AI 中文总结
本文提出基于RKHS的条件中心化检验，用于非线性自回归过程中的格兰杰非因果性，通过核岭回归和条件中心化核构造统计量，实现无需重采样的谱校准，模拟和实证验证了其有效性。
AI 中文摘要
格兰杰因果通常通过向量自回归模型中的线性预测来表述，这限制了其检测非线性预测关系的能力。我们提出了一种基于再生核希尔伯特空间（RKHS）的检验方法，用于非线性自回归过程中条件均值意义上的非线性格兰杰非因果性。关键思想是将目标回归函数进行条件中心化分解，分解为自身历史分量和一个正交分量，后者捕捉潜在源历史的额外预测贡献。非因果性由后一分量的消失来刻画。自身历史分量通过核岭回归估计，残差使用条件中心化核嵌入到第二个RKHS中。这产生了一个RKHS值残差矩，其平方范数构成检验统计量。我们建立了加权卡方零分布，并证明了总体中心化统计量对固定备择假设的一致性。经验中心化版本被证明保持零分布，并允许在不重采样的情况下进行临界值和$p$值的谱校准。模拟研究展示了准确的尺寸控制和针对非线性备择假设的功效，实际数据应用说明了该方法在检测时间序列中非线性预测关系的实用性。
英文摘要
Granger causality is commonly formulated through linear prediction in vector autoregressive models, limiting its ability to detect nonlinear predictive relationships. We propose a reproducing kernel Hilbert space (RKHS)-based test for nonlinear Granger non-causality in conditional mean for nonlinear autoregressive processes. The key idea is a conditional-centering decomposition of the target regression function into an own-history component and an orthogonal component capturing the additional predictive contribution of the potential source history. Non-causality is characterized by the vanishing of the latter component. The own-history component is estimated by kernel ridge regression, and the residuals are embedded in a second RKHS using a conditionally centered kernel. This yields an RKHS-valued residual moment whose squared norm forms the test statistic. We establish a weighted chi-square null limit and consistency against fixed alternatives for the population-centered statistic. An empirically centered version is shown to retain the null limit and enables spectral calibration of critical values and $p$-values without resampling. Simulation studies demonstrate accurate size control and power against nonlinear alternatives, and real-data applications illustrate the usefulness of the method for detecting nonlinear predictive relationships in time series.
Comments58 pages, 4 figures
