  Using the method of ``reduced parallaxes'' for the Halo RR Lyrae stars in the
Hipparcos catalogue we derive a zero point of 0.77 $\pm$ 0.26 mag for an
assumed slope of 0.18 in the $M_{\rm V}$-[Fe/H] relation. This is 0.28
magnitude brighter than the value Fernley et al. (1998a) derived by employing
the method of statistical parallax for the {\it identical} sample and using the
same slope.
  We point out that a similar difference exists between the ``reduced
parallaxes'' method and the statistical parallax method for the Cepheids in the
Hipparcos catalogue.
  We also determine the zero point for the $M_{\rm K}$-$\log P_{0}$ relation,
and obtain a value of -1.16 $\pm$ 0.27 mag (for a slope of -2.33). The distance
moduli to the Hipparcos RR Lyrae stars derived from the two relations agree
well.
  The derived distance scale is in good agreement with the results from the
Main Sequence fitting distances of Galactic globular clusters and with the
results of theoretical Horizontal Branch models, and implies a distance modulus
to the LMC of 18.61 $\pm$ 0.28 mag.
