Specifying cluster or robust does not seem to change the predicted
probabilities from oprobit. Does it? Shouldn't it?
Intuitively/naively, I am thinking that for an observation for which
the variance of the random part of the latent index is high, there is
a greater chance of ending up further away from what the
deterministic part of the latent index alone might suggest.
Interesting point about the intuition, but it is not right in this
case. The predict command does allow you to also estimate the
standard error of the linear prediction, which I think gets at the
issue you are raising, and the standard error of the prediction is
affected by whether or not you use cluster or robust.