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Identification & Information in Monotone Binary Models


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  • 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.

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Paper provided by Laboratoire d'Economie Appliquee, INRA in its series Research Unit Working Papers with number 0309.

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Length: 53 pages
Date of creation: Jun 2003
Date of revision:
Handle: RePEc:lea:leawpi:0309

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Keywords: Binary models; semiparametric methods; efficiency bounds;

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Cited by:
  1. Songnian Chen & Shakeeb Khan & Xun Tang, 2013. "Informational Content of Special Regressors in Heteroskedastic Binary Response Models," PIER Working Paper Archive 13-021, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
  2. Lewbel, Arthur, 2007. "Endogenous selection or treatment model estimation," Journal of Econometrics, Elsevier, vol. 141(2), pages 777-806, December.
  3. 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.
  4. 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.
  5. Magnac, Thierry & Maurin, Eric, 2003. "Identification and Information in Monotone Binary Models," IDEI Working Papers 180, Institut d'Économie Industrielle (IDEI), Toulouse, revised Oct 2004.


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