find no significant differences between the QSO proper motion components $\mu_\alpha$ and $\mu_\delta$, so we group both components together (i.e. $\mu=[\mu_\alpha, \mu_\delta]$) in the figure. However, we do show the $\mu_\alpha$ and $\mu_\delta$ components separately (green and blue dashed lines in the top-right panel) when we show the median proper motions to illustrate that these components *individually* have no significant systematics.

The proper motion errors should roughly scale as $\sigma (\mu) \propto 1/\Delta T$, where $\Delta T$ is the timescale between the first epoch SDSS measurements and the second epoch *Gaia* data[^2]. The SDSS photometry was taken over a significant period of time, and data from later releases have shorter time baselines. Thus, this variation in astrometry timespan is an important parameter when quantifying the proper motion uncertainties in our SDSS-*Gaia* catalog. The top-right panel of Fig. \[fig:qso\_mjd\] shows a (normalised) histogram of the time baselines ($\Delta T$). There is a wide range of time baselines, but most of the SDSS data were taken $\sim 10-15$ years ago. In the bottom-right panel of Fig. \[fig:qso\_mjd\] we show the dispersion in QSO proper motion measurements (defined as $\sigma = 1.48$ times the median absolute deviation) as a function of $\Delta T$, 