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INFERENCE ON A DISTRIBUTION FROM NOISY DRAWS

Published online by Cambridge University Press:  18 August 2022

Koen Jochmans  and
Martin Weidner
Koen Jochmans*
Affiliation:
Université Toulouse 1 Capitole
Martin Weidner
Affiliation:
University of Oxford
*
Address correspondence to Koen Jochmans, Toulouse School of Economics, Université Toulouse 1 Capitole, 1 esplanade de l’Université, 31080 Toulouse, France; e-mail: [email protected].
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Abstract

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We consider a situation where the distribution of a random variable is being estimated by the empirical distribution of noisy measurements of that variable. This is common practice in, for example, teacher value-added models and other fixed-effect models for panel data. We use an asymptotic embedding where the noise shrinks with the sample size to calculate the leading bias in the empirical distribution arising from the presence of noise. The leading bias in the empirical quantile function is equally obtained. These calculations are new in the literature, where only results on smooth functionals such as the mean and variance have been derived. We provide both analytical and jackknife corrections that recenter the limit distribution and yield confidence intervals with correct coverage in large samples. Our approach can be connected to corrections for selection bias and shrinkage estimation and is to be contrasted with deconvolution. Simulation results confirm the much-improved sampling behavior of the corrected estimators. An empirical illustration on heterogeneity in deviations from the law of one price is equally provided.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 1 , February 2024 , pp. 60 - 97
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We are grateful to Isaiah Andrews, Stéphane Bonhomme, Bo Honoré, Ryo Okui, and Peter Schmidt for comments, and to Arthur Lewbel and three referees for feedback on an earlier version. We also greatly appreciate the help of Ryo Okui and Mototsugu Shintani in providing us with the data used in our empirical illustration. Jochmans gratefully acknowledges financial support from the European Research Council through grant ERC-2016-StG-715787-MiMo and from the French Government and the ANR under the Investissements d’ Avenir program, grant ANR-17-EURE-0010. Weidner gratefully acknowledges support from the Economic and Social Research Council through the ESRC Centre for Microdata Methods and Practice grant RES-589-28-0001 and from the European Research Council grants ERC-2014-CoG-646917-ROMIA and ERC-2018-CoG-819086-PANEDA.

