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Nonparametric Identification and Semiparametric Estimation of Classical Measurement Error Models Without Side Information


  • S. M. Schennach
  • Yingyao Hu


Virtually all methods aimed at correcting for covariate measurement error in regressions rely on some form of additional information (e.g., validation data, known error distributions, repeated measurements, or instruments). In contrast, we establish that the fully nonparametric classical errors-in-variables model is identifiable from data on the regressor and the dependent variable alone, unless the model takes a very specific parametric form. This parametric family includes (but is not limited to) the linear specification with normally distributed variables as a well-known special case. This result relies on standard primitive regularity conditions taking the form of smoothness constraints and nonvanishing characteristic functions' assumptions. Our approach can handle both monotone and nonmonotone specifications, provided the latter oscillate a finite number of times. Given that the very specific unidentified parametric functional form is arguably the exception rather than the rule, this identification result should have a wide applicability. It leads to a new perspective on handling measurement error in nonlinear and nonparametric models, opening the way to a novel and practical approach to correct for measurement error in datasets where it was previously considered impossible (due to the lack of additional information regarding the measurement error). We suggest an estimator based on non/semiparametric maximum likelihood, derive its asymptotic properties, and illustrate the effectiveness of the method with a simulation study and an application to the relationship between firm investment behavior and market value, the latter being notoriously mismeasured. Supplementary materials for this article are available online.

Suggested Citation

  • S. M. Schennach & Yingyao Hu, 2013. "Nonparametric Identification and Semiparametric Estimation of Classical Measurement Error Models Without Side Information," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 177-186, March.
  • Handle: RePEc:taf:jnlasa:v:108:y:2013:i:501:p:177-186
    DOI: 10.1080/01621459.2012.751872

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    References listed on IDEAS

    1. Susanne M. Schennach, 2004. "Estimation of Nonlinear Models with Measurement Error," Econometrica, Econometric Society, vol. 72(1), pages 33-75, January.
    2. Yingyao Hu & Susanne M. Schennach, 2008. "Instrumental Variable Treatment of Nonclassical Measurement Error Models," Econometrica, Econometric Society, vol. 76(1), pages 195-216, January.
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    Cited by:

    1. Pierre‐André Chiappori & Ju Hyun Kim, 2017. "A note on identifying heterogeneous sharing rules," Quantitative Economics, Econometric Society, vol. 8(1), pages 201-218, March.
    2. Aurélie Bertrand & Ingrid Van Keilegom & Catherine Legrand, 2019. "Flexible parametric approach to classical measurement error variance estimation without auxiliary data," Biometrics, The International Biometric Society, vol. 75(1), pages 297-307, March.
    3. Kato, Kengo & Sasaki, Yuya, 2019. "Uniform confidence bands for nonparametric errors-in-variables regression," Journal of Econometrics, Elsevier, vol. 213(2), pages 516-555.
    4. 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.
    5. Susanne M. Schennach, 2014. "Entropic Latent Variable Integration via Simulation," Econometrica, Econometric Society, vol. 82(1), pages 345-385, January.
    6. Chesher, Andrew, 2017. "Understanding the effect of measurement error on quantile regressions," Journal of Econometrics, Elsevier, vol. 200(2), pages 223-237.
    7. Garcia, Tanya P. & Ma, Yanyuan, 2017. "Simultaneous treatment of unspecified heteroskedastic model error distribution and mismeasured covariates for restricted moment models," Journal of Econometrics, Elsevier, vol. 200(2), pages 194-206.
    8. Lance Lochner & Youngki Shin, 2014. "Understanding Earnings Dynamics: Identifying and Estimating the Changing Roles of Unobserved Ability, Permanent and Transitory Shocks," University of Western Ontario, Centre for Human Capital and Productivity (CHCP) Working Papers 20142, University of Western Ontario, Centre for Human Capital and Productivity (CHCP).
    9. Ben-Moshe, Dan & D’Haultfœuille, Xavier & Lewbel, Arthur, 2017. "Identification of additive and polynomial models of mismeasured regressors without instruments," Journal of Econometrics, Elsevier, vol. 200(2), pages 207-222.
    10. Bustamante, M. Cecilia, 2016. "How Do Frictions Affect Corporate Investment? A Structural Approach," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 51(6), pages 1863-1895, December.
    11. Susanne M. Schennach, 2012. "Measurement error in nonlinear models - a review," CeMMAP working papers CWP41/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Gospodinov, Nikolay & Komunjer, Ivana & Ng, Serena, 2017. "Simulated minimum distance estimation of dynamic models with errors-in-variables," Journal of Econometrics, Elsevier, vol. 200(2), pages 181-193.
    13. Ben-Moshe, Dan, 2018. "Identification Of Joint Distributions In Dependent Factor Models," Econometric Theory, Cambridge University Press, vol. 34(1), pages 134-165, February.
    14. Hahn, Jinyong & Ridder, Geert, 2017. "Instrumental variable estimation of nonlinear models with nonclassical measurement error using control variables," Journal of Econometrics, Elsevier, vol. 200(2), pages 238-250.
    15. Daniel Wilhelm, 2015. "Identification and estimation of nonparametric panel data regressions with measurement error," CeMMAP working papers CWP34/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    16. Stefan Hoderlein & Bettina Siflinger & Joachim Winter, 2015. "Identification of structural models in the presence of measurement error due to rounding in survey responses," Boston College Working Papers in Economics 869, Boston College Department of Economics.
    17. Hu, Yingyao, 2017. "The econometrics of unobservables: Applications of measurement error models in empirical industrial organization and labor economics," Journal of Econometrics, Elsevier, vol. 200(2), pages 154-168.
    18. Nikolay Gospodinov & Ivana Komunjer & Serena Ng, 2014. "Minimum Distance Estimation of Dynamic Models with Errors-In-Variables," FRB Atlanta Working Paper 2014-11, Federal Reserve Bank of Atlanta.
    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. Yingyao Hu, 2015. "Microeconomic models with latent variables: applications of measurement error models in empirical industrial organization and labor economics," CeMMAP working papers CWP03/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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