Is it possible have an SES where the component equations are probabilistic, say, logit or probit? I am evaluating a number of quality metrics of services provided by a number of providers. The metrics are binary (pass/fail quality certification). The obvious approach would appear to be to estimate the logit/probit equations, conditional upon provider characteristics, jointly in something like an SUR, but I can't find anything like that in the discrete choice literature.
This is possible. The type of model you need is called multivariate probit. For a textbook treatment you can refer to Greene's Econometric Analysis.
However, from the computational point of view, these models can be laborious. Convergence can be slow or can even fail.
I am also trying to find a solution to it. But the current available (user-written in stata) command is so slow that I got no solution. A simple solution I am using is the propensity score matching method that solves selective treatment effect. it works only when you have large observations. The estimated variance is tricky. so you have to estimate relative risk that has a neat variance estimator and convert the relative risk (rr) to probability change as a result of the other binary variable (the treatment variable). The confidence interval is not available although you can test the significance of the using rr. The Heckman model only estimate effect of a treatment (binary variable) on continuous output.