Notes
Unreported results show these differences to become even smaller relative to the low-frequency unit root tests derived in Müller and Watson (Citation2008), which are based on the same cosine transforms as Sq.
The approximate least favorable distribution under c ⩾ 1 puts a lot of weight on c = 1. Under the null hypothesis and c = 1, the unconditional variance of y1 is approximately σ2T/(1 − ρ2T) ≈ σ2T/(2c) = σ2T/2. This value corresponds more or less to the variance of y1 + μ under the alternative under c ≈ 5, explaining the dip in weighted average power.
Once this approximate linearity fails the implementation details of the Sq tests in a regression context become potentially important; for instance, a variant of (15) such as may lead to substantially different empirical results.