IDEAS home Printed from https://ideas.repec.org/p/edn/esedps/321.html

OLS Estimation when Two Noisy Measures of a Regressor are Available: Instruments, Tests and Pitfalls

Author

Listed:
  • Andy Snell

    (School of Economics, University of Edinburgh)

Abstract

It is well known that if the regression coefficient of y on w (beta say) has a constant probability limit but we only have two noisy measures of w - x and z - then we may obtain consistent estimates of beta as long as a) the measurement errors are classical and b) the measurement errors are uncorrelated. We propose a simple test of a) and a test for b) as part of a composite null. To effect the latter we instrument x with z and functions of z and vice versa to obtain two sets of overidentifying restrictions tested via a standard J test of instrument validity. If no test in this sequence rejects we then combine the orthogonality conditions to obtain a single efficient estimate of beta. We discuss the likely prior validity of the various instruments and the pitfalls in using the test procedure. Unlike standard overidentification tests which diverge in heterogeneous response settings even when each instrument is valid, our tests only diverge when one or more instrument is invalid. We apply the test sequence and estimation procedure to analyse i) the cyclical component of wages and ii) the effect of state level unemployment on burglaries in the US. Correcting for measurement error raises the estimates of beta in both applications.

Suggested Citation

  • Andy Snell, 2025. "OLS Estimation when Two Noisy Measures of a Regressor are Available: Instruments, Tests and Pitfalls," Edinburgh School of Economics Discussion Paper Series 321, Edinburgh School of Economics, University of Edinburgh.
    • Handle: RePEc:edn:esedps:321
    as

    Download full text from publisher

    File URL: http://www.econ.ed.ac.uk/papers/id321_esedps.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    2. Raphael, Steven & Winter-Ember, Rudolf, 2001. "Identifying the Effect of Unemployment on Crime," Journal of Law and Economics, University of Chicago Press, vol. 44(1), pages 259-283, April.
    3. Daniel Wilhelm, 2018. "Testing for the presence of measurement error," CeMMAP working papers CWP45/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Gary Solon & Robert Barsky & Jonathan A. Parker, 1994. "Measuring the Cyclicality of Real Wages: How Important is Composition Bias?," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 109(1), pages 1-25.
    5. Bound, John & Brown, Charles & Mathiowetz, Nancy, 2001. "Measurement error in survey data," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 5, chapter 59, pages 3705-3843, Elsevier.
    6. Dagenais, Marcel G. & Dagenais, Denyse L., 1997. "Higher moment estimators for linear regression models with errors in the variables," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 193-221.
    7. Arthur Lewbel, 1997. "Constructing Instruments for Regressions with Measurement Error when no Additional Data are Available, with an Application to Patents and R&D," Econometrica, Econometric Society, vol. 65(5), pages 1201-1214, September.
    8. Erickson, Timothy & Whited, Toni M., 2002. "Two-Step Gmm Estimation Of The Errors-In-Variables Model Using High-Order Moments," Econometric Theory, Cambridge University Press, vol. 18(3), pages 776-799, June.
    9. Joshua D. Angrist & Kathryn Graddy & Guido W. Imbens, 2000. "The Interpretation of Instrumental Variables Estimators in Simultaneous Equations Models with an Application to the Demand for Fish," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 67(3), pages 499-527.
    10. Bonggeun Kim & Gary Solon, 2005. "Implications of Mean-Reverting Measurement Error for Longitudinal Studies of Wages and Employment," The Review of Economics and Statistics, MIT Press, vol. 87(1), pages 193-196, February.
    11. Robert A. Hart, 2006. "Worker–Job Matches, Job Mobility and Real Wage Cyclicality," Economica, London School of Economics and Political Science, vol. 73(290), pages 287-298, May.
    12. Morten O. Ravn & Harald Uhlig, 2002. "On adjusting the Hodrick-Prescott filter for the frequency of observations," The Review of Economics and Statistics, MIT Press, vol. 84(2), pages 371-375.
    13. James J. Heckman, 1999. "Instrumental Variables: Response to Angrist and Imbens," Journal of Human Resources, University of Wisconsin Press, vol. 34(4), pages 828-837.
    14. M. Hashem Pesaran, 2015. "Testing Weak Cross-Sectional Dependence in Large Panels," Econometric Reviews, Taylor & Francis Journals, vol. 34(6-10), pages 1089-1117, December.
    15. Young Jun Lee & Daniel Wilhelm, 2020. "Testing for the presence of measurement error in Stata," Stata Journal, StataCorp LLC, vol. 20(2), pages 382-404, June.
    16. John Grigsby & Erik Hurst & Ahu Yildirmaz, 2021. "Aggregate Nominal Wage Adjustments: New Evidence from Administrative Payroll Data," American Economic Review, American Economic Association, vol. 111(2), pages 428-471, February.
    17. Xiaohong Chen & Han Hong & Denis Nekipelov, 2011. "Nonlinear Models of Measurement Errors," Journal of Economic Literature, American Economic Association, vol. 49(4), pages 901-937, December.
    Full references (including those not matched with items on IDEAS)

