We focus on purely autoregressive (AR)-type models defined on the bounded range \(\{0,1,\ldots , n\}\) with a fixed upper limit \(n \in \mathbb {N}\). 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.

中文翻译：

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有界数的自回归过程模型：它们有何不同？

我们专注于在有固定上限\（n \ in \ mathbb {N} \）的有界范围\（\ {0,1，\ ldots，n \} \）上定义的纯自回归（AR）型模型。这些模型包括二项式AR模型，二项式AR条件异方差（ARCH）模型，具有线性条件均值的二项式变量AR模型，非线性最大二项式AR模型以及二项式logit-ARCH模型。我们考虑识别这些AR类型模型中的哪一个是真正的数据生成过程的关键问题。尽管有大量关于模型选择的文献，但是在非嵌套和非线性时间序列计数模型的背景下，对该程序知之甚少。我们考虑了用于模型识别的最流行方法，Akaike信息准则和贝叶斯信息准则，并使用广泛的蒙特卡洛模拟进行比较。此外，我们调查了使用最大似然估计获得的拟合模型（正确模型和错误模型）的属性。实际数据示例证明了我们的发现。