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Ill-posed inverse problems in economics

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  • Joel L. Horowitz

    (Institute for Fiscal Studies and Northwestern University)

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 where the parameter of interest is a function. This paper explains how ill-posedness arises and why it causes problems for estimation. The need to modify or 'regularise' the identifying mapping is explained, and methods for regularisation and estimation are discussed. Methods for forming confidence intervals and testing hypotheses are summarised. 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, 2013. "Ill-posed inverse problems in economics," CeMMAP working papers CWP37/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:37/13
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    File URL: http://www.cemmap.ac.uk/wps/cwp371313.pdf
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    References listed on IDEAS

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    1. Xiaohong Chen & Yin Jia Jeff Qiu, 2016. "Methods for Nonparametric and Semiparametric Regressions with Endogeneity: A Gentle Guide," Annual Review of Economics, Annual Reviews, vol. 8(1), pages 259-290, October.
    2. Luc Clair, 2022. "Nonparametric Instrumental Variable Estimation using Complex Survey Data," Departmental Working Papers 2022-01, The University of Winnipeg, Department of Economics.

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    Keywords

    regularisation; nonparametric estimation; density estimation; deconvolution; nonparametric instrumental variables; Fredholm equation;
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