[note to myself; see correspondence with salim rashid april 11,12 2010 for further details]
We (or at least I) have been conditioned deeply by positivism, as I realized, while I was writing the Normative Foundations of Scarcity. Even as we shake off the big parts, there are many small ways in which it conditions our thoughts.
There is a huge literature on testing for unit roots, all of it inconclusive. For a given time series, we want to find out whether or not it is a unit root process. We do a test and make a decision on the basis of the test.
This implicitly assumes that THERE IS A TRUE MODEL and that IT CAN BE DISCOVERED. Suppose that we accept from the outset the idea that even with infinite amounts of data there will be indeterminacy. This is a particularly good case since there are "observational equivalence theorems" which state that there are good unit root approximations to stationary processes and vice versa.
Then we can never find out from the data whether there is a unit root or not. This means that we must make the decision on other grounds.
This is exactly what the mathematicians discovered following Godel Uncertainty. We cannot prove or disprove the parallel postulate, the Axiom of Choice, the Continuum Hypothesis. The traditional methods for settling mathematical truth do not work. We must decide on basis of mathematical intuition.
It would be a substantial revolution in econometric/economic methodology if we were to make this clear that the decision about whether or not a series is unit root is NOT a data based decision at all. One must make it on a priori theoretical grounds.