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Semiparametric binary regression models under shape constraints with an application to Indian schooling data


  • Banerjee, Moulinath
  • Mukherjee, Debasri
  • Mishra, Santosh


We consider estimation of the regression function in a semiparametric binary regression model defined through an appropriate link function (with emphasis on the logistic link) using likelihood-ratio based inversion. The dichotomous response variable [Delta] is influenced by a set of covariates that can be partitioned as (X,Z) where Z (real valued) is the covariate of primary interest and X (vector valued) denotes a set of control variables. For any fixed X, the conditional probability of the event of interest ([Delta]=1) is assumed to be a non-decreasing function of Z. The effect of the control variables is captured by a regression parameter [beta]. We show that the baseline conditional probability function (corresponding to X=0) can be estimated by isotonic regression procedures and develop a likelihood ratio based method for constructing asymptotic confidence intervals for the conditional probability function (the regression function) that avoids the need to estimate nuisance parameters. Interestingly enough, the calibration of the likelihood ratio based confidence sets for the regression function no longer involves the usual [chi]2 quantiles, but those of the distribution of a new random variable that can be characterized as a functional of convex minorants of Brownian motion with quadratic drift. Confidence sets for the regression parameter [beta] can however be constructed using asymptotically [chi]2 likelihood ratio statistics. The finite sample performance of the methods are assessed via a simulation study. The techniques of the paper are applied to data sets on primary school attendance among children belonging to different socio-economic groups in rural India.

Suggested Citation

  • Banerjee, Moulinath & Mukherjee, Debasri & Mishra, Santosh, 2009. "Semiparametric binary regression models under shape constraints with an application to Indian schooling data," Journal of Econometrics, Elsevier, vol. 149(2), pages 101-117, April.
  • Handle: RePEc:eee:econom:v:149:y:2009:i:2:p:101-117

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    References listed on IDEAS

    1. Banerjee Moulinath & Wellner Jon A., 2005. "Score Statistics for Current Status Data: Comparisons with Likelihood Ratio and Wald Statistics," The International Journal of Biostatistics, De Gruyter, vol. 1(1), pages 1-29, August.
    2. Thierry Magnac & Eric Maurin, 2008. "Partial Identification in Monotone Binary Models: Discrete Regressors and Interval Data," Review of Economic Studies, Oxford University Press, vol. 75(3), pages 835-864.
    3. Charles F. Manski & Elie Tamer, 2002. "Inference on Regressions with Interval Data on a Regressor or Outcome," Econometrica, Econometric Society, vol. 70(2), pages 519-546, March.
    4. Magnac, Thierry & Maurin, Eric, 2007. "Identification and information in monotone binary models," Journal of Econometrics, Elsevier, vol. 139(1), pages 76-104, July.
    5. Lavy, Victor, 1996. "School supply constraints and children's educational outcomes in rural Ghana," Journal of Development Economics, Elsevier, vol. 51(2), pages 291-314, December.
    6. Dreze, Jean & Kingdon, Geeta Gandhi, 2001. "School Participation in Rural India," Review of Development Economics, Wiley Blackwell, vol. 5(1), pages 1-24, February.
    7. P. Groeneboom & G. Jongbloed, 2003. "Density estimation in the uniform deconvolution model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 57(1), pages 136-157.
    8. Handa, Sudhanshu, 2002. "Raising primary school enrolment in developing countries: The relative importance of supply and demand," Journal of Development Economics, Elsevier, vol. 69(1), pages 103-128, October.
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    Cited by:

    1. Shively, Thomas S. & Walker, Stephen G. & Damien, Paul, 2011. "Nonparametric function estimation subject to monotonicity, convexity and other shape constraints," Journal of Econometrics, Elsevier, vol. 161(2), pages 166-181, April.
    2. Delgado, Miguel A. & Escanciano, Juan Carlos, 2012. "Distribution-free tests of stochastic monotonicity," Journal of Econometrics, Elsevier, vol. 170(1), pages 68-75.


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