{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T22:53:17Z","timestamp":1780613597602,"version":"3.54.1"},"reference-count":41,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T00:00:00Z","timestamp":1676332800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,11,15]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>The single-regression Granger\u2013Geweke causality estimator has previously been shown to solve known problems associated with the more conventional likelihood ratio estimator; however, its sampling distribution has remained unknown. We show that, under the null hypothesis of vanishing Granger causality, the single-regression estimator converges to a generalized \u03c72 distribution, which is well approximated by a \u0393 distribution. We show that this holds too for Geweke\u2019s spectral causality averaged over a given frequency band, and derive explicit expressions for the generalized \u03c72 and \u0393-approximation parameters in both cases. We present a Neyman\u2013Pearson test based on the single-regression estimators, and discuss how it may be deployed in empirical scenarios. We outline how our analysis may be extended to the conditional case, point-frequency spectral Granger causality and the important case of state-space Granger causality.<\/jats:p>","DOI":"10.1093\/biomet\/asad009","type":"journal-article","created":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T05:17:19Z","timestamp":1676351839000},"page":"933-952","source":"Crossref","is-referenced-by-count":6,"title":["Sampling distribution for single-regression Granger causality estimators"],"prefix":"10.1093","volume":"110","author":[{"given":"A J","family":"Gutknecht","sequence":"first","affiliation":[{"name":"Campus Institute for Dynamics of Biological Networks, Georg-August University , Kellnerweg 6 , 37077 Goettingen, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2329-9351","authenticated-orcid":false,"given":"L","family":"Barnett","sequence":"additional","affiliation":[{"name":"Sussex Centre for Consciousness Science, Department of Informatics, University of Sussex , Falmer, Brighton BN1 9RH, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,2,14]]},"reference":[{"key":"2023111615233489600_asad009-B1","first-page":"1","volume-title":"2012 IEEE International Workshop on Machine Learning for Signal Processing","author":"Amblard","year":"2012"},{"key":"2023111615233489600_asad009-B2","doi-asserted-by":"crossref","first-page":"0138105","DOI":"10.1103\/PhysRevLett.109.138105","article-title":"Transfer entropy as a log-likelihood ratio","volume":"109","author":"Barnett","year":"2013","journal-title":"Phys. Rev. Lett"},{"key":"2023111615233489600_asad009-B3","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.jneumeth.2011.08.010","article-title":"Behaviour of Granger causality under filtering: theoretical invariance and practical application","volume":"201","author":"Barnett","year":"2011","journal-title":"J. Neurosci. Meth."},{"key":"2023111615233489600_asad009-B4","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.jneumeth.2013.10.018","article-title":"The MVGC multivariate Granger causality toolbox: a new approach to Granger-causal inference","volume":"223","author":"Barnett","year":"2014","journal-title":"J. Neurosci. Meth"},{"issue":"R","key":"2023111615233489600_asad009-B5","doi-asserted-by":"crossref","first-page":"040101","DOI":"10.1103\/PhysRevE.91.040101","article-title":"Granger causality for state-space models","volume":"91","author":"Barnett","year":"2015","journal-title":"Phys. Rev. E"},{"key":"2023111615233489600_asad009-B6","doi-asserted-by":"crossref","first-page":"0238701","DOI":"10.1103\/PhysRevLett.103.238701","article-title":"Granger causality and transfer entropy are equivalent for Gaussian variables","volume":"103","author":"Barnett","year":"2009","journal-title":"Phys. Rev. Lett"},{"key":"2023111615233489600_asad009-B7","doi-asserted-by":"crossref","first-page":"E6676","DOI":"10.1073\/pnas.1714497115","article-title":"Misunderstandings regarding the application of Granger causality in neuroscience","volume":"115","author":"Barnett","year":"2018","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"2023111615233489600_asad009-B8","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1016\/j.neuroimage.2018.05.067","article-title":"Solved problems for Granger causality in neuroscience: a response to Stokes and Purdon","volume":"178","author":"Barnett","year":"2018","journal-title":"NeuroImage"},{"key":"2023111615233489600_asad009-B9","doi-asserted-by":"crossref","first-page":"1926","DOI":"10.1126\/science.1099745","article-title":"Neuronal