Inverse problems can be described as functional equations where the value of the function is known or easily estimable but the argument is unknown. Many problems in econometrics can be stated in the form of inverse problems where the argument itself is a function. For example, consider a nonlinear regression where the functional form is the object of interest. One can readily estimate the conditional expectation of the dependent variable given a vector of instruments. From this estimate, one would like to recover the unknown functional form. This chapter provides an introduction to the estimation of the solution to inverse problems. It focuses mainly on integral equations of the first kind. Solving these equations is particularly challenging as the solution does not necessarily exist, may not be unique, and is not continuous. As a result, a regularized (or smoothed) solution needs to be implemented. We review different regularization methods and study the properties of the estimator. Integral equations of the first kind appear, for example, in the generalized method of moments when the number of moment conditions is infinite, and in the nonparametric estimation of instrumental variable regressions. In the last section of this chapter, we investigate integral equations of the second kind, whose solutions may not be unique but are continuous. Such equations arise when additive models and measurement error models are estimated nonparametrically.
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ReDIF This chapter was published in: J.J. Heckman & E.E. Leamer (ed.) Handbook of Econometrics, , chapter 77, pages , 2007.
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This chapter was published in the following book, which is listed on IDEAS: J.J. Heckman & E.E. Leamer (ed.), 2007.
"Handbook of Econometrics,"
Handbook of Econometrics,
Elsevier,
edition 1, volume 6, number 6b, September.
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