orbit. We leave this task, and other applications of the Sgr proper motions, to a future study.\
We have now shown, using both spectroscopically confirmed QSOs and stars belonging to the Sgr stream, that our SDSS-*Gaia* proper motions are free of any significant systematic uncertainties. In the following Section we use this exquisite sample to infer the rotation signal of the stellar halo.

Halo Rotation {#sec:like}
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In this Section, we use the SDSS-*Gaia* sample of RRL, BHB and K giant stars to measure the average rotation of the Galactic stellar halo. Below we describe our rotating halo model, and outline our likelihood analysis.

In order to convert observed heliocentric velocities into Galactocentric ones, we adopt a distance to the Galactic centre of $R_0=8.3 \pm 0.3$ kpc [@schonrich12; @reid14], and we marginalise over the uncertainty in this parameter in our analysis. Given $R_0$, the total solar azimuthal velocity in the Galactic rest frame is strongly constrained by the observed proper motion of Sgr A$^*$, i.e. $V_{g, \odot} = \mu (\mathrm{Sgr \, A^*}) \times R_0$. We adopt the [@reid04] proper motion measurement of Sgr A$^*$, which gives a solar azimuthal velocity of $V_{g, \odot} = 250 \pm 9$ km s$^{-1}$. Finally, 