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Ill-Posed Inverse Problems in Economics

Author

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  • Joel L. Horowitz

    (Department of Economics, Northwestern University, Evanston, Illinois 60208)

Abstract

A parameter of an econometric model is identified if there is a one-to-one or many-to-one mapping from the population distribution of the available data to the parameter. Often, this mapping is obtained by inverting a mapping from the parameter to the population distribution. If the inverse mapping is discontinuous, then estimation of the parameter usually presents an ill-posed inverse problem. Such problems arise in many settings in economics and other fields in which the parameter of interest is a function. This article explains how ill-posedness arises and why it causes problems for estimation. The need to modify or regularize the identifying mapping is explained, and methods for regularization and estimation are discussed. Methods for forming confidence intervals and testing hypotheses are summarized. It is shown that a hypothesis test can be more precise in a certain sense than an estimator. An empirical example illustrates estimation in an ill-posed setting in economics.

Suggested Citation

  • Joel L. Horowitz, 2014. "Ill-Posed Inverse Problems in Economics," Annual Review of Economics, Annual Reviews, vol. 6(1), pages 21-51, August.
    • Handle: RePEc:anr:reveco:v:6:y:2014:p:21-51
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    File URL: http://www.annualreviews.org/doi/abs/10.1146/annurev-economics-080213-041213
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    Citations

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    Cited by:

    1. Steven Berry & Philip Haile, 2016. "Identification in Differentiated Products Markets," Annual Review of Economics, Annual Reviews, vol. 8(1), pages 27-52, October.
    2. Enache, Andreea & Florens, Jean-Pierre, 2020. "Quantile Analysis of "Hazard-Rate" Game Models," TSE Working Papers 20-1117, Toulouse School of Economics (TSE).
    3. Marie-Hélène Felt, 2018. "A Look Inside the Box: Combining Aggregate and Marginal Distributions to Identify Joint Distributions," Staff Working Papers 18-29, Bank of Canada.
    4. Samuele CENTORRINO & Jeffrey S. RACINE, 2017. "Semiparametric Varying Coefficient Models with Endogenous Covariates," Annals of Economics and Statistics, GENES, issue 128, pages 261-295.
    5. Ziyu Wang & Yucen Luo & Yueru Li & Jun Zhu & Bernhard Scholkopf, 2022. "Spectral Representation Learning for Conditional Moment Models," Papers 2210.16525, arXiv.org, revised Dec 2022.
    6. Ziyu Wang & Yuhao Zhou & Jun Zhu, 2022. "Fast Instrument Learning with Faster Rates," Papers 2205.10772, arXiv.org, revised Oct 2022.
    7. Yu, Ping & Phillips, Peter C.B., 2018. "Threshold regression with endogeneity," Journal of Econometrics, Elsevier, vol. 203(1), pages 50-68.
    8. 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.
    9. Jean-Jacques Forneron, 2019. "A Sieve-SMM Estimator for Dynamic Models," Papers 1902.01456, arXiv.org, revised Jan 2023.

    More about this item

    Keywords

    regularization; nonparametric estimation; density estimation; deconvolution; nonparametric instrumental variables; Fredholm equation;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C26 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Instrumental Variables (IV) Estimation

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