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Demand Analysis as an Ill-Posed Inverse Problem with Semiparametric Specification


  • Stefan Hoderlein

    (Boston College)

  • Hajo Holzmann

    () (Karlsruhe University)


In this paper we are concerned with analyzing the behavior of a semiparametric estimator which corrects for endogeneity in a nonparametric regression by assuming mean independence of residuals from instruments only. Because it is common in many applications, we focus on the case where endogenous regressors and additional instruments are jointly normal, conditional on exogenous regressors. This leads to a severely ill-posed inverse problem. In this setup, we show first how to test for conditional normality. More importantly, we then establish how to exploit this knowledge when constructing an estimator, and we derive results characterizing the large sample behavior of such an estimator. In addition, in a Monte Carlo experiment we analyze the finite sample behavior of the proposed estimator. Our application comes from consumer demand. We obtain new and interesting findings that highlight both the advantages, and the difficulties of an approach which leads to ill-posed inverse problems. Finally, we discuss the somewhat problematic relationship between nonparametric instrumental variable models, and the recently emphasized issue of unobserved heterogeneity in structural models.

Suggested Citation

  • Stefan Hoderlein & Hajo Holzmann, 2008. "Demand Analysis as an Ill-Posed Inverse Problem with Semiparametric Specification," Boston College Working Papers in Economics 752, Boston College Department of Economics.
  • Handle: RePEc:boc:bocoec:752

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    References listed on IDEAS

    1. Ait-Sahalia, Yacine & Bickel, Peter J. & Stoker, Thomas M., 2001. "Goodness-of-fit tests for kernel regression with an application to option implied volatilities," Journal of Econometrics, Elsevier, vol. 105(2), pages 363-412, December.
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    Cited by:

    1. Yevgeniy Kovchegov & Nese Yildiz, 2014. "Orthogonal Polynomials for Seminonparametric Instrumental Variables Model," Papers 1409.1620,
    2. 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.
    3. repec:bpj:jecome:v:6:y:2017:i:1:p:25:n:5 is not listed on IDEAS
    4. Gagliardini, Patrick & Scaillet, Olivier, 2012. "Tikhonov regularization for nonparametric instrumental variable estimators," Journal of Econometrics, Elsevier, vol. 167(1), pages 61-75.
    5. Shaw Philip & Cohen Michael Andrew & Chen Tao, 2016. "Nonparametric Instrumental Variable Estimation in Practice," Journal of Econometric Methods, De Gruyter, vol. 5(1), pages 153-177, January.
    6. Centorrino Samuele & Feve Frederique & Florens Jean-Pierre, 2017. "Additive Nonparametric Instrumental Regressions: A Guide to Implementation," Journal of Econometric Methods, De Gruyter, vol. 6(1), pages 1-25, January.


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