This paper introduces a Nearly Unstable INteger-valued AutoRegressive Conditional Heteroscedastic (NU-INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU-INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type-I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID-19 in the United Kingdom.

中文翻译：

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几乎不稳定的整数值 ARCH 过程和单位根测试

本文介绍了一种用于处理计数时间序列数据的近不稳定整数值自回归条件异方差 (NU-INARCH) 过程。证明了 NU-INARCH 过程的适当归一化弱收敛于 Skorohod 拓扑中的 Cox-Ingersoll-Ross 扩散。相关参数的条件最小二乘估计量的渐近分布被建立为某些随机积分的函数。提供基于蒙特卡罗模拟的数值实验来验证有限样本下渐近分布的行为。这些模拟表明，即使真实过程不那么接近非平稳性，几乎不稳定的方法也能比基于平稳性假设的方法提供令人满意且更好的结果。提出了单位根检验，并通过蒙特卡罗模拟检查其 I 类误差和功效。作为说明，拟议的方法适用于英国每日因 COVID-19 造成的死亡人数。