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Edit: Of course the cross is necessary. Given that the resulting dataset of 200,000 obs * 1,000 iterations becomes unmanageable, would you have any suggestions on how this could still be plotted?

If you wanted to avoid cross you could probably do the calculation in Mata, as in my previous example, but this is a bit more complicated.

George's code already reduces the size of the dataset used in the prediction by throwing away duplicate combinations in the variables of interest (the duplicates drop line). Presumably you could take this further by predicting for a subset of values that cover the typical range for each variable? I suspect that Stata wouldn't be particularly happy about plotting 200,000 predicted values on a graph in any case.