Do Panama hats originate in Panama?

What we call the Panama Hat originated in Ecuador. It is woven
from a very soft fiber that grows only there.

It is called a Panama Hat because when they were building the Panama Canal,
President Theodore Roosevelt was given a hat by a Panamanian business man.
It fit him. He looked good wearing it, so he wore it the whole time he was down there.

Well, if the president wears that, so too must his staff, and they all bought and
wore the same kind of hat.

When they got back to Washington, they were asked about the hat. They said this is
the Panama Hat, we got them in Panama.

There is a real Panama Hat that is made in Panama by Panamanians and worn by
them but looks nothing like the one we call the Panama Hat.