The case of a dichotomous dependent variable is treated in Dubin & Rivers,
"Selection Bias in Linear Regression, Logit, and Probit Models,"
Sociological Methods & Research (1989), reprinted in Fox & Long, Modern
Methods of Data Analysis (1990).

No new issues arise with an ordered probit model. (The logit model involves
a non-standard and somewhat restrictive specification of the covariance
parameter.) We provide the likelihood for FIML estimation (which was
implemented in SST) as well as a suggestion for a method of moments
estimator. Also, there is an easy specification test for selection bias
that does not require estimation of the full likelihood. All of these could
be easily adapted for an ordered response outcome equation.

Nowadays, there are semi-parametric alternatives. As well as a raging debate
on what the right specification is for this problem.

Doug Rivers

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