How to do a South Park 'Real Head' character? The head moves up and down not the jaw.

Hey all, I have a client who wants South Park style 'real head' character designs for one of his clients (probably the most famous example is the Saddam Hussein character concept). The nutcracker jaw would be perfect but the SP character's head moves up and down and the jaw stays fixed, 

Is it possible to do this in Character Animator?