There are many examples for this feedback indeed spoiling the predictions of a model. One of the best known is maybe the experiment conducted at Hawthorne Works from 1924 to 1932, where it was studied (among other things) how monetary incentives affect workers' productivity. Surprisingly, the productivity decreased. It has been suspected that this happened because the workers had heard of the study and were afraid an increase in their productivity would later result in lay-offs or a lowering of the base rate. Another example is Nobel-prize winners or other experts and authorities commenting on the economy. It is well known that consumer behavior is influenced by whether the outlook is pessimistic or optimistic, though in this case it's of course more difficult to identify the causes.
In any case, the argument that feedback necessarily spoils any model and thus such efforts are in vain has never made much sense to me. While this may be for some models, there's no reason a model can't remain unmodified under the feedback or that the feedback must be such to necessarily spoil the accuracy of the model. Take the previous example about a prediction affecting consumer behavior. If it's an optimistic outlook it (ideally) causes people to spend more. This doesn't spoil the prediction. On the contrary: it may turn it into a self-fulfilling prophecy. Or take the model of supply and demand. Most people know it, yet they don't go and buy the most expensive crap just to prove economists wrong. And why is that? Because they have no reason to. Instead, they believe everything is working in their favor as long as they continue to do what the model says they'll do anyway.
This of course lead me to wonder if there's fixed points in the set of models. There is arguably a trivial fixed point. That's the one when nobody knows of a model or nobody believes it, thus there's no feedback. But one could say it's not an attractive fixed point in the sense that it's unstable: The more successful a model is the more people will know of it and believe it. So, I'm posing the question to you: is there an attractive fixed-point? Because if there is one, that might be where we're going.