), pages 1784-1796, October.
    4. Stelios Michalopoulos & Elias Papaioannou, 2016. "The Long-Run Effects of the Scramble for Africa," American Economic Review, American Economic Association, vol. 106(7), pages 1802-1848, July.
    5. 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.
    6. Christian N. Brinch & Magne Mogstad & Matthew Wiswall, 2017. "Beyond LATE with a Discrete Instrument," Journal of Political Economy, University of Chicago Press, vol. 125(4), pages 985-1039.
    7. Jeremiah E. Dittmar, 2011. "Information Technology and Economic Change: The Impact of The Printing Press," The Quarterly Journal of Economics, Oxford University Press, vol. 126(3), pages 1133-1172.
    8. Olsen, Randall J, 1980. "A Least Squares Correction for Selectivity Bias," Econometrica, Econometric Society, vol. 48(7), pages 1815-1820, November.
    9. Matthias Parey & Fabian Waldinger, 2011. "Studying Abroad and the Effect on International Labour Market Mobility: Evidence from the Introduction of ERASMUS," Economic Journal, Royal Economic Society, vol. 121(551), pages 194-222, March.
    10. Kato, Ryutah & Sasaki, Yuya, 2017. "On Using Linear Quantile Regressions For Causal Inference," Econometric Theory, Cambridge University Press, vol. 33(3), pages 664-690, June.
    11. Peter Hull, 2018. "Estimating Treatment Effects in Mover Designs," Papers 1804.06721, arXiv.org.
    12. Joshua D. Angrist & Jörn-Steffen Pischke, 2009. "Mostly Harmless Econometrics: An Empiricist's Companion," Economics Books, Princeton University Press, edition 1, number 8769.
    13. Gustavo J. Bobonis & Luis R. Cámara Fuertes & Rainer Schwabe, 2016. "Monitoring Corruptible Politicians," American Economic Review, American Economic Association, vol. 106(8), pages 2371-2405, August.
    14. Nicole Maestas & Kathleen J. Mullen & Alexander Strand, 2013. "Does Disability Insurance Receipt Discourage Work? Using Examiner Assignment to Estimate Causal Effects of SSDI Receipt," American Economic Review, American Economic Association, vol. 103(5), pages 1797-1829, August.
    15. Jeffrey M. Wooldridge, 2005. "Fixed-Effects and Related Estimators for Correlated Random-Coefficient and Treatment-Effect Panel Data Models," The Review of Economics and Statistics, MIT Press, vol. 87(2), pages 385-390, May.
    16. Liyang Sun & Sarah Abraham, 2018. "Estimating Dynamic Treatment Effects in Event Studies with Heterogeneous Treatment Effects," Papers 1804.05785, arXiv.org, revised Sep 2020.
    17. Patrick Kline & Christopher R. Walters, 2019. "On Heckits, LATE, and Numerical Equivalence," Econometrica, Econometric Society, vol. 87(2), pages 677-696, March.
    18. Joseph G. Altonji & Todd E. Elder & Christopher R. Taber, 2005. "An Evaluation of Instrumental Variable Strategies for Estimating the Effects of Catholic Schooling," Journal of Human Resources, University of Wisconsin Press, vol. 40(4), pages 791-821.
    19. Brantly Callaway & Pedro H. C. Sant'Anna, 2018. "Difference-in-Differences with Multiple Time Periods and an Application on the Minimum Wage and Employment," DETU Working Papers 1804, Department of Economics, Temple University.
    20. Richard Blundell & Rosa L. Matzkin, 2014. "Control functions in nonseparable simultaneous equations models," Quantitative Economics, Econometric Society, vol. 5, pages 271-295, July.
    21. Joshua D. Angrist, 1998. "Estimating the Labor Market Impact of Voluntary Military Service Using Social Security Data on Military Applicants," Econometrica, Econometric Society, vol. 66(2), pages 249-288, March.
    22. Victor Chernozhukov & Iván Fernández‐Val & Jinyong Hahn & Whitney Newey, 2013. "Average and Quantile Effects in Nonseparable Panel Models," Econometrica, Econometric Society, vol. 81(2), pages 535-580, March.
