It is not because it is pre-defined, that it is exported. For instance, on my machine , both PATH and HOST (not HOSTNAME) are defined with bash or zsh, but only PATH is exported.

I have:

$ export | grep '^PATH'
PATH=/opt/bin:/usr/local/bin:/bin:/usr/bin:/sbin:/usr/sbin
$ export | grep '^HOST'
$

You can add the export directive in your shell “rc” file.