At about 45 minutes in, where Jay's mouse begins to move by itself, you can clearly see the wire as Jay is pulling it with his right hand that causes the mouse to move "by itself". Watching it in slow motion, it is clearly evident because the "haunted mouse" moves at the exact same time and rate as Jay's right hand.
CORRECTION: The user above is mistaken. On closer inspection, Jay isn't in contact with the wire. He is actually holding a remote control. The "mouse" and wire move completely by themselves.