My idea was to prove the impossibility of observing an i.i.d. sequence. However, George Judge suggest that we go after the big target. My response to his email is below:
I agree that that is the target, My view of appropriate strategy is to build up the general case by means of several sharp and concrete special cases where details can be spelled out. Even when writing a general paper I would put down a theme and then illustrate several instances of it. I am not sure how much we have in the way of illustration of the general theme -- that is the big picture is not clear in my mind, though I do knoiw that struggling with it is the best way to obtain clarity.
So for example, we could list particular cases:
A: Impossiblity of verifying the parsing of observations as X = Model + i.i.d. Error
B: Your E.S.P problem (which I still dont understand fully, but I still havent read the more recent paper you sent me)
C: Max Entropy and the Ill-posed problem and how solutions cannot really be found.
Are there others? Freedmans Impossiblity paper handles a variant and generalization of A. His book on Statistical Models has a detailed discussion of Instrumental Variables and how these cannot be found in realistic cases.
I recently started a book by Spyros Makridakis: Dance with Chance, which gives an excellent theme which we could use as a basis for our big picture. By the way, get a copy and read it, you will enjoy it.
The theme is: we human beings seek the Illusion of Certainty and the Illusion of Control. WE like to think that things are predictable when they are not. This illusion is very costly. The book illustrates how costly these illusions can be in various contexts.
In terms of econometrics, we face the same problem. We make horrendous assumptions and pretend that our results are correct, when they may not even be in the ballpark. Let us try to be realistic about what can and cannot be done. Obviously data is not useless, but at the same time, it cannot give us the sharp point estimates that we would like to have. Exploring the boundary of what can and cannot be done is very worthwhile.
That reminds that we can add to the list above:
D: Makridakis is the editor of the (International?) Journal of Forecasting. They ran a forecasting competition for sevearl years. In a survey he summarized what he learned from this competition. As I recall, he basically said that mathematical complexity and sophistication usually detracts from performance in real world live forecast environments.