References

REFERENCES

Ahn, D., Choi, S., Gale, D., & Kariv, S. (2014) Estimating ambiguity aversion in a portfolio choice experiment. Quantitative Economics 5, 195–223.CrossRefGoogle Scholar
Alvarez, J. & Arellano, M. (2003) The time series and cross-section asymptotics of dynamic panel data estimators. Econometrica 71, 1121–1159.CrossRefGoogle Scholar
Barras, L., Gagliardini, P., & Scaillet, O. (2021) Skill, scale, and value creation in the mutual fund industry. Journal of Finance 77(1), 601–638.CrossRefGoogle Scholar
Bonhomme, S., Jochmans, K., & Robin, J.-M. (2016a) Estimating multivariate latent-structure models. Annals of Statistics 44, 540–563.CrossRefGoogle Scholar
Bonhomme, S., Jochmans, K., & Robin, J.-M. (2016b) Nonparametric estimation of finite mixtures from repeated measurements. Journal of the Royal Statistical Society, Series B 78, 211–229.CrossRefGoogle Scholar
Browning, M., Ejrnæs, M., & Alvarez, J. (2010) Modeling income processes with lots of heterogeneity. Review of Economic Studies 77, 1353–1381.CrossRefGoogle Scholar
Carroll, R.J. & Hall, P. (1988) Optimal rates of convergence for deconvoluting a density. Journal of the American Statistical Association 83, 1184–1186.CrossRefGoogle Scholar
Chamberlain, G. (1984) Chapter 22: Panel data. In Griliches, Z. & Intriligator, M. (eds.), Handbook of Econometrics . Handbooks in Economics, 2, pp. 1247–1315. Elsevier.Google Scholar
Chesher, A. (1991) The effect of measurement error. Biometrika 78, 451–462.CrossRefGoogle Scholar
Chesher, A. (2017) Understanding the effect of measurement error on quantile regressions. Journal of Econometrics 200, 223–237.CrossRefGoogle Scholar
Chetty, R., Friedman, J.N., & Rockoff, J.E. (2014) Measuring the impacts of teachers I: Evaluating bias in teacher value-added estimates. American Economic Review 104, 2593–2632.CrossRefGoogle Scholar
Crucini, M.J., Shintani, M., & Tsuruga, T. (2015) Noisy information, distance and law of one price dynamics across US cities. Journal of Monetary Economics 74, 52–66.CrossRefGoogle Scholar
Delaigle, A. & Meister, A. (2008) Density estimation with heteroscedastic error. Bernoulli 14, 562–579.CrossRefGoogle Scholar
Dhaene, G. & Jochmans, K. (2015) Split-panel jackknife estimation of fixed-effect models. Review of Economic Studies 82, 991–1030.CrossRefGoogle Scholar
Doss, H. & Gill, R.D. (1992) An elementary approach to weak convergence for quantile processes, with applications to censored survival data. Journal of the American Statistical Association 87(419), 869–877.CrossRefGoogle Scholar
Efron, B. (2011) Tweedie’s formula and selection bias. Journal of the American Statistical Association 106, 1602–1614.CrossRefGoogle ScholarPubMed
Efron, B. (2016) Empirical Bayes deconvolution estimates. Biometrika 103, 1–20.CrossRefGoogle Scholar
Evdokimov, K. & Zeleneev, A. (2020) Simple Estimation of Semiparametric Models with Measurement Errors. CeMMAP Working paper 08/22, Mimeo.Google Scholar
Fernández-Val, I. & Lee, J. (2013) Panel data models with nonadditive unobserved heterogeneity: Estimation and inference. Quantitative Economics 4, 453–481.CrossRefGoogle Scholar
Guvenen, F. (2009) An empirical investigation of labor income processes. Review of Economic Dynamics 12, 58–79.CrossRefGoogle Scholar
Hahn, J. & Kuersteiner, G. (2002) Asymptotically unbiased inference for a dynamic panel model with fixed effects when both $n$  and  $T$  are large. Econometrica 70, 1639–1657.CrossRefGoogle Scholar
Hahn, J. & Newey, W.K. (2004) Jackknife and analytical bias reduction for nonlinear panel models. Econometrica 72, 1295–1319.CrossRefGoogle Scholar
Horowitz, J.L. & Markatou, M. (1996) Semiparametric estimation of regression models from panel data. Review of Economic Studies 63, 145–168.CrossRefGoogle Scholar
Hu, Y. (2008) Identification and estimation of nonlinear models with misclassification error using instrumental variables: A general solution. Journal of Econometrics 144, 27–61.CrossRefGoogle Scholar
Hu, Y. & Schennach, S.M. (2008) Instrumental variable treatment of nonclassical measurement error models. Econometrica 76, 195–216.CrossRefGoogle Scholar
Jackson, C.K., Rockoff, J.E., & Staiger, D.O. (2014) Teacher effects and teacher related policies. Annual Review of Economics 6, 801–825.CrossRefGoogle Scholar
James, W. & Stein, C. (1961) Estimation with quadratic loss. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , vol. I, pp. 361–379. University of California Press.Google Scholar
Komlós, J., Major, P., & Tusnády, G. (1975) An approximation of partial sums of independent RV’-s, and the sample DF. I. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 32, 111–131.CrossRefGoogle Scholar
Li, T. & Vuong, Q. (1998) Nonparametric estimation of measurement error models using multiple indicators. Journal of Multivariate Analysis 65, 139–165.CrossRefGoogle Scholar
Magnac, T. & Roux, S. (2021) Heterogeneity and wage inequalities over the life cycle. European Economic Review 134, 103715.CrossRefGoogle Scholar
Maritz, J.S. & Jarrett, R.G. (1978) A note on estimating the variance of the sample median. Journal of the American Statistical Association 73, 194–196.CrossRefGoogle Scholar
Mason, D.M. (1981) Bounds for weighted empirical distribution functions. Annals of Probability 9, 881–884.CrossRefGoogle Scholar
Neyman, J. & Scott, E. (1948) Consistent estimates based on partially consistent observations. Econometrica 16, 1–32.CrossRefGoogle Scholar
Okui, R. & Yanagi, T. (2019) Panel data analysis with heterogeneous dynamics. Journal of Econometrics 212, 451–475.CrossRefGoogle Scholar
Okui, R. & Yanagi, T. (2020) Kernel estimation for panel data with heterogenous dynamics. The Econometrics Journal 23, 156–175.CrossRefGoogle Scholar
Parsley, D.C. & Wei, S.-J. (2001) Convergence to the law of one price without trade barriers or currency fluctuations. Quarterly Journal of Economics 111, 1211–1236.CrossRefGoogle Scholar
Robbins, H. (1956) An empirical Bayes approach to statistics. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability , vol. I, pp. 157–163. University of California Press.Google Scholar
Rockoff, J.E. (2004) The impact of individual teachers on student achievement: Evidence from panel data. American Economic Review: Papers & Proceedings 94, 247–252.CrossRefGoogle Scholar
Rosenthal, H.P. (1970) On the subspaces of ${L}_p$ ( $p>2$ ) spanned by sequences of independent random variables. Israel Journal of Mathematics 8, 273–303.CrossRefGoogle Scholar
Vivalt, E. (2015) Heterogeneous treatment effects in impact evaluation. American Economic Review: Papers & Proceedings 105, 467–470.CrossRefGoogle Scholar
Weinstein, A., Ma, Z., Brown, L.D., & Zhang, C.-H. (2018) Group-linear empirical Bayes estimates for a heteroscedastic normal mean. Journal of the American Statistical Association 113, 698–710.CrossRefGoogle Scholar