We analyze models for panels of endogenous length. Agents decide on exiting the panel based on noisy observations on an underlying persistent state process. The econometrician has noisy data on the agent's state observations and observes the exit decisions. We show that the survival data facilitate the deconvolution of the econometrician's state data in the agent's state observations and the econometrician's noise. Then, we explore how the assumptions of (i) a single unobserved state variable, and (ii) rational expectations restrict the agent's state observation process and decision rule. We develop numerically feasible estimation methods. Applications to models of industry dynamics are discussed.