  It is commonly accepted that quasar redshifts have a cosmological character
and that most of the quasars are at Gigaparsec distances. However, there are
some cases where several quasars with completely different redshifts and a
nearby active galaxy are aligned in a certain way or occupy a very small patch
on the sky, which is claimed by some authors to be unlikely to happen by
chance. Is there a small subset of quasars with non-cosmological redshifts? For
quasars apparently associated with galaxies, we consider two scenarios for the
origin of their redshift: 1. a standard, cosmological scenario, 2. a
velocity-induced Doppler shift of a nearby object's spectrum (local, ejection
scenario). We argue for a simple astrometric test which can distinguish between
these two sources of quasar redshifts by constraining their proper motions.
  We give the predictions for the maximum possible proper motions of a quasar
for the cosmological and local scenarios of the origin of their redshifts. We
apply these theoretical results to the Bukhmastova (2001) catalog, which
contains more than 8000 close QSO-galaxy associations. In the standard
interpretation of quasar redshifts, their typical proper motions are a fraction
of micro arc-second, and beyond the reach of planned astrometric missions like
GAIA and SIM. On the other hand, the quasars ejected from local AGNs at
velocities close to the speed of light would have proper motions 5-6 orders of
magnitude larger, which would easily be measurable with future astrometric
missions. The distributions of proper motions for the cosmological and local
scenarios are very well separated. Moreover, the division corresponds nicely to
the expected accuracy from GAIA and SIM.
