That said "I can imagine" isn't terribly good science. Articles like https://en.wikipedia.org/wiki/Dynamo_theory don't make mention any role of the Moon, and googling around drowned out the actual question by talking about how the Moon might have once had a magnetosphere, along with articles about how smaller planets might be able to.
It doesn't. The planet was spinning when it coalesced from a planetary nebula and conservation of angular momentum means it sped up as it coalesced. As the moon moves further away it _slows down_ the rotation of the planet.
I tried to find some discussion on this subject. I found this [1] quite recent paper (2016-03-31) that states that influence of Moon is necessary to maintain Earth's magnetosphere. Just food for thought. There are articles that quote this paper on the Web, but they have this suspicious date of 1st April (but paper was published day before) :)
[1] http://www.sciencedirect.com/science/article/pii/S0012821X16...