  Count data are often subject to underreporting, especially in infectious
disease surveillance. We propose an approximate maximum likelihood method to
fit count time series models from the endemic-epidemic class to underreported
data. The approach is based on marginal moment matching where underreported
processes are approximated through completely observed processes from the same
class. Moreover, the form of the bias when underreporting is ignored or taken
into account via multiplication factors is analysed. Notably, we show that this
leads to a downward bias in model-based estimates of the effective reproductive
number. A marginal moment matching approach can also be used to account for
reporting intervals which are longer than the mean serial interval of a
disease. The good performance of the proposed methodology is demonstrated in
simulation studies. An extension to time-varying parameters and reporting
probabilities is discussed and applied in a case study on weekly rotavirus
gastroenteritis counts in Berlin, Germany.
