For the most part the correlation between administrations and the economy is arbitrary. But in this case I would make a case that it is causative.
In addition, it's the circles that person runs in and the circles that person's people run in. Do you think judge or cabinet appointments are decided on by one person? Sure, the President is a figurehead in this position and ultimately has to say yes or no, but there's a large pool of candidates out there and it's up to the staffers, maybe friends, maybe people in the know to propose those people to the President.
So while on paper the President doesn't, or shouldn't have that much power--in actuality with our current political process it's certainly much more.
Now, that said, just like the billionaires--they can only control so much. At least in the United States, there are competing interests even among the wealthy class, and sometimes shit just sort of happens--like a meteor killing the dinosaurs....or the release of decades of e-mails, videos, documents, and communications surrounding their pedophile behavior on a secret island in the Caribbean.
I do believe that there could be a small causative effect, but there is usually a very long delay between cause and effect.