  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.
