Ill-Posed Inverse Problems in Economics

Joel L. Horowitz ()
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Joel L. Horowitz: Department of Economics, Northwestern University, Evanston, Illinois 60208

Annual Review of Economics, 2014, vol. 6, issue 1, 21-51

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.

Keywords: regularization; nonparametric estimation; density estimation; deconvolution; nonparametric instrumental variables; Fredholm equation (search for similar items in EconPapers)
JEL-codes: C14 C26 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (13)

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