Modeling volatility is one of the prime objectives of financial time‐series analysis. A significant feature encountered in the modeling of financial data is the asymmetric response to the volatility process of unanticipated shocks. With improvements in data acquisition, functional versions of the heteroskedastic models have emerged to deal with the high‐frequency observations. Although previous studies have developed some functional time‐series methods, it remains a necessity to analyze the variations in the asymmetry of the discrete model and the function model. In this study, we propose a functional threshold GARCH (fTGARCH) model and extend the news impact curve (NIC) and the cumulative impact response function (CIRF) within the functional heteroskedastic framework. We find that the fTGARCH model can describe the asymmetry of the observation data, which are revealed by the sample cross‐correlation functions. The slope of the NIC changes with time for functional GARCH class models, and the changes are asymmetrical for the fTGARCH model. Using the generalized CIRF, we can explore the persistent effects of volatility for the functional GARCH class models. By fitting the models to the S&P 500 stock market index, we conclude that the fTGARCH model has some flexibility and superiority in regard to volatility asymmetry.

中文翻译：

---

功能阈值 GARCH 模型中的波动性不对称

对波动性建模是金融时间序列分析的主要目标之一。金融数据建模中遇到的一个重要特征是对意外冲击的波动过程的不对称响应。随着数据采集的改进，出现了异方差模型的函数版本来处理高频观测。虽然之前的研究已经开发了一些函数时间序列方法，但仍然有必要分析离散模型和函数模型的不对称性变化。在这项研究中，我们提出了一个功能阈值 GARCH (fTGARCH) 模型，并在功能异方差框架内扩展了新闻影响曲线 (NIC) 和累积影响响应函数 (CIRF)。我们发现 fTGARCH 模型可以描述观测数据的不对称性，这是由样本互相关函数揭示的。功能 GARCH 类模型的 NIC 斜率随时间变化，而 fTGARCH 模型的变化是不对称的。使用广义 CIRF，我们可以探索波动性对函数 GARCH 类模型的持续影响。通过将模型与标准普尔 500 股票市场指数拟合，我们得出结论，fTGARCH 模型在波动率不对称方面具有一定的灵活性和优势。我们可以探索波动性对函数 GARCH 类模型的持续影响。通过将模型与标准普尔 500 股票市场指数拟合，我们得出结论，fTGARCH 模型在波动率不对称方面具有一定的灵活性和优势。我们可以探索波动性对函数 GARCH 类模型的持续影响。通过将模型与标准普尔 500 股票市场指数拟合，我们得出结论，fTGARCH 模型在波动率不对称方面具有一定的灵活性和优势。