I am looking to install an exhaust fan in a bathroom where one does not already exist. I've read that it is preferable to run exhaust ducts down instead of up, but does that apply here? We would be talking about 20-some feet to run an exhaust duct from the 2nd floor ceiling down to the first floor ground level. The alternative would be to run the exhaust duct up about 5 or 6 feet and out the roof fascia (seems easier to drill through wood fascia than brick wall, but that may be a separate question). I'm not even sure I can get a duct down through the wall yet, but thought I'd ask before I spend too much time figuring it out. It's an exterior wall, if that makes any difference.
One purpose for the vent is to remove lingering odors. Venting high has to do with peoples noses and, once upon a time, truly noxious gas; restrooms today are not as toxic as they once were. Still, if the gases are vented downward, they are less likely to be further evacuated from the area because there is less wind on the ground than in the sky, especially next to a wall that has at least some wind shielding.
Technically, moisture, heavier gases, and odors settle; so it's understandable that someone might think it's more efficient to move odors (etc.) downward, but the amount of power or efficiency is really negligible. A far more important factor of efficiency would be the length and size of the vent (fan and duct). In your case it's a no-brainer, 6ft vs 20ft.
But if the lengths were the same then it really wouldn't make a huge difference, besides the lingering smell.
edit- Historically, (before 1900) fires were lit/used to create a thermal draft (perhaps this is the true source of lighting a match when making a stink, or why candles are used in the bathroom... it's a cultural memory). Exhaust was pulled through vents via a stack effect. The stacks were then powered by fans when electricity became common. Source: http://aceee.org/files/proceedings/2004/data/papers/SS04_Panel7_Paper21.pdf
Impetus (handed down) is one reason that the exhaust vent goes up, but the fact is, it still usually works better to go up for the other reasons mentioned above.