  The paper introduces a novel conditional independence (CI) based method for
linear and nonlinear, lagged and contemporaneous causal discovery from
observational time series in the causally sufficient case. Existing CI-based
methods such as the PC algorithm and also common methods from other frameworks
suffer from low recall and partially inflated false positives for strong
autocorrelation which is an ubiquitous challenge in time series. The novel
method, PCMCI$^+$, extends PCMCI [Runge et al., 2019b] to include discovery of
contemporaneous links. PCMCI$^+$ improves the reliability of CI tests by
optimizing the choice of conditioning sets and even benefits from
autocorrelation. The method is order-independent and consistent in the oracle
case. A broad range of numerical experiments demonstrates that PCMCI$^+$ has
higher adjacency detection power and especially more contemporaneous
orientation recall compared to other methods while better controlling false
positives. Optimized conditioning sets also lead to much shorter runtimes than
the PC algorithm. PCMCI$^+$ can be of considerable use in many real world
application scenarios where often time resolutions are too coarse to resolve
time delays and strong autocorrelation is present.
