I just a look at the last time Moyes' contract was turning into a saga. It was the beginning of 2008/2009. Here are the results until he signed:
http://www.evertonresults.com/200809.htm
He signed on October 13. Up until that point the record was 5 losses, 2 wins and 3 draws. After he signed, we went the rest of the year losing a further six times. My point is, I think the contract dragging on has a negative effect.