Probably this question is better placed at stackoverflow. Technically, in order to predict out of the sample, you need the full set of variables in the estimation. So after carrying out the estimation, you could do something like this. I assume your last observed year is 2010.
* expand US data for the last observed data point.
expand 20 if country == "US" & year == 2010
bys country year: replace year = . if _n > 1
bys country (year): replace year = year[_n-1] if year == .
estimates restore panel_estimation
* check whether that worked
tw line outcome_var year if country == "US"
This will predict a value based on your panel estimation also for the newly created observations. I am not judging whether it makes sense to extrapolate a time-trend over three years to twenty years ;)