You need to specify a list of allowed mime types.
You could make it easy by just getting the allowed mime types like:
$file = $_FILES['the-file'];
$upload_file = wp_handle_upload($file, array(
'test_form' => false,
'mimes' => get_allowed_mime_types()
));
If you look at the codex for Default allowed mime types, you could manually specify which ever mime types you want in that format.
An example would be like this answer.