Partial Identification in Triangular Systems of Equations With Binary Dependent Variables


  • Azeem M. Shaikh,

    1. Dept. of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637, U.S.A.;
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  • Edward J. Vytlacil

    1. Dept. of Economics, Yale University, New Haven, CT 06520-8281, U.S.A.;
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    • An earlier version of this paper titled “Threshold Crossing Models and Bounds on Treatment Effects: A Nonparametric Analysis” appeared in May 2005 as NBER Technical Working Paper 307. We would like to thank Hide Ichimura, Jim Heckman, Whitney Newey, and Jim Powell for very helpful comments on this paper. This research was conducted in part while Edward Vytlacil was in residence at Hitotsubashi University. This research was supported by NSF SES-05-51089 and DMS-08-20310.


This paper studies the special case of the triangular system of equations in Vytlacil and Yildiz (2007), where both dependent variables are binary but without imposing the restrictive support condition required by Vytlacil and Yildiz (2007) for identification of the average structural function (ASF) and the average treatment effect (ATE). Under weak regularity conditions, we derive upper and lower bounds on the ASF and the ATE. We show further that the bounds on the ASF and ATE are sharp under some further regularity conditions and an additional restriction on the support of the covariates and the instrument.