This paper examines dynamic binary response and multi-spell duration model approaches to analyzing longitudinal discrete-time binary outcomes. Prototypical dynamic binary response models specify low-order Markovian state dependence and restrict the effects of observed and unobserved heterogeneity on the probability of transitioning into and out of a state to have the same magnitude and opposite signs. In contrast, multi-spell duration models typically allow for state-specific duration dependence, and allow the probability of entry into and exit from a state to vary flexibly. We show that both of these approaches are special cases within a general framework. We compare specific dynamic binary response and multi-spell duration models empirically using a case study of poverty transitions. In this example, both the specification of state dependence and the restrictions on the state-specific transition probabilities imposed by the simpler dynamic binary response models are severely rejected against the more flexible multi-spell duration models. Consistent with recent literature, we conclude that the standard dynamic binary response model is unacceptably restrictive in this context.