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Well-posedness of measurement error models for self-reported data

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  • An, Yonghong
  • Hu, Yingyao

Abstract

This paper considers the widely admitted ill-posed inverse problem for measurement error models: estimating the distribution of a latent variable X∗ from an observed sample of X, a contaminated measurement of X∗. We show that the inverse problem is well-posed for self-reporting data under the assumption that the probability of truthful reporting is nonzero, which is supported by empirical evidences. Comparing with ill-posedness, well-posedness generally can be translated into faster rates of convergence for the nonparametric estimators of the latent distribution. Therefore, our optimistic result on well-posedness is of importance in economic applications, and it suggests that researchers should not ignore the point mass at zero in the measurement error distribution when they model measurement errors with self-reported data. We also analyze the implications of our results on the estimation of classical measurement error models. Then by both a Monte Carlo study and an empirical application, we show that failing to account for the nonzero probability of truthful reporting can lead to significant bias on estimation of the latent distribution.

Suggested Citation

  • An, Yonghong & Hu, Yingyao, 2012. "Well-posedness of measurement error models for self-reported data," Journal of Econometrics, Elsevier, vol. 168(2), pages 259-269.
  • Handle: RePEc:eee:econom:v:168:y:2012:i:2:p:259-269
    DOI: 10.1016/j.jeconom.2012.01.036
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    5. Kassas, Bachir & Palma, Marco A. & Anderson, David P., 2018. "Fine-tuning willingness-to-pay estimates in second price auctions for market goods," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 77(C), pages 50-61.
    6. Juan Carlos Escanciano & Stefan Hoderlein & Arthur Lewbel & Oliver Linton & Sorawoot Srisuma, 2010. "Nonparametric Euler Equation Identification and Estimation," Boston College Working Papers in Economics 757, Boston College Department of Economics, revised 15 Mar 2020.
    7. Andrews, Donald W.K., 2017. "Examples of L2-complete and boundedly-complete distributions," Journal of Econometrics, Elsevier, vol. 199(2), pages 213-220.
    8. Toru Kitagawa & Martin Nybom & Jan Stuhler, 2018. "Measurement error and rank correlations," CeMMAP working papers CWP28/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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    More about this item

    Keywords

    Well-posed; Ill-posed; Inverse problem; Fredholm integral equation; Deconvolution; Rate of convergence; Measurement error model; Self-reported data; Survey data;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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