How would you test instantaneous causality? I'm working with two time series over the same time period, and by plotting the time series graphically there is an obvious relationship that exists between the two. When one's values peak, the other's instantaneously drop. 
Is there a way I could calculate a significance test for such relationship? I know the Granger causality test models a series' current state as a function the previous states of itself and the other series. In my case the relationship is instantaneous, so it's not giving me anything tangible.
 A: Granger causality is not causality. Granger causality is actually prediction of a time series based on distributed lags from that time series as well as other time series. Causality is the ability to infer a counterfactual difference in outcomes given you experimentally manipulate ("do") an exposure in a hypothetical research setting. 
Instead, if you wish to measure how "instantaneously related" two time series are, calculate the cross-correlation of the two time series. This test can be non-specific, since it's possible that two ARMA processes simply follow the same seasonal trends. You can expand on the idea of cross correlation by fitting the following model:
$$E(Y_{(t)}, | Y_{(s<t)}, X_{(s<t)}, X_t) = \sum_{j=1}^s \beta_j Y_{(j)} + \sum_{j=1}^n\gamma_j X_{(j)} + \gamma_t X_{(t)}$$
Here the series are either discrete or pseudo-discrete and the linear combinations may be generalized to some form of GAM. The goal is performing semi-parametric inference on the term $\gamma_t$ which estimates a mean difference in the outcome, $Y_{t}$ controlling for all Granger predictive values (Granger causal is misleading in light of modern causal inference). If the $\gamma_t$ is statistically significant, we can infer that the apparent cross correlation of time series owes to an instantaneous value, rather than spurious synchronization, or a shared historical autocorrelation.
A: A quite recent method to assess causality between two times series is the Convergent Cross Mapping Method [1] available through the official R package rEDM, with its comprehensive tutorial [3].
And other one is the PC-MCI method [4] available through the Tigramite python package [5].
Some description and explanation about the above methods are discussed in a similar topic on earth stackexchange : Correlation and causation.

*

*[1] Sugihara, G., R. May, H. Ye, C.-h. Hsieh, E. Deyle, M. Fogarty, and S. Munch. “Detecting Causality in Complex Ecosystems.” Science 338, no. 6106 (October 26, 2012): 496–500.


*[2] rEDM: Applications of Empirical Dynamic Modeling from Time Series


*[3] rEDM tutorial


*[4] Runge, Jakob, Dino Sejdinovic, and Seth Flaxman. “Detecting Causal Associations in Large Nonlinear Time Series Datasets .” arXiv:1702.07007 [Physics, Stat], February 22, 2017.


*[5] TIGRAMITE – Causal discovery for time series datasets
A: In the empirical sciences, you can't prove anything, just provide evidence supporting something. Worse, time series can't provide any direct evidence of a causal relationship, because any association between measured variables could be due to a causal effect of unmeasured variables. Finally, beware of testing hypotheses suggested by the data: a typical hypothesis test works on the assumption that you came up with the hypothesis before you saw the data you're testing.
