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TERM-058: Granger causality: the past of one series improving the prediction of another's next value beyond the past of everything else in a stated information set, a predictive relation relative to that set and its lags, and not by itself the effect of an intervention

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Properties4
StatusActive
Tagscausality, mathematical-statistics
Lineformal-apparatus
Used byCLAIM-333 — Measuring every relevant variable is necessary but not sufficient for a Granger-causal analysis to say anything about interventions: the intervention must also be an isolated exogenous change and no influence may act within one sampling step, and then Granger noncausality implies no intervention effect and the effect is identified from the observational predictor, but a Granger-causal relation is still only potentially causal, CLAIM-334 — Granger causation from a macro variable to the micro variables it supervenes on, Seth's downward Granger causation, is not evidence of a macro-to-micro intervention effect, even with every micro variable measured: supervenience leaves no intervention on the macro variable that holds its micro base fixed at the same time, CLAIM-335 — Transfer entropy measures directed predictive improvement, not information flow: as a conditional mutual information it is the source's unique information about the target's next value plus its synergy with the target's past, so it can credit a source with a bit the source carries only jointly and read zero from sources that jointly fix the target, and conditioning on more series does not remove the synergy

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Definition

Let X_V be a stationary multivariate time series and A, B disjoint sets of its components. X_A Granger-causes X_B relative to X_V if the past of X_A improves the prediction of X_B(t + 1) beyond the past of the other components of V. Strong Granger noncausality is the distributional form: X_B(t + 1) ⊥⊥ X_A’s past | X_{V∖A}‘s past (Eichler and Didelez, Def. 4.1, NOTE-941). The usual measure is linear. For a univariate predictee it is the log ratio of residual variances of the restricted and full regressions. For a multivariate predictee it is Geweke’s log ratio of their determinants (NOTE-934, Eqs. 8–9).

Three things are always relative and should be stated with it:

  1. The information set V. Adding or removing a series can create or remove a Granger-causal relation (NOTE-941, Example 4.7).
  2. The lags and the sampling interval. An influence faster than one sampling step shows as contemporaneous dependence, not as Granger causation (CLAIM-333).
  3. The model class, for the measured quantity. A linear Granger statistic is blind to dependence a linear predictor cannot use (Seth, §2.3, NOTE-935).

What it must not be confused with

  • Intervention (Pearl-type) causation. The effect on X_B of setting X_A. Granger noncausality implies its absence only under Eichler and Didelez’s conditions (THEORY-704). A Granger-causal relation is at most “potentially causal”.
  • Transfer entropy. H(X_B(t + 1) | rest’s past) − H(X_B(t + 1) | all past). It is the same quantity, halved, for jointly Gaussian processes (THEORY-700). In general it is the likelihood-ratio limit of a well-specified model (THEORY-641). Granger causality is the linear case, not an approximation to it, when the data are Gaussian.
  • Directed information (Massey). For Gaussian data Granger causality corresponds to its lagged part, full-history transfer entropy; the instantaneous part, which Geweke’s measure of instantaneous feedback tracks, is outside it (THEORY-711, TERM-057).
  • Interventional information flow (Ay and Polani), the information-theoretic counterpart of an intervention effect (THEORY-710).
  • Downward (Granger) causation in Seth’s sense: Granger causation from a macro variable to its micro variables. A use of the term, not a kind of causation (CLAIM-334).
  • Granger’s own definition (1969, LIT-1173), which the record has not read. The citing works agree that it is predictive and relative to the relevant information. The record’s definition is theirs.

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