oscillations in cortical networks","volume":"304","author":"Buzs\u00e1ki","year":"2004","journal-title":"Science"},{"key":"2023111615233489600_asad009-B10","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.jneumeth.2005.06.011","article-title":"Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data","volume":"150","author":"Chen","year":"2006","journal-title":"J. Neurosci. Meth"},{"key":"2023111615233489600_asad009-B11","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.neuroimage.2018.04.043","article-title":"Granger-Geweke causality: estimation and interpretation","volume":"175","author":"Dhamala","year":"2018","journal-title":"NeuroImage"},{"key":"2023111615233489600_asad009-B12","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1016\/j.neuroimage.2008.02.020","article-title":"Analyzing information flow in brain networks with nonparametric Granger causality","volume":"41","author":"Dhamala","year":"2008","journal-title":"Neuroimage"},{"key":"2023111615233489600_asad009-B13","doi-asserted-by":"crossref","first-page":"018701","DOI":"10.1103\/PhysRevLett.100.018701","article-title":"Estimating Granger causality from Fourier and wavelet transforms of time series data","volume":"100","author":"Dhamala","year":"2008","journal-title":"Phys. Rev. Lett"},{"key":"2023111615233489600_asad009-B14","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1002\/9783527609970.ch17","volume-title":"Handbook of Time Series Analysis","author":"Ding","year":"2006"},{"key":"2023111615233489600_asad009-B15","volume-title":"Stochastic Processes","author":"Doob","year":"1953"},{"key":"2023111615233489600_asad009-B16","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1080\/07350015.2021.1904960","article-title":"Practical methods for modeling weak VARMA processes: identification, estimation and specification with a macroeconomic application","volume":"40","author":"Dufour","year":"2022","journal-title":"J. Bus. Econ. Statist"},{"key":"2023111615233489600_asad009-B17","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.jeconom.2009.06.008","article-title":"Short and long run causality measures: theory and inference","volume":"154","author":"Dufour","year":"2010","journal-title":"J. Economet"},{"key":"2023111615233489600_asad009-B18","doi-asserted-by":"publisher","DOI":"10.12688\/f1000research.12694.1.","article-title":"On the interpretability and computational reliability of frequency-domain Granger causality","author":"Faes","year":"2017","journal-title":"F1000Res."},{"key":"2023111615233489600_asad009-B19","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1080\/01621459.1982.10477803","article-title":"Measurement of linear dependence and feedback between multiple time series","volume":"77","author":"Geweke","year":"1982","journal-title":"J. Am. Statist. Assoc"},{"key":"2023111615233489600_asad009-B20","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1080\/01621459.1984.10477110","article-title":"Measures of conditional linear dependence and feedback between time series","volume":"79","author":"Geweke","year":"1984","journal-title":"J. Am. Statist. Assoc"},{"key":"2023111615233489600_asad009-B21","doi-asserted-by":"crossref","DOI":"10.1515\/9780691218632","volume-title":"Time Series Analysis","author":"Hamilton","year":"1994"},{"key":"2023111615233489600_asad009-B22","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611972191","volume-title":"The Statistical Theory of Linear Systems","author":"Hannan","year":"2012"},{"key":"2023111615233489600_asad009-B23","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1111\/j.2517-6161.1979.tb01072.x","article-title":"The determination of the order of an autoregression","volume":"41","author":"Hannan","year":"1979","journal-title":"J. R. Statist. Soc. B"},{"key":"2023111615233489600_asad009-B24","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S0024-3795(99)00045-2","article-title":"Construction of the exact Fisher information matrix of Gaussian time series models by means of matrix differential rules","volume":"321","author":"Klein","year":"2000","journal-title":"Lin. Algeb. Appl"},{"key":"2023111615233489600_asad009-B25","volume-title":"Testing Statistical Hypotheses","author":"Lehmann","year":"2005","edition":"3rd ed"},{"key":"2023111615233489600_asad009-B26","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/978-3-642-51514-9_4","volume-title":"Studies in Applied