This paper studies the autoregressive and moving average (ARMA) model with time-functional variance (TFV) noises, called the ARMA-TFV model. We first establish the consistency and asymptotic normality of its least squares estimator (LSE). The Wald tests and portmanteau tests are constructed based on the theory for variable selection and model checking. A simulation study is carried out to assess the performance of our approach in finite samples, and two real examples are given. It should be mentioned that the process generated from the ARMA-TFV model is not stationary, and the technique in this paper is nonstandard and may provide insights for future research in this area.

中文翻译：

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具有时间函数方差噪声的 ARMA 模型的渐近推理

本文研究带有时间函数方差 (TFV) 噪声的自回归和移动平均 (ARMA) 模型，称为 ARMA-TFV 模型。我们首先建立其最小二乘估计量（LSE）的一致性和渐近正态性。 Wald检验和混合检验是基于变量选择和模型检验理论构建的。进行了模拟研究以评估我们的方法在有限样本中的性能，并给出了两个真实的例子。应该指出的是，ARMA-TFV 模型生成的过程不是平稳的，本文中的技术是非标准的，可以为该领域的未来研究提供见解。