  We propose a new class of unit root tests that exploits invariance properties
in the Locally Asymptotically Brownian Functional limit experiment associated
to the unit root model. The invariance structures naturally suggest tests that
are based on the ranks of the increments of the observations, their average,
and an assumed reference density for the innovations. The tests are
semiparametric in the sense that they are valid, i.e., have the correct
(asymptotic) size, irrespective of the true innovation density. For a correctly
specified reference density, our test is point-optimal and nearly efficient.
For arbitrary reference densities, we establish a Chernoff-Savage type result,
i.e., our test performs as well as commonly used tests under Gaussian
innovations but has improved power under other, e.g., fat-tailed or skewed,
innovation distributions. To avoid nonparametric estimation, we propose a
simplified version of our test that exhibits the same asymptotic properties,
except for the Chernoff-Savage result that we are only able to demonstrate by
means of simulations.
