The claim
Granger causality is meant as TERM-058 defines it: predictive, relative to a stated information set. NOTE-010 records the ΦID review’s C5, §IV A: if all relevant variables are measured, Granger- and Pearl-type analyses coincide. It has no citation and no conditions. The record holds instead:
- Necessary. An unobserved relevant process can produce a Granger-causal relation. In Eichler and Didelez’s Example 4.2 a latent Z adds α₁α₂/(1 + α₂²) to X2’s one-step coefficient on X1, which the reading checked (NOTE-941). That is why they need V to include every relevant process, observed or not (THEORY-704).
- Not sufficient. Their Assumption 2.3(i) also needs the intervention independent of the other variables at the same time, which excludes influences that act within one sampling step. Without it, the inference fails with nothing hidden (below).
- What the conditions buy. A missing edge a → b means no effect on b at the next step. Non-ancestry means none at any horizon. The effect is identified from the observational predictor, with S = V always enough. A present edge is “potentially causal”. So the two analyses agree on zero effects and share an estimand. They are still not one analysis.
The counterexample
Take two series, nothing hidden: a(t) = ε_a(t) and b(t) = ρ b(t − 1) + γ a(t) + ε_b(t), with independent white noises, |ρ| < 1 and γ ≠ 0.
- a is strongly Granger-noncausal for b relative to {a, b}. b(t + 1) = ρ b(t) + γ ε_a(t + 1) + ε_b(t + 1), and ε_a(t + 1) is independent of the whole past. So given b’s past, a’s past adds nothing.
- Intervening on a(t) changes b(t + 1). Setting a(t) = x gives b(t) = ρ b(t − 1) + γx + ε_b(t), so E b(t + 1) = ργx.
- What fails. Assumption 2.3(i): b(t) responds to the intervention on a(t). In the mixed graph a and b are joined by a dashed edge, not an arrow.
Checked by simulation on 2026-10-16 (ρ = 0.6, γ = 0.8, n = 2·10⁵). OLS of b(t + 1) on b(t), b(t − 1), a(t), a(t − 1) gives coefficients 0.601, −0.004, −0.004 and 0.004. The residual correlates 0.62 with a(t + 1), which is the contemporaneous dependence. The intervention effect is ργ = 0.48.
The process has zero full-history transfer entropy from a to b and positive directed information, all of it in the instantaneous term I(a(t); b(t) | past) (THEORY-711). That term is what Granger causality and transfer entropy leave out.
The example is a sampling problem, not a pathology. Any influence faster than the sampling interval, or any aggregation over time, puts the example’s structure into the data.
What it does not say
- That Granger analysis is useless for causal questions. Under the conditions it is how the effect is identified.
- That the conditions above are the only sufficient ones. Eichler and Didelez give sufficient conditions, and the front-door criterion identifies some effects with a confounder unobserved.
- That ΦID’s quantities are interventional or not. That is NOTE-010’s C4 and THEORY-497’s footnote-3 condition, which this claim leaves standing.