Maybe it's even enough if you simply just sell it via mail order, you can then look at the addresses.
There's probably a natural information spread in any market (word of mouth, trade magazines), and there's probably a physical dispersion of the target group of people too (people move, visitors/tourists saw the ad/product and order it at home), but it still should be a valuable to see how much effect just one campaign has.
Maybe one of the best products for this could be a car. They are pretty standard, really not much difference between them, they are in all price ranges, and regularly new models come out. Advertise one in a few major US cities but don't in others.
To see this in action - spend a day watching pitbull videos on youtube and see how many spanish language ads you'll get.
Plus, maybe it's possible to somehow offer a coupon for those who buy via the ad, and so on.
After all the ad/tracking industry probably have a thousand tricks to increase accuracy of this.
But, yeah, I'm not holding my breath for a conclusive answer on this.
It's not currently possible to run an A/B experiment with a hold out group of potential customers across all channels, let alone for any longer duration experiment. So how can we separate cause and effect? (although pay per conversion channels do get gamed left and right)