The 2018 Runge et al paper titled "Detecting causal associations in large non linear time series datasets" describes the PCMCI method. It compares the new PCMCI method with another method they call Full Conditional Indepence testing (FullCI). FullCI works by testing for independence between two variables conditioned on all other variables (it's like a brute force version of PC, if I understand it correctly).
They highlight the need for PCMCI by using FullCI to test for the causal relationship between weather patterns ($Nino$) and British Columbia surface temperature ($BCT$). First they test for (and correctly find) the link $Nino \rightarrow BCT$ using FullCI. Then they add a variable $Z$ that is driven by $Nino + \mu $ with some lag where $\mu$ is noise. Importantly, $Z$ does not influence $BCT$.
Adding $Z$ causes the effect and significance of the $Nino \rightarrow BCT$ relationship to reduce. They explain part of this reduction by pointing out that $Z$ will "explain away" some of the effect of $BCT$ since it will be part of the conditioning set, which makes sense to me. But they also state that part of the reason for the reduction in significance is due to the increased dimensionality.
Why/how does dimensionality reduce the effect size / significance of causal relationships identified by this "FullCI" method?
