Measured aggregate U.S. consumption does not behave like a martingale. This paper develops and tests two variants of the permanent income model that are consistent with this fact. In both variants, we assume agents make decisions on a continuous time basis. According to the first variant, the martingale hypothesis holds in continuous time and serial persistence in measured consumption reflects only the effects of time aggregation. We investigate this variant using both structural and atheoretical econometric models. The evidence against these models is far from overwhelming. This suggests that the martingale hypothesis may yet be a useful way to conceptualize the relationship between aggregate quarterly U.S. consumption and income. According to the second variant of the permanent income model, serial persistence in measured consumption reflects the effects of exogenous technology shocks and time aggression. In this model, continuous time consumption does not behave like a martingale. We find little evidence against this variance of the permanent income model. It is difficult, on the basis of aggregate quarterly U.S. data, to convincingly distinguish between the different continuous time models considered in the paper.