arXiv 13 Aug 2026 · Econometrics
arXiv:2608.13431 · PDF · Extracted main text
Many questions across the sciences take the same form: several coupled series are observed together, and the analyst wants to know not merely that they move together but which one moves first, and how strongly. This paper sets out a complete method built on one organising idea: the direction of a coupled system is exactly the part of its behaviour that changes when the record is played backwards. Tools built on contemporaneous covariance alone (correlation matrices, distance measures, spanning trees, undirected centralities, principal components) carry no information about direction: a reversible system and a circulating one can share identical covariance at every sampling of the same point-in-time record. Formally, direction is a circulation matrix carried by the lagged covariance. Its vanishing is exactly statistical time reversibility for linear systems, feature maps carry the characterisation to nonlinear ones, and under the Gaussian benchmark its magnitude is an entropy-production functional of the identified circulation, the quadratic component of the divergence per unit time between the forward and reversed records. Around this estimand we build a cross-fitted estimator removing first-order bias, delete-block jackknife standard errors, and a randomisation test exact under its stated block null, with a familywise correction and a nonlinear extension. A sampling theory says when the arrow is measurable at all, and a design layer separates transmission from the ordering of clocks. A laboratory of four systems with known answers compares the method with correlation networks, Granger causality, transfer entropy, and connectedness indices, reporting the failures of each when read as a measure of direction, including our own. Complete algorithms and worked examples in two languages make the paper the base reference for a series of applications.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Joseph P. Romano and Michael Wolf (2005) Stepwise multiple testing as formalized data snooping | 0.737 | 4 | 4 | 50% |
| 2 | Lionel Barnett, Adam B. Barrett, and Anil K. Seth (2009) Granger causality and transfer entropy are equivalent for Gaussian variables | 0.737 | 3 | 3 | 67% |
| 3 | Claude Godrèche and Jean-Marc Luck Characterising the nonequilibrium stationary states of Ornstein–Uhlenbeck processes | 0.737 | 3 | 3 | 67% |
| 4 | Lars Peter Hansen and Thomas J. Sargent (1983) The dimensionality of the aliasing problem in models with rational spectral densities | 0.737 | 3 | 3 | 67% |
| 5 | Peter C. B. Phillips (1973) The problem of identification in finite parameter continuous time models | 0.737 | 3 | 3 | 67% |
| 6 | Gideon Weiss (1975) Time-reversibility of linear stochastic processes | 0.737 | 3 | 3 | 67% |
| 7 | John Geweke (1982) Measurement of linear dependence and feedback between multiple time series | 0.644 | 2 | 2 | 100% |
| 8 | Da-Quan Jiang, Min Qian, and Min-Ping Qian (2004) Mathematical Theory of Nonequilibrium Steady States, volume 1833 of Lecture Notes in Mathematics | 0.644 | 2 | 2 | 100% |
| 9 | Frank P. Kelly (1979) Reversibility and Stochastic Networks | 0.644 | 2 | 2 | 100% |
| 10 | Chulan Kwon, Ping Ao, and David J. Thouless (2005) Structure of stochastic dynamics near fixed points | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 51 scored citations.