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On the estimation of a binary response model in a selected population.

Francesco Claudio Stingo1, Elena Stanghellini2, Rosa Capobianco3

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Summary

This study introduces a generalized Probit model using an extended skew-normal distribution to handle binary data with selectivity bias. The method allows for parameter estimation and inference, even when selection affects unmeasured factors.

Keywords:
directed acyclic graphextended skew-normal distributionhidden truncationinstrumental variablesself-selectionunobserved confounder

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Area of Science:

  • Econometrics
  • Biostatistics
  • Statistical Modeling

Background:

  • Selectivity bias can distort findings in binary response models.
  • Traditional Probit models may not adequately address complex selection mechanisms.
  • Unobserved factors influencing selection require careful consideration.

Purpose of the Study:

  • To propose a generalized Probit model incorporating an extended skew-normal distribution.
  • To model binary response variables in the presence of selectivity bias.
  • To provide methods for parameter estimation and inference on selection parameters.

Main Methods:

  • Generalization of the Probit model using extended skew-normal cumulative distribution as a link function.
  • Maximum Likelihood (ML) estimation for model parameters.
  • Inference on parameters quantifying the degree of selection.

Main Results:

  • The proposed model effectively handles selectivity bias when the selection mechanism influences unmeasured factors.
  • The model remains valid even when the standard assumption of selection independence from explanatory variables is violated, provided other conditional independencies hold.
  • Instrumental variable approaches are applicable within this framework, particularly at the second stage of estimation.

Conclusions:

  • The extended skew-normal Probit model offers a flexible approach to binary data analysis with selection bias.
  • The methodology provides a robust framework for estimating selection effects and their impact.
  • The model's derivation ensures applicability in scenarios where instrumental variables are relevant for addressing endogeneity.