Ill-posed inverse problems in economics

Joel L. Horowitz*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)21-51
Number of pages31
JournalAnnual Review of Economics
StatePublished - Aug 2014


  • Fredholm equation
  • deconvolution
  • density estimation
  • nonparametric estimation
  • nonparametric instrumental variables
  • regularization

ASJC Scopus subject areas

  • Economics and Econometrics

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