a function of $r$-band magnitude (left panel) and $g-r$ colour (right panel). The top and bottom panels show the median and standard deviation of the QSO proper motions. The dashed green and blue lines in the top panels show the median $\mu_{\alpha}$ and $\mu_\delta$ proper motions, and the grey shaded region indicates median offsets from zero of $\pm 0.1$ mas/yr. The dotted line in the bottom panels indicates the median proper motion error of 2 mas/yr. There is a slight correlation of $\sigma(\mu)$with $r$-band magnitude, but this is very minor over the magnitude range probed in this study ($ r \lesssim 19$). Furthermore, there is no variation with colour. []{data-label="fig:qso_mag_col"}](qsos_mag_color.pdf){width="8.5cm" height="4.25cm"}

We quantify the uncertainties in the SDSS-*Gaia* proper motion measurements using spectroscopically confirmed QSOs from SDSS DR12 [@paris17]. This QSO sample is cross-matched with the SDSS-*Gaia* catalog by searching for the nearest neighbour within $1\arcsec$. There are $N=71, 799$ QSOs in the catalog with $r < 20$, and we show the distribution of QSO proper motions in the left-hand panel of Fig. \[fig:qso\_mjd\]. The QSO proper motions are nicely centred around $\mu =0$ mas/yr, and there are no significant high proper motion tails to the distribution. Note that we 