School accountability systems increasingly hold schools to account for their performances using value-added models purporting to measure the effects of schools on student learning. The most common approach is to fit a linear regression of student current achievement on student prior achievement, where the school effects are the school means of the predicted residuals. In the literature, further adjustments are usually made for student sociodemographics and sometimes school composition and "non-malleable" characteristics. However, accountability systems typically make fewer adjustments: for transparency to end users, because data are unavailable or of insufficient quality, or for ideological reasons. There is therefore considerable interest in understanding the extent to which simpler models give similar school effects to more theoretically justified but complex models. We explore these issues via a case study and empirical analysis of England's "Progress 8" secondary school accountability system.