is likely to pass an informed expert can be manipulated by an uninformed expert, and so is equally likely to pass an uninformed expert even if he is restricted to produce forecasts based on an exchangeable process (Proposition 1 below). 

Now consider the case of infinite data sets, but with no restrictions on the types of sto- chastic processes that may generate the data. Shmaya (2008) showed that the impossibility results still hold: Any prequential test that is likely to pass an informed expert is also likely to pass an uninformed, but strategic, expert (Proposition 2 below). 

Thus, neither a restriction to exchangeable processes, nor infinite sequences of forecasts and outcomes, can, by itself, rule out the impossibility theorems. The question addressed in this paper is whether the impossibility theorem still hold if it is assumed simultaneously that the data generating process is exchangeable and that the tester can obtain large, perhaps infinite, sequences of forecasts and outcomes. Under these conditions, we show that the impossibility theorems do not hold. We construct a prequential test that is likely to pass an informed expert and cannot be manipulated by an uninformed expert; no matter which forecasting scheme he uses, he cannot be assured that he is likely to pass the test. 

Here is the basic idea of our test: By De-Finetti’s Theorem, every exchangeable process has a representation as a sequence of i.i.d. coin tosses with an unknown parameter q . By the law of large numbers, the infinite realization pins down the parameter q . In addition, we show that after observing Bob’s forecasts along the realization Alice can deduce Bob’s prior distribution on the parameter. So, the sequential framework reduces to a one-shot framework in which Bob delivers a prior that (he claims) generates a parameter q that is observed by Alice. The existence of non-manipulable test can now be obtained by adapting a result in Olszewski and Sandroni (2009b) to this one-shot framework. 

Our argument reveals a non-intuitive distinction between the infinite and finite horizon setups. With infinite streams of forecasts and outcomes, it is possible to determine q and, hence, to screen informed and uninformed experts. With finite (perhaps arbitrarily long) streams of forecasts and outcomes, it is possible to make arbitrarily accurate statistical inferences about q and yet this is insufficient to screen informed and uninformed experts. 

