Title: A nonlinearity and model specification test for functional time series Show Chinese title

Title: 非线性与模型设定检验对于函数时间序列

Abstract: An important issue in functional time series analysis is whether an observed series comes from a purely random process. We extend the BDS test, a widely-used nonlinear independence test, to the functional time series. Like the BDS test in the univariate case, the functional BDS test can act as the model specification test to evaluate the adequacy of various prediction models and as a nonlinearity test to detect the existence of nonlinear structures in a functional time series after removing the linear structure exhibited. We show that the test statistic from the functional BDS test has the same asymptotic properties as those in the univariate case and provides the recommended range of its hyperparameters. Additionally, empirical data analysis features its applications in evaluating the adequacy of the fAR(1) and fGARCH(1,1) models in fitting the daily curves of cumulative intraday returns (CIDR) of the VIX index. We showed that the functional BDS test remedies the weakness of the existing independence test in the literature, as the latter is restricted in detecting linear structures, thus, can neglect nonlinear temporal structures. Show Chinese abstract

Abstract: 在函数时间序列分析中，一个重要问题是确定观察到的序列是否来自纯随机过程。我们扩展了BDS检验，这是一种广泛使用的非线性独立性检验，以适用于函数时间序列。与单变量情况下的BDS检验类似，函数BDS检验可以作为模型设定检验，用于评估各种预测模型的充分性，并作为非线性检验，在去除函数时间序列中表现出的线性结构后，用于检测非线性结构的存在。我们证明了函数BDS检验的检验统计量具有与单变量情况下相同的渐近性质，并提供了其超参数的推荐范围。此外，实证数据分析展示了其在评估fAR(1)和fGARCH(1,1)模型在拟合VIX指数每日累计日内收益曲线（CIDR）中的适用性方面的应用。我们表明，函数BDS检验弥补了文献中现有独立性检验的不足，因为后者仅能检测线性结构，因此可能忽略非线性时间结构。