The problem is: yes, there's only a tiny chance somebody who searches 'nike shoes' ends up in your store. But the probability that someone ends up in your store BECAUSE he saw your ad is even smaller. So if you don't correct for selection effects your going to overstate ad effects by orders of magnitude.
Just did. One study is about brand keywords, which massively heighten the risk for what you're talking about. (I specifically excluded them in my earlier comment)
Another study is about display ads, paid for per impression. The study gives an average figure of 0.02. These are also not the typical successful google or facebook ad campaign, and are more akin to old tv ads.
The 66 page FB study isn't clear, but it also seems to be talking about ads paid for per impression? This is not what people talk about when they say you can easily track online ad returns.
Ppc ads are easy to track. You send a user to a special landing page, and you know exactly when someone got there via an ad, as only people who clicked ads go to those pages.
That said, a lot of advertisers don't do this, so they'll definitely have very real measurement problems.
The problem is: yes, there's only a tiny chance somebody who searches 'nike shoes' ends up in your store. But the probability that someone ends up in your store BECAUSE he saw your ad is even smaller. So if you don't correct for selection effects your going to overstate ad effects by orders of magnitude.