The levels and growth achievement functions make extreme and diametrically opposed assumptions about the rate at which teacher inputs persist. I first show that if these assumptions are incorrect, teacher value-added estimates can be significantly biased. I then develop a tractable, cumulative model of student achievement that allows for the joint estimation of unobserved teacher quality and its persistence. The model can accommodate varying persistence rates, student heterogeneity, and time-varying teacher attributes. I implement the proposed methodology using schooling data from North Carolina, and find that only a third of the contemporaneous teacher effect survives into the next grade. (Contains 7 tables and 32 footnotes.)