    23. John Y. Campbell & Stefano Giglio & Parag Pathak, 2011. "Forced Sales and House Prices," American Economic Review, American Economic Association, vol. 101(5), pages 2108-2131, August.
    24. Damon Clark & Emilia Del Bono, 2016. "The Long-Run Effects of Attending an Elite School: Evidence from the United Kingdom," American Economic Journal: Applied Economics, American Economic Association, vol. 8(1), pages 150-176, January.
    25. Dan A. Black & Seth G. Sanders & Evan J. Taylor & Lowell J. Taylor, 2015. "The Impact of the Great Migration on Mortality of African Americans: Evidence from the Deep South," American Economic Review, American Economic Association, vol. 105(2), pages 477-503, February.
    26. Imbens,Guido W. & Rubin,Donald B., 2015. "Causal Inference for Statistics, Social, and Biomedical Sciences," Cambridge Books, Cambridge University Press, number 9780521885881.
    27. Jishnu Das & Alaka Holla & Aakash Mohpal & Karthik Muralidharan, 2016. "Quality and Accountability in Health Care Delivery: Audit-Study Evidence from Primary Care in India," American Economic Review, American Economic Association, vol. 106(12), pages 3765-3799, December.
    28. William Rhodes, 2010. "Heterogeneous Treatment Effects: What Does a Regression Estimate?," Evaluation Review, , vol. 34(4), pages 334-361, August.
    29. Dan A. Black & Jeffrey A. Smith & Mark C. Berger & Brett J. Noel, 2003. "Is the Threat of Reemployment Services More Effective Than the Services Themselves? Evidence from Random Assignment in the UI System," American Economic Review, American Economic Association, vol. 93(4), pages 1313-1327, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Strobl, Renate & Wunsch, Conny, 2018. "Identification of causal mechanisms based on between-subject double randomization designs," CEPR Discussion Papers 13028, C.E.P.R. Discussion Papers.
    2. Renate Strobl & Conny Wunsch, 2017. "Does Voluntary Risk Taking Affect Solidarity? Experimental Evidence from Kenya," CESifo Working Paper Series 6578, CESifo.
    3. Sant’Anna, Pedro H.C. & Zhao, Jun, 2020. "Doubly robust difference-in-differences estimators," Journal of Econometrics, Elsevier, vol. 219(1), pages 101-122.
    4. Bruce E. Hansen & Seojeong Lee, 2021. "Inference for Iterated GMM Under Misspecification," Econometrica, Econometric Society, vol. 89(3), pages 1419-1447, May.
    5. Alberto Abadie & Susan Athey & Guido W. Imbens & Jeffrey M. Wooldridge, 2020. "Sampling‐Based versus Design‐Based Uncertainty in Regression Analysis," Econometrica, Econometric Society, vol. 88(1), pages 265-296, January.
    6. Graham, Bryan S. & Pinto, Cristine Campos de Xavier, 2022. "Semiparametrically efficient estimation of the average linear regression function," Journal of Econometrics, Elsevier, vol. 226(1), pages 115-138.
    7. Nathan Lane, 2021. "Manufacturing Revolutions: Industrial Policy and Industrialization in South Korea," SoDa Laboratories Working Paper Series 2021-10, Monash University, SoDa Laboratories.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Słoczyński, Tymon, 2012. "New Evidence on Linear Regression and Treatment Effect Heterogeneity," MPRA Paper 39524, University Library of Munich, Germany.
    3. Valentin Verdier, 2020. "Average treatment effects for stayers with correlated random coefficient models of panel data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 35(7), pages 917-939, November.
    4. Sant’Anna, Pedro H.C. & Zhao, Jun, 2020. "Doubly robust difference-in-differences estimators," Journal of Econometrics, Elsevier, vol. 219(1), pages 101-122.
    5. Jeffrey Smith & Arthur Sweetman, 2016. "Viewpoint: Estimating the causal effects of policies and programs," Canadian Journal of Economics, Canadian Economics Association, vol. 49(3), pages 871-905, August.
