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Identification & Information in Monotone Binary Models


  • Thierry Magnac
  • Eric Maurin



Let, y, a binary outcome, v a continuous explanatory variable and x some other explanatory variables. We study inference on the parameter b of the semiparametric binary regression model y=1(xb+v+e>0). We show that the set-up introduced by Lewbel (2000) that is, an uncorrelated-error restriction (E(x'e)=0) combined with a partial-independance assumption (F(e/v,x)=F(e/x)) and a large support assumption (Supp(-xb-e) c Supp(v)) provides exact identification of b and F(e/x). The two restrictions that the population distribution of the random variable w=(y,v,x) should satisfy are Monotone (1) and Large Support (2) conditions: (1) E(y/v,x) is monotone in v and (2) E(y/v,x) varies from 0 to 1 when v varies over its support. Moreover, we show that Lewbel's moment estimator attains the semi-parametric efficiency bound in the set of latent models that he considers. Yet, the uncorrelated-error and partial-independence assumptions are not sufficient to identify b when the support of v is not sufficiently rich. We propose intuitive additional restrictions on the tails of the conditional distribution of e under which b remains exactly identified even when condition(2) is not satisfied. In such a case, Monte-Carlo experiments show that the estimation performs well in moderately small samples. An extension to ordered choice models is provided.

Suggested Citation

  • Thierry Magnac & Eric Maurin, 2003. "Identification & Information in Monotone Binary Models," Research Unit Working Papers 0309, Laboratoire d'Economie Appliquee, INRA.
  • Handle: RePEc:lea:leawpi:0309

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    Cited by:

    1. Lewbel, Arthur, 2007. "Endogenous selection or treatment model estimation," Journal of Econometrics, Elsevier, vol. 141(2), pages 777-806, December.
    2. Magnac, Thierry & Maurin, Eric, 2007. "Identification and information in monotone binary models," Journal of Econometrics, Elsevier, vol. 139(1), pages 76-104, July.
    3. Bontemps, Christophe & Nauges, Céline, 2017. "Endogenous Variables in Binary Choice Models: Some Insights for Practitioners," TSE Working Papers 17-855, Toulouse School of Economics (TSE).
    4. Stewart, Mark B., 2005. "A comparison of semiparametric estimators for the ordered response model," Computational Statistics & Data Analysis, Elsevier, vol. 49(2), pages 555-573, April.
    5. Chen, Songnian & Khan, Shakeeb & Tang, Xun, 2016. "Informational content of special regressors in heteroskedastic binary response models," Journal of Econometrics, Elsevier, vol. 193(1), pages 162-182.
    6. Lewbel, Arthur & Schennach, Susanne M., 2007. "A simple ordered data estimator for inverse density weighted expectations," Journal of Econometrics, Elsevier, vol. 136(1), pages 189-211, January.

    More about this item


    Binary models; semiparametric methods; efficiency bounds;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities

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