Structural vector autoregressive (SVAR) processes are commonly used time series models to identify and quantify causal interactions between dynamically interacting processes from observational data. The causal relationships between these processes can be effectively represented by a finite directed process graph - a graph that connects two processes whenever there is a direct delayed or simultaneous effect between them. Recent research has introduced a framework for quantifying frequency domain causal effects along paths on the process graph. This framework allows to identify how the spectral density of one process is contributing to the spectral density of another. In the current work, we characterise the asymptotic distribution of causal effect and spectral contribution estimators in terms of algebraic relations dictated by the process graph. Based on the asymptotic distribution we construct approximate confidence intervals and Wald type hypothesis tests for the estimated effects and spectral contributions. Under the assumption of causal sufficiency, we consider the class of differentiable estimators for frequency domain causal quantities, and within this class we identify the asymptotically optimal estimator. We illustrate the frequency domain Wald tests and uncertainty approximation on synthetic data, and apply them to analyse the impact of the 10 to 11 year solar cycle on the North Atlantic Oscillation (NAO). Our results confirm a significant effect of the solar cycle on the NAO at the 10 to 11 year time scale.

翻译：结构向量自回归（SVAR）过程是常用的时间序列模型，用于从观测数据中识别和量化动态交互过程之间的因果相互作用。这些过程之间的因果关系可以通过一个有向过程图（一种连接两个过程的有向图，当且仅当它们之间存在直接延迟或瞬时效应）来有效表示。近期研究引入了一个框架，用于量化过程图上沿路径的频域因果效应。该框架能够识别一个过程的谱密度如何影响另一个过程的谱密度。在当前工作中，我们依据过程图所决定的代数关系，刻画了因果效应与谱贡献估计量的渐近分布。基于该渐近分布，我们为估计的效应和谱贡献构建了近似置信区间以及Wald型假设检验。在因果充分性假设下，我们考虑了对频域因果量可微的估计量类，并在此类中识别出了渐近最优估计量。我们在合成数据上演示了频域Wald检验与不确定性近似方法，并将其应用于分析10至11年太阳周期对北大西洋涛动（NAO）的影响。我们的结果证实了在10至11年时间尺度上，太阳周期对NAO存在显著效应。