Evaluating the school effect: adjusting for pre-test or using gain scores?

We consider the issue of estimating the effect of schools on student achievement, when a pre-test is available. Based on Invalsi data, our focus is on the causal effect of the lower secondary school type (public versus private) on test scores at the 8th grade (post-test), accounting for the student test scores at the 5th grade (pre-test). The treatment effect can be estimated by either conditioning on the pre-test score or substituting the post-test score with the difference between post-test and pretest scores (gain score). The performance of the two approaches, in terms of bias and efficiency, depends on several factors, such as pre-test reliability and validity of the common trend assumption. We compare the two approaches by an application using Invalsi data and by a simulation study.