Causality in Time Series - ClopiNet

Popescuwhich would correspond to a vanishing partial correlation. If all A’s are lower triangularG non-causality is satisfied (in one direction but not the converse). It is however veryrare that the physical mechanism we are observing is indeed the embodiment of a VAR,and therefore even in the case in which G non-causality can be safely rejected, it isnot likely that the best VAR approximation of the data observed is strictly lower/uppertriangular. The necessity of a distinction between strict causality, which has a structuralinterpretation, and a causality statistic, which does not measure independence inthe sense of Granger-non causality, but rather relative degree of dependence in bothdirections among two signals (driving) is most evident in this case. If the VAR in questionhad very small (and statistically observable) upper triangular elements would adiscussion of causality of the observed time series be rendered moot?One of the most common physical mechanisms which is incompatible with VARis aliasing, i.e. dynamics which are faster than the (shortest) sampling interval. Thestandard interpretation of aliasing is the false representation of frequency componentsof a signal due to sub-Nyquist frequency sampling: in the multivariate time-series casethis can also lead to spurious correlations in the observed innovations process (Phillips,1973). Consider a continuous bivariate VAR of order 1 with Gaussian innovations inwhich the sampling frequency is several orders of magnitude smaller than the Nyquistfrequency. In this case we would observe a covariate time independent Gaussian processsince for all practical purposes the information travels ‘instantaneously’. In economics,this effect could be due to social interactions or market reactions to news whichhappen faster than the sampling interval (be it daily, hourly or monthly). In fMRI analysissub- sampling interval brain dynamics are observed over a relatively slow timeconvolution process of hemodynamic response of neural activity (for a detailed expositionof causality inference in fMRI see Roebroeck et al. (2011) in this volume).Although ‘aliasing’ normally refers to temporal aliasing, the same process can occurspatially. In neuroscience and in economics the observed variables are summations(dimensionality reductions) of a far larger set of interacting agents, be they individualsor neurons. In electroencephalography (EEG) the propagation of electrical potentialfrom cortical axons arrives via multiple pathways to the same recording location onthe scalp: the summation of micrometer scale electric potentials on the scalp at centimeterscale. Once again there are spurious observable correlations: this is known asthe mixing problem. Such effects can be modeled, albeit with significant informationloss, by the same DGP class which is a superset of VAR and known in econometrics asSVAR (structural vector auto-regression, the time series equivalent of structural equationmodeling (SEM), often used in static causality inference (Pearl, 2000)). Anotherbasic problem in dynamic system identification is that we not only discard much informationfrom the world in sampling it, but that our observations are susceptible toadditive noise, and that the randomness we see in the data is not entirely the randomnessof the mechanism we intend to study. One of the most problematic of additivenoise models is mixed colored noise, in which there are structured correlations both intime and across elements of the time-series, but not in any causal way: there is only alinear transformation of colored noise, sometimes called mixing, due to spatial aliasing.40