is it possible to apply a Difference-in-Differences method for a quasi-experiment that determines treatment by a threshold? All schools below a certain API rank are treated the rest is not (control). The original paper uses RDD and I would try to estimate the causal effect with a DiD using the WHOLE range of X (not the one slightly above and slightly below the cutoff). According to Angrist and Pischke (2009) the treated and control schools can differ as long as the difference is captured by the school fixed effects. Does this mean all differences cancel out, if they are constant over time? For example in control schools there are simple people with higher IQ compared to treatment schools, but if the average IQ in treat and control is constant over time this is not a problem?
Therefore, I would conclude that only time varying factors that influence the API rank (determinant of treatment) and test scores are a problem. Do you see a problem with time varying factors in this setup? What could they look like?