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 :)