Contribution
Granger noncausality had been read informally as “not causal”, and Granger causality as “potentially causal”. This paper states when the first reading is a theorem. It defines causation for time series by intervention, links interventions to the Granger-causal Markov property of a mixed graph, and gives graphical identification criteria for time series, analogues of Pearl’s back-door and front-door criteria.
Key insight
An intervention indicator can be treated as one more node, with a single edge into the intervened component. If the indicator is Granger-noncausal for b given S at all horizons, the observational predictor of b from S’s past is also the post-intervention predictor. The graph’s paths say when that holds.
Assumptions
- X is stationary and mixing, with regular, a.s. absolutely continuous conditional distributions (§2).
- Assumption 2.3 (intervention). (i) (X_V(t−1) history, X_{V∖a}(t)) ⊥⊥ σ_a(t). Whether one intervenes is neither informed by the past nor informative about the other variables at t, which “excludes instantaneous causality”. (ii) {X_V(t + j), j ∈ ℕ} ⊥⊥ σ_a(t) | X_V(t) history. The future is tied to the intervention only through the present and past. Together they make an intervention “an isolated exogenous change of the system”. They must be checked in each application.
- V is rich enough. The system V includes every relevant process, observable or not, so that Assumption 2.3 can be believed. The observed processes S ⊆ V may be fewer (§2.1).
- σ is a decision variable, not a random one (Remark 2.2).
- The global Granger-causal Markov property for a mixed graph G (Def. 4.6) for the graphical results.
- Effects are defined for h > 0. Contemporaneous effects are outside the framework.
Key results
- Theorem 3.1, back-door. If Assumption 2.3 holds, a, b ∈ S, and X_b(t + h) ⊥⊥ σ_a(t) | X_S(t) history for all h, then S identifies the effect: E_s X_b(t + h) = E_∅ E_s[E_∅(X_b(t + h) | X_S(t) history)]. “Under Assumptions 2.3, V always has to identify the causal effect due to condition (ii).”
- Example 3.2, Gaussian. For a purely nondeterministic stationary Gaussian process with a well-conditioned spectrum, the effect of setting X_a(t) = x* is Φ_ba^{(h)}(1)x*, the coefficient of the best linear h-step predictor from S’s autoregression, computed recursively.
- Def. 4.1, strong Granger noncausality. X_A ↛ X_B [X_V] if X_B(t + 1) ⊥⊥ X_A history at t | X_{V∖A} history at t: conditional independence, not just absence of correlation.
- Theorem 4.8. If every an(B)-pointing path between A and an(B) is m-blocked given B ∪ C, X_A is Granger-noncausal for X_B at all horizons.
- Lemma 4.10, Theorem 4.11. If every an(b)-pointing back-door path between a and an(b) is m-blocked given S, X_b(t + h) ⊥⊥ σ_a(t) | X_S(t) history, so Theorem 3.1 applies.
- Theorem 4.13, front-door. Three m-blocking conditions through a mediating set C identify the effect. Proof deferred.
- Conclusion. No edge a → b implies no effect of X_a(t) on X_b(t + 1); a ∉ an(b) implies no effect at any horizon. Granger-causal edges are “potentially causal”, given Assumption 2.3.
- Example 4.2, ctd. With Z latent, the effect of X3(t) on X1(t + 2) is β₁₂β₂₃x₃*. Back-door with S = {1, 2, 3}, from the observed autoregression, gives φ₁₂(1)φ₂₃(1) + φ₁₃(2) = β₁₂β₂₃. Checked: projecting Z(t − 2) on the observed past gives coefficient c = α₁α₂/(1 + α₂²) on X2(t − 1) and −cβ₂₃ on X3(t − 2), so (β₁₂ + c)β₂₃ − cβ₂₃ = β₁₂β₂₃. The latent confounder biases the one-step coefficients but cancels in the two-step one.
