non-classical measurement error in a binary outcome model I have a binary outcome model that I am estimating with a probit, so
$$\Pr(Y=1\vert x,z)=\Phi(\alpha +\beta\cdot x^* + z'\gamma)$$
I am interested in the marginal effect of $x^*$ on $\Pr(Y=1\vert x,z)$. The $z$s are other covariates that don't have measurement error.
Unfortunately, I don't get to observe $x^*$. I get to see $x \le x^*$. That is, $x$ is a downward-biased version of $x^*$. In my setting, $x$ is a duration since an event, but people are inattentive in heterogeneous ways, so the actual duration that they respond to is always longer than the measured one. That gap varies across individuals. 


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*From some simulations I have done, the marginal effect is attenuated (biased towards zero). Is there an analytic proof of this somewhere in the literature that I can cite?

*Is there anything I can do to remedy the bias or to quantify its magnitude?


I would prefer to stick with the probit setting, but going to a linear probability model would be an acceptable if it makes things easier. Dropping the other covariates would be OK as well.
 A: Citing from the survey article by Chen et al. (2011) "Nonlinear Models of Measurement Errors", Journal of Economic Literature:

The approximate bias depends on the derivatives of the regression
  function with respect to the mismeasured regressor and the curvature
  of the distribution functions of the true regressor and the
  mismeasured regressors. Locally the conditional mean function of the
  dependent variable given the mismeasured regressors is smoother than
  the conditional mean function given the true regressors, in analog to
  the attenuation bias on the regression coefficient in linear models.

This was a result from Chesher (1991). Also Carroll et al (1984) derive the bias in binary regression models with measurement error. The survey article discusses several types of measurement error in nonlinear models and potential ways around it.
For a practical implementation in Stata have a look at the Stata site titled "Stata software for generalized linear measurement error models" [link]. If you have an idea about the variance of the measurement error then things seem to become a little less complicated.
