and their recalibrated position in SDSS.

Proper motion errors {#sec:pmerr}
--------------------

![ *Left panel:* The distribution of measured proper motions of SDSS DR12 spectroscopically confirmed QSOs. We find very similar distributions for $\mu_{\alpha}$ and $\mu_\delta$ (also $\mu_\ell$ and $\mu_b$), so for simplicity we use both proper motion measurements in this plot (i.e. $\mu=[\mu_\alpha, \mu_\delta]$). *Top right panel:* A histogram of the time baseline between first epoch SDSS and second epoch *Gaia* measurements ($\Delta T$). *Middle right panel:* Median proper motion of QSOs as a function of time baseline. The median $\mu_{\alpha}$ and $\mu_\delta$ values are shown with the dashed green and blue lines respectively. The median proper motions are consistent with zero at the 0.1 mas/yr level. The grey shaded region indicates median offsets from zero of $\pm 0.1$ mas/yr. *Bottom right panel:* The dispersion in QSO proper motions as a function of $\Delta T$. Here, $\sigma$ is 1.48 times the median absolute deviation. The red dashed line shows the best-fit model for $\sigma(\mu)$, where $\sigma= A+B/\Delta T$. We use this relation to assign proper motion uncertainties to stars in the SDSS-*Gaia* sample as a function of $\Delta T$.[]{data-label="fig:qso_mjd"}](qsos_mjd.pdf){width="8.5cm" height="7.08cm"}

![ Proper motion errors estimated from SDSS DR12 QSOs as 