    6. Cornelissen, Thomas & Dustmann, Christian & Raute, Anna & Schönberg, Uta, 2016. "From LATE to MTE: Alternative methods for the evaluation of policy interventions," Labour Economics, Elsevier, vol. 41(C), pages 47-60.
    7. Maclean, J. Catherine & Pichler, Stefan & Ziebarth, Nicolas R., 2020. "Mandated Sick Pay: Coverage, Utilization, and Welfare Effects," IZA Discussion Papers 13132, Institute of Labor Economics (IZA).
    8. Sloczynski, Tymon, 2020. "Interpreting OLS Estimands When Treatment Effects Are Heterogeneous: Smaller Groups Get Larger Weights," IZA Discussion Papers 13283, Institute of Labor Economics (IZA).
    9. Patrick Kline & Christopher R. Walters, 2019. "On Heckits, LATE, and Numerical Equivalence," Econometrica, Econometric Society, vol. 87(2), pages 677-696, March.
    10. Dustmann, Christian & Ku, Hyejin & Kwak, Do Won, 2018. "Why Are Single-Sex Schools Successful?," Labour Economics, Elsevier, vol. 54(C), pages 79-99.
    11. Susan Athey & Guido Imbens, 2018. "Design-based Analysis in Difference-In-Differences Settings with Staggered Adoption," Papers 1808.05293, arXiv.org, revised Sep 2018.
    12. Bryan S. Graham & Cristine Campos de Xavier Pinto, 2018. "Semiparametrically efficient estimation of the average linear regression function," Papers 1810.12511, arXiv.org.
    13. Liyang Sun & Sarah Abraham, 2018. "Estimating Dynamic Treatment Effects in Event Studies with Heterogeneous Treatment Effects," Papers 1804.05785, arXiv.org, revised Sep 2020.
    14. Jianfei Cao & Shirley Lu, 2019. "Synthetic Control Inference for Staggered Adoption: Estimating the Dynamic Effects of Board Gender Diversity Policies," Papers 1912.06320, arXiv.org.
    15. Huber, Martin, 2019. "An introduction to flexible methods for policy evaluation," FSES Working Papers 504, Faculty of Economics and Social Sciences, University of Freiburg/Fribourg Switzerland.
    16. Tamara Bischof & Boris Kaiser, 2021. "Who cares when you close down? The effects of primary care practice closures on patients," Health Economics, John Wiley & Sons, Ltd., vol. 30(9), pages 2004-2025, September.
    17. Dmitry Arkhangelsky & Guido Imbens, 2018. "The Role of the Propensity Score in Fixed Effect Models," NBER Working Papers 24814, National Bureau of Economic Research, Inc.
    18. Isaiah Andrews & Emily Oster, 2017. "A Simple Approximation for Evaluating External Validity Bias," NBER Working Papers 23826, National Bureau of Economic Research, Inc.
    19. Phillip Heiler, 2020. "Efficient Covariate Balancing for the Local Average Treatment Effect," Papers 2007.04346, arXiv.org.
    20. Patrick Kline & Christopher R. Walters, 2016. "Evaluating Public Programs with Close Substitutes: The Case of HeadStart," The Quarterly Journal of Economics, Oxford University Press, vol. 131(4), pages 1795-1848.

    More about this item

    Keywords

    heterogeneity; ordinary least squares; propensity score; two-stage least squares; treatment effects;
    All these keywords.

    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C24 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Truncated and Censored Models; Switching Regression Models; Threshold Regression Models
    • C26 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Instrumental Variables (IV) Estimation
    • C31 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models; Quantile Regressions; Social Interaction Models

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:iza:izadps:dp11866. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: https://edirc.repec.org/data/izaaade.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Holger Hinte (email available below). General contact details of provider: https://edirc.repec.org/data/izaaade.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.