redirect time to file

[Erik Xavior]

$ time echo hi > file

real    0m0.000s
user    0m0.000s
sys     0m0.000s
$

why doesn't it redirect the:

real    0m0.000s
user    0m0.000s
sys     0m0.000s

to the "file"?

even
time echo hi 2>&1 > file
doesnt work :O