How to define variables in psql?

I know that one can store multiple psql commands and SQL statements in
a separate "script" file, and then "source" the script file, e.g. with
the \i command.

My question is, can I define variables in such a script file?
E.g. can I do something like the following unix-shell-like script?

-- beginning of script

DB=frobozz
DIR=/foo/bar/baz/quux

\c $frobozz
\i "$DIR/other_script.sql"
\i "$DIR/one_more_script.sql"
\i "$DIR/yas.sql"

\echo "done with $DB"

-- end of script

Thanks!

kj