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THEORY-704: In a stationary multivariate time series whose components include every relevant process, with interventions that are isolated exogenous changes and no instantaneous causation, a missing Granger-causal edge from a to b means intervening on a has no effect on b at the next step, non-ancestry means no effect at any horizon, and a set whose observed past blocks every back-door path into b's ancestors identifies the effect from observational data, as a multi-step linear-predictor coefficient in the Gaussian case

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Properties4
StatusActive
Tagscausality, mathematical-statistics
SourceLIT-1164 — Causal Reasoning in Graphical Time Series Models
SupportsCLAIM-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

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Source

Eichler & Didelez (2007), “Causal Reasoning in Graphical Time Series Models”, UAI 2007: 109–116, arXiv 1206.5246, LIT-1164, read in full in NOTE-941.

What was actually shown

Causation by intervention. The effect of Xa(t) on Xb(t + h), h > 0, is E_{σa(t)=s} Xb(t + h) for an atomic, conditional or random intervention s (Defs. 2.1, 2.5).

Assumption 2.3. (i) The intervention is independent of the system’s past and of the other components at t, which excludes instantaneous causality. (ii) The future depends on the intervention only through the present and past. This is believable only if the system V includes every relevant process, observed or not.

Identification (Theorem 3.1). If the observed past of S makes Xb(t + h) independent of σa(t), the effect is an observational quantity. Under Assumption 2.3, S = V always qualifies.

Graphs (Lemma 4.10, Theorems 4.8, 4.11, 4.13). If X satisfies the global Granger-causal Markov property for a mixed graph, m-blocking of every an(b)-pointing back-door path given S suffices. A front-door criterion handles a latent confounder of the cause.

Granger to intervention (Conclusion). No directed edge a → b means no effect of Xa(t) on Xb(t + 1); a ∉ an(b) means no effect at any horizon. A present edge is “potentially causal”.

Gaussian case (Example 3.2). The effect of setting Xa(t) = x* is Φ_ba^{(h)}(1)x*. Examples 4.2 (back-door, latent Z) and 4.14 (front-door) both give β₁₂β₂₃x₃*, which the reading checked against the defining equations.

What could have come out otherwise: in Example 4.2 the latent Z biases the one-step regression coefficients, and the effect is still recovered, because the bias cancels in the two-step predictor.

What this does not say

  • That measuring every relevant process is enough. Assumption 2.3(i) is a separate condition. A process with every variable measured but a causal influence faster than the sampling step violates it, and Granger noncausality can then coexist with an intervention effect (CLAIM-333).
  • That a Granger-causal edge is a causal effect. Only the absence of an edge is given an interventional meaning. A present edge is potential.
  • When an effect is not identifiable. All criteria are sufficient.
  • That Assumption 2.3 can be checked from data. It rests on background knowledge.
  • Anything about contemporaneous effects, which the definitions exclude (h > 0).

Connections

  • It states the conditions that NOTE-010’s C5 omits. The record’s statement is CLAIM-333.
  • Its Assumption 2.3 is the time-series form of THEORY-497’s condition that p(X_t′ | X_t) be a do()-distribution. Its back-door theorem needs no faithfulness.
  • Supervenience rules out the intervention 2.3(i) requires for a macro variable, so macro-to-micro Granger causation is not an effect in this sense (CLAIM-334).
  • Ay and Polani’s information flow (THEORY-710) is the information-theoretic measure of the same interventional notion. It assumes the mechanisms are known; this paper supplies conditions for identifying effects from observational data.

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