Several models for count time series have been developed during the last decades, often inspired by traditional autoregressive moving average (ARMA) models for real-valued time series, including integer-valued ARMA (INARMA) and integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models. Both INARMA and INGARCH models exhibit an ARMA-like autocorrelation function (ACF). To achieve negative ACF values within the class of INGARCH models, log and softplus link functions are suggested in the literature, where the softplus approach leads to conditional linearity in good approximation. However, the softplus approach is limited to the INGARCH family for unbounded counts, i.e. it can neither be used for bounded counts, nor for count processes from the INARMA family. In this paper, we present an alternative solution, named the Tobit approach, for achieving approximate linearity together with negative ACF values, which is more generally applicable than the softplus approach. A Skellam--Tobit INGARCH model for unbounded counts is studied in detail, including stationarity, approximate computation of moments, maximum likelihood and censored least absolute deviations estimation for unknown parameters and corresponding simulations. Extensions of the Tobit approach to other situations are also discussed, including underlying discrete distributions, INAR models, and bounded counts. Three real-data examples are considered to illustrate the usefulness of the new approach.

翻译：近几十年来，受传统实值时间序列自回归滑动平均(ARMA)模型的启发，学者们发展出多种计数时间序列模型，包括整数值ARMA(INARMA)模型和整数值广义自回归条件异方差(INGARCH)模型。INARMA与INGARCH模型均呈现类ARMA的自相关函数(ACF)特征。为在INGARCH模型框架内实现负ACF值，文献中提出了log和softplus链接函数，其中softplus方法可良好近似条件线性关系。然而，softplus方法仅适用于无界计数的INGARCH族，既不能用于有界计数，也无法应用于INARMA族的计数过程。本文提出名为Tobit方法的替代方案，该方法在实现近似线性与负ACF值方面具有比softplus更广泛的适用性。我们详细研究了针对无界计数的Skellam-Tobit INGARCH模型，涵盖平稳性、矩的近似计算、未知参数的最大似然估计与删失最小绝对偏差估计及其对应仿真。同时探讨了Tobit方法在包括基础离散分布、INAR模型及有界计数等其他情形下的扩展应用。通过三个真实数据案例验证了新方法的实用性。