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Say I am evaluating a null hypothesis such that

H0: |p| = 1

Ha: |p| < 1

And want to understand how this differs in implementation and evaluation from the test:

H0: p = 1

Ha: p < 1

Is this analogous to testing the individual hypotheses H0 p = 1, H0 p = -1, and introducing a multiple hypothesis correction, or is this something entirely different?

The basis of this question comes from a theoretical "fix" to the convention dickey-fuller test with the null hypothesis specified as p = 1. This original implementation seems to fail to account for the case of p = -1, which in turn is also nonstationary, and I'm trying to understand the change called for if I introduce an absolute value into the test. But I also want to understand more generally how a hypothesis test about an absolute value of a given parameter affects its evaluation.

Any guidance is greatly appreciated. Thank you for your time to read this regardless :)

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