John Geweke (1982), “Measurement of Linear Dependence and Feedback between Multiple Time Series”, Journal of the American Statistical Association 77(378): 304–313 — doi:10.1080/01621459.1982.10477803
Key takeaways
Unread. From the abstract and citing works, to be checked against the text.
- Three-way split. The abstract, as Semantic Scholar gives it: the measure of linear dependence is the sum of linear feedback from the first series to the second, from the second to the first, and instantaneous linear feedback.
- Generalized variance. For a multivariate predictee the residual variance is replaced by its determinant, F = ln(|Σ(ε)|/|Σ(ε′)|). Barnett, Barrett and Seth adopt this form, say Geweke lists several motivations for it, and add the Gaussian equivalence with transfer entropy as another (NOTE-934).
- Inference. The maximum-likelihood estimator is asymptotically χ² under the null and non-central χ² under the alternative, as Barnett, Barrett and Seth cite it ([14]).
- By frequency. The directed measures decompose additively over frequency, which is the spectral Granger causality later work uses.
Standing in the record
Filed on 2026-10-16 as the form of Granger causality that the record’s two transfer-entropy readings rest on. Barnett, Barrett and Seth’s equivalence is stated for this measure, not for Granger’s univariate one. Until a copy is supplied the record holds only what they and the abstract say. That includes the non-central χ² claim, which NOTE-053 argues can only be a local-alternative approximation.
Its three-way split, two directed feedbacks and an instantaneous one, has the shape of the decomposition of Massey’s directed information into full-history transfer entropy plus instantaneous terms (THEORY-711). For Gaussian processes the directed parts are the Granger measures, twice the transfer entropies (THEORY-700). That the correspondence is exact term by term is the record’s inference, unchecked against the paper.