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We focus on purely autoregressive (AR)-type models defined on the bounded range $$\{0,1,\ldots , n\}$$ { 0 , 1 , … , n } with a fixed upper limit $$n \in \mathbb {N}$$ n ∈ N . These include the binomial AR model, binomial AR conditional heteroscedasticity (ARCH) model, binomial-variation AR model with their linear conditional mean, nonlinear max-binomial AR model, and binomial logit-ARCH model. We consider the key problem of identifying which of these AR-type models is the true data-generating process. Despite the volume of the literature on model selection, little is known about this procedure in the context of nonnested and nonlinear time series models for counts. We consider the most popular approaches used for model identification, Akaike’s information criterion and the Bayesian information criterion, and compare them using extensive Monte Carlo simulations. Furthermore, we investigate the properties of the fitted models (both the correct and wrong models) obtained using maximum likelihood estimation. A real-data example demonstrates our findings.

Models for autoregressive processes of bounded counts: How different are they?