  Real count data time series often show the phenomenon of the underdispersion
and overdispersion. In this paper, we develop two extensions of the first-order
integer-valued autoregressive process with Poisson innovations, based on
binomial thinning, for modeling integer-valued time series with equidispersion,
underdispersion and overdispersion. The main properties of the models are
derived. The methods of conditional maximum likelihood, Yule-Walker and
conditional least squares are used for estimating the parameters, and their
asymptotic properties are established. We also use a test based on our
processes for checking if the count time series considered is overdispersed or
underdispersed. The proposed models are fitted to time series of number of
weekly sales and of cases of family violence illustrating its capabilities in
challenging cases of overdispersed and underdispersed count data.
