Rejoinder

Click to increase image sizeClick to decrease image size

Notes

Unreported results show these differences to become even smaller relative to the low-frequency unit root tests derived in Müller and Watson (2008), which are based on the same cosine transforms as S_q.

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 − ρ²_T) ≈ 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 S_q tests in a regression context become potentially important; for instance, a variant of (15) such as may lead to substantially different empirical results.