The literature on this topic is almost embarrassingly huge, a situation that probably reflects two faces of reality:
To be honest, many any people have essentially given up on testing of stationarity.
They realize that (1) the test we have all assume some restricted model and some restricted set of alternative hypotheses and (2) even with such strong assumptions the tests are not particularly powerful.
In such a macabre situation, it is natural to turn to humor. The famous "Prize for the Best Abstract for a Paper in Economics" is a favorite example.
In essence, one faces what I call Foster's Boundary.
When you cross Fosters Boundary you make your assumption not because "it's just clearly so" but because "it's just clearly impossible to test."
When we step over this line, we just go ahead and "assume what we need to assume. "
Naturally, if one is smart, this step is taken with the kind of attitude one is encouraged to have by signs that say things like "Caution: Trail Narrows" or "Caution: Falling Rocks."
I'll add more to this later. For the moment I'll just append some references.
Give me time and I will add prioritization and comments. If this area interests you, it is definitely pays to start with a good survey. The stream is a muddy one, if not down-right polluted.
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