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Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization

In: Handbook of Econometrics

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  • Carrasco, Marine
  • Florens, Jean-Pierre
  • Renault, Eric

Abstract

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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This chapter was published in:

  • J.J. Heckman & E.E. Leamer (ed.), 2007. "Handbook of Econometrics," Handbook of Econometrics, Elsevier, edition 1, volume 6, number 6b.
    This item is provided by Elsevier in its series Handbook of Econometrics with number 6b-77.

    Handle: RePEc:eee:ecochp:6b-77

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  • J.J. Heckman & E.E. Leamer (ed.), 2007. "Handbook of Econometrics," Handbook of Econometrics, Elsevier, edition 1, volume 6, number 6b.

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    Cited by:
    1. Xiaohong Chen & Markus Reiss, 2007. "On rate optimality for ill-posed inverse problems in econometrics," CeMMAP working papers CWP20/07, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Xiaohong Chen & Yingyao Hu, 2006. "Identification and Inference of Nonlinear Models Using Two Samples with Arbitrary Measurement Errors," Cowles Foundation Discussion Papers 1590, Cowles Foundation for Research in Economics, Yale University.
    3. Peter Sandholt Jensen & Allan H. Würtz, 2006. "On determining the importance of a regressor with small and undersized samples," Economics Working Papers 2006-08, School of Economics and Management, University of Aarhus.
    4. Marek Jarociński & Albert Marcet, 2010. "Autoregressions in small samples, priors about observables and initial conditions," Working Paper Series 1263, European Central Bank.
    5. Darolles, Serge & Fan, Yanqing & Florens, Jean-Pierre & Renault, Eric, 2011. "Nonparametric Instrumental Regression," Open Access publications from Université Paris-Dauphine urn:hdl:123456789/8440, Université Paris-Dauphine.
    6. Van Keilegom, Ingrid & Vanhems, Anne, 2011. "Semiparametric transformation model with endogeneity: a control function approach," TSE Working Papers 11-243, Toulouse School of Economics (TSE).
    7. Xiaohong Chen & Demian Pouzo, 2008. "Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals," Cowles Foundation Discussion Papers 1650RR, Cowles Foundation for Research in Economics, Yale University, revised Jan 2011.
    8. Xiaohong Chen & Demian Pouzo, 2008. "Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Moments," Cowles Foundation Discussion Papers 1650, Cowles Foundation for Research in Economics, Yale University, revised Oct 2008.
    9. Joel Horowitz & Sokbae 'Simon' Lee, 2006. "Nonparametric instrumental variables estimation of a quantile regression model," CeMMAP working papers CWP09/06, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    10. Johannes, Jan & Van Bellegem, Sébastien & Vanhems, Anne, 2011. "Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator," Econometric Theory, Cambridge University Press, vol. 27(03), pages 522-545, June.
    11. Jeremy T. Fox & Kyoo il Kim, 2011. "A Simple Nonparametric Approach to Estimating the Distribution of Random Coefficients in Structural Models," NBER Working Papers 17283, National Bureau of Economic Research, Inc.
    12. Eric Gautier & Yuichi Kitamura, 2009. "Nonparametric Estimation in Random Coefficients Binary Choice Models," Cowles Foundation Discussion Papers 1721, Cowles Foundation for Research in Economics, Yale University.
    13. Peter C.B. Phillips & Liangjun Su, 2009. "Nonparametric Structural Estimation via Continuous Location Shifts in an Endogenous Regressor," Cowles Foundation Discussion Papers 1702, Cowles Foundation for Research in Economics, Yale University.
    14. Eric Gautier & Yuichi Kitamura, 2011. "Nonparamatric estimation in random coefficients binary choice models," Working Papers hal-00403939, HAL.

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