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

*Yonghong An* and
*Yingyao Hu*

*Journal of Econometrics*, 2012, vol. 168, issue 2, 259-269

**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.

**Keywords:** Well-posed; Ill-posed; Inverse problem; Fredholm integral equation; Deconvolution; Rate of convergence; Measurement error model; Self-reported data; Survey data (search for similar items in EconPapers)

**JEL-codes:** C14 (search for similar items in EconPapers)

**Date:** 2012

**References:** View references in EconPapers View complete reference list from CitEc

**Citations:** View citations in EconPapers (16)

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http://www.sciencedirect.com/science/article/pii/S0304407612000462

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**Related works:**

Working Paper: Well-posedness of measurement error models for self-reported data (2009)

Working Paper: Well-Posedness of Measurement Error Models for Self-Reported Data (2009)

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:econom:v:168:y:2012:i:2:p:259-269

**DOI:** 10.1016/j.jeconom.2012.01.036

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