Sample Splitting and Assessing Goodness-of-fit of Time Series
样本分割与时间序列拟合优度评估AI 解读 ↓
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样本分割与时间序列拟合优度评估
- 研究问题
- 拟合残差共用参数估计而产生额外依赖时，如何利用时间序列样本分割，使残差自相关和距离相关检验获得适当的参考分布？
- 方法与思路
- 作者用前段观测估计模型参数，以这些参数计算后段残差；通过选择两段样本的重叠关系，分析 ACF 与 ADCF 的极限分布。
- 材料陈述的贡献
- 摘要称，在特定重叠比例及相关条件下，两类检验可具有如同真实残差独立同分布时的极限分布，从而减少置信界调整需求。
- 阅读关注点
- 应核对重叠比例的精确定义、参数估计与序列依赖条件，以及前半段估计参数、全样本计算残差这一方案适用的模型范围。
- 阅读边界
- 所给版本为 arXiv 预印本，同行评审及接收状态未核实；摘要使用“通常”和“可以”等限定，不能解释为所有时间序列模型均成立。
原文摘要
A fundamental and often final step in time series modeling is to assess the quality of fit of a proposed model to the data. Since the underlying distribution of the innovations that generate a model is often not prescribed, goodness-of-fit tests typically take the form of testing the fitted residuals for serial independence. However, these fitted residuals are inherently dependent since they are based on the same parameter estimates and thus standard tests of serial independence, such as those based on the autocorrelation function (ACF) or distance correlation function (ADCF) of the fitted residuals need to be adjusted. The sample splitting procedure in Pfister et al.~(2018) is one such fix for the case of models for independent data, but fails to work in the dependent setting. In this paper sample splitting is leveraged in the time series setting to perform tests of serial dependence of fitted residuals using the ACF and ADCF. Here the first $f_n$ of the data points are used to estimate the parameters of the model and then using these parameter estimates, the last $l_n$ of the data points are used to compute the estimated residuals. Tests for serial independence are then based on these $l_n$ residuals. As long as the overlap between the $f_n$ and $l_n$ data splits is asymptotically 1/2, the ACF and ADCF tests of serial independence tests often have the same limit distributions as though the underlying residuals are indeed iid. In particular if the first half of the data is used to estimate the parameters and the estimated residuals are computed for the entire data set based on these parameter estimates, then the ACF and ADCF can have the same limit distributions as though the residuals were iid. This procedure ameliorates the need for adjustment in the construction of confidence bounds for both the ACF and ADCF in goodness-of-fit testing.