distributions get slightly broader at fainter magnitudes. However, most of the halo stars in this work have $ r < 19$ and there is little variation at these brighter magnitudes. Finally, we find no detectable dependence of $\sigma (\mu)$ on $g-r$ colour. It is worth remarking that the stability of these proper motion measurements to changes in magnitude and colour is a testament to the astrometric stability of the improved SDSS-*Gaia* catalog.\
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In Section \[sec:like\] we introduce a rotating velocity ellipsoid model for the Milky Way halo stars. In order to test the effects of any systematic uncertainties in the SDSS-*Gaia* proper motions, we also apply this modeling procedure to the sample of SDSS DR12 QSOs. We adopt a “distance” of 20 kpc, which is the mean distance to our halo star samples, and find the best fit rotation ($\langle V_{\rm \phi} \rangle$) value. This procedure gives a best fit value of $\langle V_{\rm \phi} \rangle \sim 10$ km s$^{-1}$. Note, that if there were no systematics present, then there would be no rotation signal. In Fig. \[fig:qso\_mjd\] we showed that the median proper motions of the QSOs was $\sim 0.1$ mas/yr. Indeed, at a distance of 20 kpc, 