Research on teacher productivity, and recently developed accountability systems for teachers, rely on value-added models to estimate the impact of teachers on student performance. The authors test many of the central assumptions required to derive value-added models from an underlying structural cumulative achievement model and reject nearly all of them. Moreover, they find that teacher value added and other key parameter estimates are highly sensitive to model specification. While estimates from commonly employed value-added models cannot be interpreted as causal teacher effects, employing richer models that impose fewer restrictions may reduce the bias in estimates of teacher productivity.