C07 — Granger Causality

This framework asks: Does knowing a component’s earlier-layer activation improve our prediction of a downstream component, beyond what the downstream component’s own history provides?

Granger causality (GC) is a statistical notion of predictive causality: X Granger-causes Y if past values of X contain information about future values of Y that is not already contained in past values of Y alone. Applied to transformer circuits, where “time” is layer depth, GC tests whether an earlier component’s output carries predictive signal for a later component’s state.

GC is attractive because it requires only observational data — no interventions. This makes it computationally cheap and applicable to any recorded activation trace. However, it captures predictive relationships, not true causal mechanisms — a limitation addressed by the interventional instruments in the causal pillar.

Theoretical grounding

Source	Year	Key contribution
Granger, “Investigating Causal Relations by Econometric Models and Cross-spectral Methods”	1969	Original Granger causality definition
Barnett et al., “Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables”	2009	Equivalence to transfer entropy
Tank et al., “Neural Granger Causality”	2022	Nonlinear GC via neural networks
Schwab & Bhatt, “CXPlain: Causal Explanations for Model Interpretation”	2019	Granger-style importance for neural network components

Core concept

Variable ( X ) Granger-causes ( Y ) if:

[ H(Y_t \mid Y_{t-1}, \ldots, Y_{t-p}) > H(Y_t \mid Y_{t-1}, \ldots, Y_{t-p}, X_{t-1}, \ldots, X_{t-p}) ]

In the linear Gaussian case, this reduces to comparing the residual variance of two autoregressive models — one with and one without the candidate cause. The F-statistic for this comparison provides a significance test.

For transformer circuits with layer index ( \ell ) as the time axis, let ( A_i^\ell ) be the activation of head ( i ) at layer ( \ell ). Head ( i ) Granger-causes head ( j ) if:

[ \mathrm{Var}(A_j^{\ell+k} \mid A_j^{\ell+k-1}) > \mathrm{Var}(A_j^{\ell+k} \mid A_j^{\ell+k-1}, A_i^\ell) ]

across the corpus, tested via the F-statistic.

Instruments under C07

OCSE Script (07_ocse.py)

Observational causal sensitivity estimation implements a form of predictive causality by measuring how natural variation in one component’s activation predicts variation in downstream components — the nonparametric analogue of Granger causality.

What it establishes: Predictive directed relationships between circuit components without intervention. What it does not establish: True causal necessity — Granger causality can be confounded by common inputs from the residual stream.

Usage:

uv run python 07_ocse.py --tasks ioi sva

Reading the scores

| Pattern | What it means | |---|---|---| | Significant GC from head A to head B | A’s activation predicts B beyond B’s own context | | Bidirectional GC | Likely driven by a shared upstream cause | | GC only at specific layer gaps | Characteristic information relay depth | | No GC despite same-circuit membership | Parallel computation without interaction |

Connection to other frameworks

Granger causality is the linear-model counterpart of C03 (Transfer Entropy) — they are equivalent for Gaussian processes but TE captures nonlinear dependencies. C08 (OCSE) implements a flexible version of predictive causality. When GC identifies a link, C09 (NOTEARS) can incorporate it as a constraint in DAG recovery, and the causal pillar validates whether the link is functionally necessary via intervention.