THEORY-700: For jointly Gaussian stationary processes, Granger causality in Geweke's generalized-variance form is exactly twice the transfer entropy, conditional and multivariate, at the regressions' lags, because least-squares residual covariance is the partial covariance and a Gaussian's conditional entropy is half its log-determinant plus a constant

Source

Barnett, Barrett & Seth (2009), “Granger causality and transfer entropy are equivalent for Gaussian variables”, arXiv 0910.4514v2 (Phys. Rev. Lett. 103, 238701), LIT-1171, read in full in NOTE-934. The PRL version was not compared.

What was actually shown

The setting. X_t, Y_t, Z_t jointly Gaussian and stationary, any dimensions, lags p, q, r fixed. G-causality is Geweke’s F_{Y→X|Z} = ln(|Σ(X|X⁻⊕Z⁻)|/|Σ(X|X⁻⊕Y⁻⊕Z⁻)|) (Eq. 9). Transfer entropy is T_{Y→X|Z} = H(X|X⁻⊕Z⁻) − H(X|X⁻⊕Y⁻⊕Z⁻) (Eq. 10).

Two identities. The OLS residual covariance of X on Y is the partial covariance Σ(X|Y) (Eq. 3). For jointly Gaussian X, Y, H(X|Y) = ½ln|Σ(X|Y)| + ½n ln(2πe), by |Σ(X⊕Y)| = |Σ(Y)|·|Σ(X|Y)| (Eq. 11). Both were checked.

The result. F_{Y→X|Z} = 2T_{Y→X|Z} (Eq. 13). With a univariate predictee it is the standard log variance ratio (Eq. 8).

What could have come out otherwise: the identity needs the determinant. With the trace (total variance) in place of the generalized variance, a multivariate predictee would not give twice the transfer entropy. That is the Letter’s argument for Geweke’s form.

What this does not say

• That either quantity measures causation in the interventional sense. The authors say a G-causality interaction is not a physically instantiated one (p. 2). When it is, is THEORY-704’s question.
• That estimates agree in sample. Covariance-based entropy estimates agree by construction. Partition, kernel and nearest-neighbour estimators need not (p. 7).
• Anything outside Gaussianity. That nonzero G-causality implies nonzero transfer entropy in general is cited, not shown. That nonlinear G-causality adds nothing for Gaussian AR processes rests on unpublished work.
• That the regressions model the process. They are predictive fits and may omit latent variables (fn. 31).

Connections

• THEORY-641 generalizes this result: transfer entropy is the limit of a log-likelihood ratio under any well-specified parametric Markov model, and the linear Gaussian VAR is its case “half the Granger statistic”.
• Geweke (1982, LIT-1169, Unavailable) is the source of Eq. 9 and of the frequency decomposition the authors propose to carry over to transfer entropy.
• TERM-058 defines Granger causality in the record’s own words.