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How Linear Models Can Mask Non-Linear Causal Relationships. An Application to Family Size and Children's Education

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Abstract

Many empirical studies specify outcomes as a linear function of endogenous regressors when conducting instrumental variable (IV) estimation. We show that commonly used tests for treatment effects, selection bias, and treatment effect heterogeneity are biased if the true relationship is non-linear. In particular, using linear models can only lead to under-rejection of the null hypothesis of no treatment effects. In light of these results, we re-examine the recent evidence suggesting that family size has no causal effect on children's education. Following common practice, a linear IV estimator has been used, assuming constant marginal effects of additional children across family sizes. We show that the conclusion of no causal effect of family size is an artifact of the specification of a linear model, which masks significant marginal family size effects. Estimating a model that is non-parametric in family size, we find that family size matters substantially for children's educational attainment, but in a non-monotonic way. Our findings illustrate that IV estimation of models which relax linearity restrictions is an important addition to empirical research, particularly when OLS estimation and theory suggests the possibility of non-linear causal effects.

Suggested Citation

  • Magne Mogstad & Matthew Wiswall, 2009. "How Linear Models Can Mask Non-Linear Causal Relationships. An Application to Family Size and Children's Education," Discussion Papers 586, Statistics Norway, Research Department.
  • Handle: RePEc:ssb:dispap:586
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    Cited by:

    1. Sandra E. Black & Paul J. Devereux & Katrine V. L�ken & Kjell G. Salvanes, 2014. "Care or Cash? The Effect of Child Care Subsidies on Student Performance," The Review of Economics and Statistics, MIT Press, vol. 96(5), pages 824-837, December.
    2. Lance Lochner & Alexander Monge-Naranjo, 2012. "Credit Constraints in Education," Annual Review of Economics, Annual Reviews, vol. 4(1), pages 225-256, July.
    3. Lundborg, Petter & Nilsson, Anton & Rooth, Dan-Olof, 2012. "Parental Education and Offspring Outcomes: Evidence from the Swedish Compulsory Schooling Reform," IZA Discussion Papers 6570, Institute of Labor Economics (IZA).
    4. Mathias Kuepie & Michel Tenikue & Samuel Nouetagni & Nicaise Misangumukini, 2014. "Number, Age Composition and School Achievements of Siblings in Two African Capital Cities," Oxford Development Studies, Taylor & Francis Journals, vol. 42(4), pages 534-552, December.
    5. Mogstad, Magne & Wiswall, Matthew, 2009. "How Much Should We Trust Linear Instrumental Variables Estimators? An Application to Family Size and Children's Education," IZA Discussion Papers 4562, Institute of Labor Economics (IZA).
    6. Gordon B. Dahl & Lance Lochner, 2012. "The Impact of Family Income on Child Achievement: Evidence from the Earned Income Tax Credit," American Economic Review, American Economic Association, vol. 102(5), pages 1927-1956, August.
    7. repec:wly:econjl:v:126:y:2016:i:596:p:f136-f183 is not listed on IDEAS
    8. Román David Zárate, 2013. "Family size and children quality: New evidence and new exogenous shocks in the case of Colombian Households," Documentos CEDE 010588, Universidad de los Andes - CEDE.
    9. Elizabeth M. Caucutt & Lance Lochner, 2012. "Early and Late Human Capital Investments, Borrowing Constraints, and the Family," NBER Working Papers 18493, National Bureau of Economic Research, Inc.
    10. Jessica Meredith & Frank Neri & Joan Rodgers, 2013. "Family Impacts on Cognitive Development of Young Children: Evidence from Australia," Economics Working Papers wp13-05, School of Economics, University of Wollongong, NSW, Australia.
    11. Daniela Del Boca & Christopher Flinn & Matthew Wiswall, 2016. "Transfers to Households with Children and Child Development," Economic Journal, Royal Economic Society, vol. 126(596), pages 136-183, October.
    12. Daniela Del Boca & Christopher Flinn & Matthew Wiswall, 2014. "Household Choices and Child Development," Review of Economic Studies, Oxford University Press, vol. 81(1), pages 137-185.
    13. Marcela Ibanez & Gerhard Riener, 2018. "Sorting through Affirmative Action: Three Field Experiments in Colombia," Journal of Labor Economics, University of Chicago Press, vol. 36(2), pages 437-478.
    14. Aaberge, Rolf & Mogstad, Magne & Peragine, Vito, 2011. "Measuring long-term inequality of opportunity," Journal of Public Economics, Elsevier, vol. 95(3-4), pages 193-204, April.
    15. Drange, Nina & Havnes, Tarjei & Sandsør, Astrid M.J., 2016. "Kindergarten for all: Long run effects of a universal intervention," Economics of Education Review, Elsevier, vol. 53(C), pages 164-181.
    16. Alexander M. Gelber & Matthew C. Weinzierl, 2012. "Equalizing Outcomes and Equalizing Opportunities: Optimal Taxation when Children's Abilities Depend on Parents' Resources," NBER Working Papers 18332, National Bureau of Economic Research, Inc.
    17. Sara Cools & Rannveig Kaldager Hart, 2015. "The effect of childhood family size on fertility in adulthood. New evidence form IV estimation," Discussion Papers 802, Statistics Norway, Research Department.
    18. Dev, Pritha & Mberu, Blessing & Pongou, Roland, 2013. "Communitarianism, Oppositional Cultures, and Human Capital Contagion: Theory and Evidence from Formal versus Koranic Education," MPRA Paper 46234, University Library of Munich, Germany, revised 15 Apr 2013.
    19. 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. 97(2), pages 387-397, May.
    20. 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.
    21. Mogstad, M. & Wiswall, M., 2012. "Instrumental variables estimation with partially missing instruments," Economics Letters, Elsevier, vol. 114(2), pages 186-189.
    22. Lance Lochner & Enrico Moretti, 2011. "Estimating and Testing Non-Linear Models Using Instrumental Variables," University of Western Ontario, Centre for Human Capital and Productivity (CHCP) Working Papers 20112, University of Western Ontario, Centre for Human Capital and Productivity (CHCP).
    23. Parfait Eloundou-Enyegue & Sarah Giroux, 2012. "Fertility Transitions and Schooling: From Micro- to Macro-Level Associations," Demography, Springer;Population Association of America (PAA), vol. 49(4), pages 1407-1432, November.
    24. repec:esx:essedp:726 is not listed on IDEAS

    More about this item

    Keywords

    Instrumental variables; variable treatment intensity; treatment effect heterogeneity; selection bias; quantity-quality; family size; child outcome;

    JEL classification:

    • 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
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • J13 - Labor and Demographic Economics - - Demographic Economics - - - Fertility; Family Planning; Child Care; Children; Youth

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