    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. Andrei Zeleneev & Kirill Evdokimov, 2023. "Simple estimation of semiparametric models with measurement errors," CeMMAP working papers 10/23, Institute for Fiscal Studies.
    2. Meijer, Erik & Spierdijk, Laura & Wansbeek, Tom, 2017. "Consistent estimation of linear panel data models with measurement error," Journal of Econometrics, Elsevier, vol. 200(2), pages 169-180.
    3. Schennach, Susanne M., 2020. "Mismeasured and unobserved variables," Handbook of Econometrics, in: Steven N. Durlauf & Lars Peter Hansen & James J. Heckman & Rosa L. Matzkin (ed.), Handbook of Econometrics, edition 1, volume 7, chapter 0, pages 487-565, Elsevier.
    4. Kirill S. Evdokimov & Andrei Zeleneev, 2023. "Simple Estimation of Semiparametric Models with Measurement Errors," Papers 2306.14311, arXiv.org, revised Nov 2025.
    5. Hu, Yingyao & Schennach, Susanne & Shiu, Ji-Liang, 2022. "Identification of nonparametric monotonic regression models with continuous nonclassical measurement errors," Journal of Econometrics, Elsevier, vol. 226(2), pages 269-294.
    6. Takahide Yanagi, 2019. "Inference on local average treatment effects for misclassified treatment," Econometric Reviews, Taylor & Francis Journals, vol. 38(8), pages 938-960, September.
    7. Dongwoo Kim & Daniel Wilhelm, 2024. "Powerful t-tests in the presence of nonclassical measurement error," Econometric Reviews, Taylor & Francis Journals, vol. 43(6), pages 345-378, July.
    8. Susanne M. Schennach, 2012. "Measurement error in nonlinear models - a review," CeMMAP working papers 41/12, Institute for Fiscal Studies.
    9. Arthur Lewbel, 2012. "Using Heteroscedasticity to Identify and Estimate Mismeasured and Endogenous Regressor Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 30(1), pages 67-80.
    10. Bonhomme, Stphane & Robin, Jean-Marc, 2009. "Consistent noisy independent component analysis," Journal of Econometrics, Elsevier, vol. 149(1), pages 12-25, April.
    11. David Welsch & David Zimmer, 2010. "The Effect of Health and Poverty on Early Childhood Cognitive Development," Atlantic Economic Journal, Springer;International Atlantic Economic Society, vol. 38(1), pages 37-49, March.
    12. Erickson, Timothy & Jiang, Colin Huan & Whited, Toni M., 2014. "Minimum distance estimation of the errors-in-variables model using linear cumulant equations," Journal of Econometrics, Elsevier, vol. 183(2), pages 211-221.
    13. Milan Hladík & Michal Černý & Jaromír Antoch, 2020. "EIV regression with bounded errors in data: total ‘least squares’ with Chebyshev norm," Statistical Papers, Springer, vol. 61(1), pages 279-301, February.
    14. Christopher F Baum & Arthur Lewbel, 2019. "Advice on using heteroskedasticity-based identification," Stata Journal, StataCorp LLC, vol. 19(4), pages 757-767, December.
    15. Madeira, Carlos, 2026. "Identification of earnings dynamics using rotating samples over short periods: the case of Chile," World Development, Elsevier, vol. 199(C).
    16. Leonardo Becchetti & Luisa Corrado & Paola Samà, 2012. "Inside the Life Satisfaction Blackbox," AICCON Working Papers 116-2012, Associazione Italiana per la Cultura della Cooperazione e del Non Profit.
    17. Joseph J. Sabia, 2007. "Reading, Writing, And Sex: The Effect Of Losing Virginity On Academic Performance," Economic Inquiry, Western Economic Association International, vol. 45(4), pages 647-670, October.
    18. Hruschka, Harald, 2010. "Considering endogeneity for optimal catalog allocation in direct marketing," European Journal of Operational Research, Elsevier, vol. 206(1), pages 239-247, October.
    19. Arthur Lewbel, 2019. "The Identification Zoo: Meanings of Identification in Econometrics," Journal of Economic Literature, American Economic Association, vol. 57(4), pages 835-903, December.
    20. Biørn, Erik, 2017. "Identification and Method of Moments Estimation in Polynomial Measurement Error Models," Memorandum 01/2017, Oslo University, Department of Economics.

    More about this item

    Keywords

    ;
    ;
    ;

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:edn:esedps:321. See general information about how to correct material in RePEc.

    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: Research Office (email available below). General contact details of provider: https://edirc.repec.org/data/deediuk.html .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.