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Granger causality and its associated frequency domain strength measures (GEMs) (due to Geweke) provide a framework for the formulation and analysis of these issues. In pursuing this setup, three significant unresolved issues emerge. First, computing GEMs involves computing submodels of vector time series models, for which reliable methods do not exist. Second, the impact of filtering on GEMs has never been definitively established. Third, the impact of downsampling on GEMs has never been established. In this work, using state-space methods, we resolve all these issues and illustrate the results with some simulations. Our analysis is motivated by some problems in (fMRI) brain imaging, to which we apply it, but it is of general applicability.<\/jats:p>","DOI":"10.1162\/neco_a_00828","type":"journal-article","created":{"date-parts":[[2016,3,4]],"date-time":"2016-03-04T19:50:18Z","timestamp":1457121018000},"page":"914-949","source":"Crossref","is-referenced-by-count":63,"title":["State-Space Analysis of Granger-Geweke Causality Measures with Application to fMRI"],"prefix":"10.1162","volume":"28","author":[{"given":"Victor","family":"Solo","sequence":"first","affiliation":[{"name":"School of Electrical Engineering, UNSW, Sydney, Australia, and MGH\/HST Martinos Center for Biomedical Imaging, Department of Radiology-MGH, Harvard Medical School, Boston, MA, U.S.A."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1007\/PL00007990"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466605050127"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1007\/s004220050556"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1976.1101368"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1975.1101008"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2012.2195327"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jneumeth.2005.06.011"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2013.2294616"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pbio.0060315"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1016\/0005-1098(95)00089-6"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2009.11.060"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1007\/s004229900137"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.2307\/2999631"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2009.06.008"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2009.07.007"},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2012.01.049"},{"key":"B18","doi-asserted-by":"publisher","DOI":"10.2307\/1914238"},{"key":"B19","doi-asserted-by":"publisher","DOI":"10.2307\/2287238"},{"key":"B20","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1984.10477110"},{"key":"B21","doi-asserted-by":"publisher","DOI":"10.1006\/nimg.1998.0419"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1016\/j.mri.2003.08.026"},{"key":"B23","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(63)90092-5"},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.2307\/1912791"},{"key":"B25","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(88)90108-X"},{"key":"B26","doi-asserted-by":"crossref","DOI":"10.1515\/9780691218632","volume-title":"Time series analysis","author":"Hamilton J.","year":"1994"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2012.02.015"},{"key":"B28","volume-title":"Statistical theory of linear systems","author":"Hannan E.","year":"1988"},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350836"},{"key":"B30","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2011.03.005"},{"key":"B31","first-page":"671","volume-title":"Time series analysis: Theory and practice","author":"Hsiao C.","year":"1982"},{"key":"B32","volume-title":"Linear estimation","author":"Kailath T.","year":"2000"},{"key":"B33","doi-asserted-by":"publisher","DOI":"10.1007\/BF00198091"},{"key":"B34","doi-asserted-by":"publisher","DOI":"10.1038\/40332"},{"key":"B35","doi-asserted-by":"publisher","DOI":"10.1016\/j.sigpro.2011.03.008"},{"key":"B36","volume-title":"Multivariate analysis","author":"Kshirsagar A.","year":"1972"},{"key":"B37","first-page":"445","volume-title":"Proc. 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