Presidents have hardly any influence over the economy. However, the economy does have influence over president's chances of re-election.
This
used to be true. Not that presidents have total influence... obviously not. But now, in a pretty evenly divided Congress, the president has a large say in economic policy, because his vetoes can't be overridden. Two big examples during Bush's first term... The major tax cuts would not have gone through were it not for Bush; also, the huge govt spending increases wouldn't have gone through if Bush was economically conservative (he'd have vetoed the budget bill, as fiscal conservatives advised, and his veto wouldn't have been overridden). And these are just two examples.
Obviously, things like this don't "control" the economy. But they clearly have a significant impact. If we were in a time with a much more heavily tilted Congress (so, vetoes could be overridden), the prez would wield very little power. In this age, the presidency is a very powerful position... economically and otherwise.
