  A popular and flexible time series model for counts is the generalized
integer autoregressive process of order $p$, GINAR($p$). These Markov processes
are defined using thinning operators evaluated on past values of the process
along with a discretely-valued innovation process. This class includes the
commonly used INAR($p$) process, defined with binomial thinning and Poisson
innovations. GINAR processes can be used in a variety of settings, including
modeling time series with low counts, and allow for more general mean-variance
relationships, capturing both over- or under-dispersion. While there are many
thinning operators and innovation processes given in the literature, less focus
has been spent on comparing statistical inference and forecasting procedures
over different choices of GINAR process. We provide an extensive study of exact
and approximate inference and forecasting methods that can be applied to a wide
class of GINAR($p$) processes with general thinning and innovation parameters.
We discuss the challenges of exact estimation when $p$ is larger. We summarize
and extend asymptotic results for estimators of process parameters, and present
simulations to compare small sample performance, highlighting how different
methods compare. We illustrate this methodology by fitting GINAR processes to a
disease surveillance series.
