Dear colleagues,

I would like to perform a statistical test of the difference between two point predictions obtained from separate regressions, but I don't seem to know how to go about it. This would seem to be a pretty basic exercise, but I seem also to ignore the statistics behind it, so I would very much appreciate someone's help.

To illustrate, consider this dataset:

Code:
clear
set seed 1234
set obs 1000
drawnorm x e
gen y1 = 2*x + 3 + e
gen y2 = 3*x + 4 + e

I run these two regressions, and I obtain the fitted values (and, for good measure, the standard error of the fitted values). To illustrate my question, let me also plot the regression lines.

Code:
reg y1 x
predict y1h, xb
predict se1, stdp
*
reg y2 x
predict y2h, xb
predict se2, stdp
*
twoway (line y1h x, sort) (line y2h x, sort)
I would like to test the null hypothesis that the difference between the conditional mean of y1 and y2, given (say) x = 2, is equal to zero. So, formally, H0: "E[y1|x=2] - E[y2|x=2] = 0".

How should I proceed?