- Example 4.14. With Z correlated with X3’s noise, back-door fails, and the front-door criterion gives φ₁₂(1)φ₂₃(1)x₃* = β₁₂β₂₃x₃*. Checked: Z(t) is untouched by the intervention, and the naive two-step coefficient carries an extra αρ.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Under Assumption 2.3, if S’s observed past makes b’s future independent of σ_a, the effect is identified from observational quantities | strong | Theorem 3.1, proved |
| C2 | Under Assumption 2.3, the whole system V always identifies the effect | strong | by condition (ii); remark after Theorem 3.1 |
| C3 | Graphical back-door blocking in the mixed graph is sufficient for C1 | strong | Lemma 4.10 via Theorem 4.8; proof sketched by augmenting the graph |
| C4 | A missing Granger edge means no effect at the next step; non-ancestry means none at any horizon | moderate | Conclusion, “by a similar reasoning as for Lemma 4.10”; not written out |
| C5 | A front-door criterion identifies effects under a latent confounder of the cause | moderate | Theorem 4.13 stated; proof deferred to work in preparation; Example 4.14 checks |
| C6 | In a Gaussian VAR the effect is a multi-step linear-predictor coefficient | strong | Example 3.2 with Box et al.’s recursion; Examples 4.2, 4.14 checked |
| C7 | Granger (1969) required the series to contain all relevant processes but did not ask whether an intervention is possible | assertion | Conclusion; about Granger’s paper, not shown from it |
Concepts
- Intervention indicator σ_a(t) — idle (∅), atomic (set to x*), conditional (a function of the observed past) or random (a known distribution) (Def. 2.1).
- ACE — E_{σ_a(t)=s} X_b(t + h) (Def. 2.5).
- Mixed graph — directed edges a → b for Granger causality, dashed undirected edges for contemporaneous dependence, up to three edges per pair (§4.2). Contemporaneous dependence may come from hidden processes.
- m-separation — path blocking where a node is an m-collider if both adjacent edges have an arrowhead or a dashed end at it (Def. 4.4).
Connections
- Builds on Granger (1969, LIT-1173), Florens and Mouchart’s strong noncausality, Eichler’s graphical time-series models, Pearl’s back-door and front-door criteria, and Dawid’s decision-theoretic intervention indicators. White (2006) is named as the nearest work on interventions in time series.
- NOTE-010 C5, “if all relevant variables are measured, Granger- and Pearl-type analyses coincide”. This paper is a statement of conditions for it. Measuring everything gives C2, but C2 also needs Assumption 2.3(i), which excludes instantaneous causation. My counterexample, filed in CLAIM-333, has every variable measured and a Granger-noncausal for b, yet a lagged intervention effect of a on b. It works because a acts on b within one sampling step, which the graph shows only as a dashed edge.
- NOTE-010 C4 and THEORY-497 say ΦID’s quantities are interventional if p(X_t′ | X_t) is a do()-distribution, with faithfulness and the causal Markov condition. Assumption 2.3 is the time-series form of the first condition. Theorem 3.1 needs neither faithfulness nor a Markov condition. The graphical criteria need the global Granger-causal Markov property, an analogue of the causal Markov condition. No result here needs faithfulness.
- NOTE-935 (Seth 2010). A macro variable that is a function of the micro variables at the same time cannot be intervened on as Assumption 2.3(i) requires. So Granger causation from it to its micro variables is not an effect in this paper’s sense. My inference; filed as CLAIM-334.
Bearing on the record
- Filed as THEORY-704. It answers NOTE-010’s C5, which is annotated, and grounds CLAIM-333.
- NOTE-934’s C4 says only perturbation identifies physical causation “unambiguously”. By this paper observational data identify it too, but only under Assumption 2.3, which the data cannot test. So “unambiguously” is the word that keeps C4 true.
- For the Anthology of the SOTA: no instruction for ML practice.
Limitations
- Every result is a sufficient condition. Nothing says when an effect is not identifiable.
- Assumption 2.3 is untestable from observational data. The paper says it must be believed “based on substantive background knowledge”.
- C4 is stated in the conclusion as easily seen, and C5’s proof is not in the paper.
- The examples are linear Gaussian. The definitions are general, but the computation of effects is shown only for the Gaussian case.
- Excluding instantaneous causality makes the results depend on the sampling interval, which the paper notes is “often chosen arbitrarily” (§4.1) while declining a per-time-point graph for that reason.
Open questions
- Sequential interventions at several times (the authors’, §5).
- Necessary conditions for identification, the analogue of completeness results for DAGs.
- What survives when the sampling interval is coarser than the causal one, so that Assumption 2.3(i) fails by aggregation.