The INARCH(1) model for overdispersed time series of counts has a simple structure, a parsimonious parametrization, and a great potential for applications in practice. We analyze two approaches to approximate the marginal process distribution: a Markov chain approach and the Poisson–Charlier expansion. Then approaches for estimating the two model parameters are discussed. We derive explicit expressions for the asymptotic distribution of the maximum likelihood and conditional least squares estimators. They are used for constructing simultaneous confidence regions, the finite-sample performance of which is analyzed in a simulation study. A real-data example from economics illustrates the application of the INARCH(1) model.