This paper discusses a model for regression analysis with a time series of counts. Correlation is assumed to arise from an unobservable process added to the linear predictor in a log linear model. An estimating equation approach used for parameter estimation leads to an iterative weighted and filtered least-squares algorithm. Asymptotic properties for the regression coefficients are presented. We illustrate the technique with an analysis of trends in U.S. polio incidence since 1970.