[1509.07636] Validity of time reversal for testing Granger causality
While further compelling intuitive ideas for robust causality measures have been presented [11], [23], [18], our result provides, to the best of our knowledge, the first proof of the correctness of one of such techniques (Diff-TRGC) for a relatively general class of timeseries models

Abstract: Inferring causal interactions from observed data is a challenging problem,
especially in the presence of measurement noise. To alleviate the problem of
spurious causality, Haufe et al. (2013) proposed to contrast measures of
information flow obtained on the original data against the same measures
obtained on time-reversed data. They show that this procedure, time-reversed
Granger causality (TRGC), robustly rejects causal interpretations on mixtures
of independent signals. While promising results have been achieved in
simulations, it was so far unknown whether time reversal leads to valid
measures of information flow in the presence of true interaction. Here we prove
that, for linear finite-order autoregressive processes with unidirectional
information flow, the application of time reversal for testing Granger
causality indeed leads to correct estimates of information flow and its
directionality. Using simulations, we further show that TRGC is able to infer
correct directionality with similar statistical power as the net Granger
causality between two variables, while being much more robust to the presence
of measurement noise.

Fig. 1. Performance of Granger causality and different variants of timereversed Granger causality (TRGC). (a) True positive rate in the noiseless case as a function of the number of samples T for fixed standard deviation σA = 0.2 of the AR coefficients, and as a function of σA for fixed T = 2000. (b) True and false positive rates as a function of the SNR for additive mixed autocorrelated noise (according to (??)) for T = 2000 and σA = 0.2. (Comparison of TRGC variants under interaction)Fig. 2. False positive rates of Granger causality (standard GC and NetGC) and difference-based time-reversed Granger causality (Diff-TRGC) as a function of the SNR for two signals lacking any causal connection. (A) Instantaneous linear mixture of two independent univariate AR(5) processes. (B) Common unobserved cause. xt and yt. (C) Superposition of two independent univariate AR(5) processes with additive Gaussian noise. (Impact of latent variables and measurement noise in the absence of causal interaction)Fig. 3. Performance of Granger causality (standard GC and Net-GC) and difference-based time-reversed Granger causality (Diff-TRGC) for two signals with unidirectional information flow from xt to yt. Shown are the fractions of true positives (xt → y detected) and false positives (yt → xt detected), when xt and yt are corrupted by noise (A-D), downsampling (E), and temporal aggregation (F). The underlying latent signals x(l) and y(l) were generated from VAR(5) processes with random AR coefficients, except for D, in which signals follow a VAR(1) process with long memory according to (??). (Impact of noise in the presence of causal interaction)›