We consider situations in which a significant proportion of observations in a time series are zero, but the remaining observations are positive and measured on a continuous scale. We propose a new dynamic model in which the conditional distribution of the observations is constructed by shifting a distribution for non-zero observations to the left and censoring negative values. The key to generalizing the censoring approach to the dynamic case is to have (the logarithm of) the location/scale parameter driven by a filter that depends on the score of the conditional distribution.  An exponential link function means that seasonal effects can be incorporated into the model and this is done by means of a cubic spline (which can potentially be time- varying). The model is fitted to daily rainfall in northern Australia and compared with a dynamic zero-augmented model.