Spectral Granger Causality: Mode-by-Mode Causal Bandwidth

Granger causality (Granger, 1969) tests whether the past of \(X\) helps predict \(Y\) beyond \(Y\)'s own past. The test produces a single \(p\)-value and F-statistic, answering "does \(X\) cause \(Y\)?" but not "how?" or "through which mechanism?" We introduce spectral Granger causality: project both the restricted model (\(Y\) on its own lags) and the unrestricted model (\(Y\) on its own lags plus \(X\)'s lags) onto the eigenmode basis, and compare the spectral information states mode by mode.

For each eigenmode \(k\), the causal information flow is:

\[\Delta I_k = \frac{1}{2}\log\frac{\sigma^2_{k,R}}{\sigma^2_{k,U}}\]

where \(\sigma^2_{k,R}\) and \(\sigma^2_{k,U}\) are the per-mode uncertainties in the restricted and unrestricted models. This is the transfer entropy per mode — the bits of information flowing from \(X\) to \(Y\) through eigenmode \(k\). The total transfer entropy \(\sum_k \Delta I_k\) recovers the classical Granger statistic. The per-mode decomposition reveals the causal mechanism: which frequencies, which patterns, which timescales carry the causal influence.

The causal bandwidth \(\Delta K^ = K^_U - K^_R\) counts how many new signal modes emerge when \(X\)'s information is added. If \(\Delta K^ = 0\): \(X\) adds no new resolvable structure. If \(\Delta K^* = 3\): \(X\) opens three new channels of predictability in \(Y\).

The causal direction is determined by the asymmetry: \(\Delta K^(X \to Y) \neq \Delta K^(Y \to X)\). If \(X\) truly causes \(Y\): adding \(X\) to \(Y\)'s model creates new modes, but adding \(Y\) to \(X\)'s model does not. The spectral framework makes this asymmetry visible mode by mode.

Advantages over classical Granger: (1) structural — identifies WHICH aspects of \(X\) influence WHICH aspects of \(Y\); (2) robust — multicollinearity is handled by spectral shrinkage; (3) multi-output — \(p\)-values, confidence intervals, Bayesian posteriors, and MDL costs are all available per causal channel from the spectral information state; (4) quantitative — transfer entropy per mode gives the information flow in bits, not just a binary yes/no.