Econometrics","author":"L\u00fctkepohl","year":"1993"},{"key":"2023111615233489600_asad009-B27","first-page":"351","volume-title":"Multivariate Analysis","author":"Masani","year":"1966"},{"key":"2023111615233489600_asad009-B28","doi-asserted-by":"crossref","DOI":"10.1142\/3573","volume-title":"Regression and Time Series Model Selection","author":"McQuarrie","year":"1998"},{"key":"2023111615233489600_asad009-B29","volume-title":"Observed Brain Dynamics","author":"Mitra","year":"2008"},{"key":"2023111615233489600_asad009-B30","first-page":"289","article-title":"On the problem of the most efficient tests of statistical hypotheses","volume":"231","author":"Neyman","year":"1933","journal-title":"Phil. Trans. R. Soc. A"},{"key":"2023111615233489600_asad009-B31","volume-title":"Stationary Random Processes","author":"Rozanov","year":"1967"},{"key":"2023111615233489600_asad009-B32","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring information transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Rev. Lett"},{"key":"2023111615233489600_asad009-B33","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1109\/TNNLS.2011.2178327","article-title":"Assessing Granger non-causality using nonparametric measure of conditional independence","volume":"23","author":"Seth","year":"2012","journal-title":"IEEE Trans. Neural Networks Learn. Syst"},{"key":"2023111615233489600_asad009-B34","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1162\/NECO_a_00828","article-title":"State-space analysis of Granger-Geweke causality measures with application to fMRI","volume":"28","author":"Solo","year":"2016","journal-title":"Neural Comp"},{"key":"2023111615233489600_asad009-B35","doi-asserted-by":"crossref","first-page":"7063","DOI":"10.1073\/pnas.1704663114","article-title":"A study of problems encountered in Granger causality analysis from a neuroscience perspective","volume":"114","author":"Stokes","year":"2017","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"2023111615233489600_asad009-B36","first-page":"E6964","article-title":"Correction for Stokes and Purdon, A study of problems encountered in Granger causality analysis from a neuroscience perspective","volume":"115","author":"Stokes","year":"2018","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"2023111615233489600_asad009-B37","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1090\/S0002-9947-1943-0012401-3","article-title":"Tests of statistical hypotheses concerning several parameters when the number of observations is large","volume":"54","author":"Wald","year":"1943","journal-title":"Trans. Am. Math. Soc"},{"key":"2023111615233489600_asad009-B38","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1093\/biomet\/50.1-2.129","article-title":"On the fitting of multivariate autoregressions, and the approximate caonical factorization of a spectral density matrix","volume":"50","author":"Whittle","year":"1963","journal-title":"Biometrika"},{"key":"2023111615233489600_asad009-B39","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF02404472","article-title":"The prediction theory of multivariate stochastic processes: I. The regularity condition","volume":"98","author":"Wiener","year":"1957","journal-title":"Acta Math."},{"key":"2023111615233489600_asad009-B40","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1214\/aoms\/1177732360","article-title":"The large-sample distribution of the likelihood ratio for testing composite hypotheses","volume":"6","author":"Wilks","year":"1938","journal-title":"Ann. Math. Statist."},{"key":"2023111615233489600_asad009-B41","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1137\/0123044","article-title":"The factorization of matricial spectral densities","volume":"23","author":"Wilson","year":"1972","journal-title":"SIAM J. Appl. Math."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asad009\/50729418\/asad009.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/110\/4\/933\/53471826\/asad009.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/110\/4\/933\/53471826\/asad009.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,13]],"date-time":"2024-10-13T23:58:15Z","timestamp":1728863895000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/110\/4\/933\/7035939"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,14]]},"references-count":41,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,2,14]]},"published-print":{"date-parts":[[2023,11,15]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asad009","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,12,1]]},"published":{"date-parts":[[2023,2,14]]}}}