Testing Martingale Difference Hypothesis for Functional Time Series

Testing Martingale Difference Hypothesis for Functional Time Series | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Testing Martingale Difference Hypothesis for Functional Time Series Kai Lou, Guochang Wang, Pengfei Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9139516/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract Testing the predictability of time series is crucial for determining whether forecasting models should be developed. Existing methodologies for this problem generally fall into three categories based on the null hypothesis: white noise, independently identically distributed (i.i.d.) and martingale difference hypothesis (MDH). While numerous tests within these categories have been established for multivariate time series, research on functional time series (FTS) has predominantly concentrated on testing the white noise and i.i.d. hypotheses. This paper proposes a novel MDH test for FTS by introducing functional auto-martingale difference divergence (FAMDD), a new metric capable of effectively detecting nonlinear dependence within dependent functional data. Compared with the existing approaches, our test can detect a wider range of functional sequences with nonlinear dependence. The asymptotic behaviors of the test statistics are established under some suitable conditions. Since the limiting distribution is non-pivotal, a wild bootstrap procedure is introduced to obtain the critical values for conducting inference. Monte Carlo simulations and two real data applications are analyzed to illustrate the effectiveness of our methods. Functional time series Martingale difference hypothesis Spectral test Wild bootstrap Full Text Additional Declarations No competing interests reported. Supplementary Files Supplements.pdf Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 13 May, 2026 Reviewers agreed at journal 30 Mar, 2026 Reviewers agreed at journal 22 Mar, 2026 Reviewers agreed at journal 20 Mar, 2026 Reviewers invited by journal 20 Mar, 2026 Editor assigned by journal 17 Mar, 2026 Submission checks completed at journal 17 Mar, 2026 First submitted to journal 16 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9139516","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":610115777,"identity":"63f96734-448f-480c-8375-98ccf78cf203","order_by":0,"name":"Kai Lou","email":"","orcid":"","institution":"jinan university","correspondingAuthor":false,"prefix":"","firstName":"Kai","middleName":"","lastName":"Lou","suffix":""},{"id":610115778,"identity":"07754b3a-eced-4384-835d-3eb64f5494ff","order_by":1,"name":"Guochang Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA40lEQVRIiWNgGAWjYPACCTl59gbStNgYG/YcIE1LWmLDjQQi1RocP3v45Zeaw4yNMx8/3fjjD4M8fwPzswd4tZzJS7OWOXaYmV06zew2bxuD4YwDbOYG+LSYHcgxM5ZsOMzGODvB7DZjAwPjBgYeNgm8Ws6/AWvhYbh5/NtNoMPsCWu5kWP88GNDmgTDDR6zGzxsDIkEtdjfeGPGzHDMxsCwJ6cM6BeJ5BmH2czwapHszzH++KNGon4++/FtQIfZ2Pa3Nz/DqwUI2KR5EBygYmYC6oGA+eMPwopGwSgYBaNgJAMAUy1Ka6UwVUAAAAAASUVORK5CYII=","orcid":"","institution":"jinan university","correspondingAuthor":true,"prefix":"","firstName":"Guochang","middleName":"","lastName":"Wang","suffix":""},{"id":610115779,"identity":"da5b2b03-d1e2-40f7-bc55-0311b29e9e15","order_by":2,"name":"Pengfei Liu","email":"","orcid":"","institution":"Jiangsu Normal University","correspondingAuthor":false,"prefix":"","firstName":"Pengfei","middleName":"","lastName":"Liu","suffix":""}],"badges":[],"createdAt":"2026-03-16 15:08:52","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9139516/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9139516/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105564674,"identity":"47a81aaf-d7bf-4e7b-ad0c-79ba54040fa3","added_by":"auto","created_at":"2026-03-27 12:50:27","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":886829,"visible":true,"origin":"","legend":"","description":"","filename":"template.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9139516/v1_covered_1911bab5-9e6b-401f-9432-c50d06b567eb.pdf"},{"id":105270520,"identity":"14b88107-471a-4494-abde-bd62577ff68c","added_by":"auto","created_at":"2026-03-24 08:30:47","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":243473,"visible":true,"origin":"","legend":"","description":"","filename":"Supplements.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9139516/v1/a49592170de0f1105bf0a4c5.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eTesting Martingale Difference Hypothesis for Functional Time Series\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"statistics-and-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stco","sideBox":"Learn more about [Statistics and Computing](http://link.springer.com/journal/11222)","snPcode":"11222","submissionUrl":"https://submission.nature.com/new-submission/11222/3","title":"Statistics and Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Functional time series, Martingale difference hypothesis, Spectral test, Wild bootstrap","lastPublishedDoi":"10.21203/rs.3.rs-9139516/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9139516/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Testing the predictability of time series is crucial for determining whether forecasting models should be developed. Existing methodologies for this problem generally fall into three categories based on the \n\tnull hypothesis: white noise, independently identically distributed (i.i.d.) and martingale difference hypothesis (MDH). While numerous tests within these categories have been established for multivariate time series, research on functional time series (FTS) has predominantly concentrated on testing the white noise and i.i.d. hypotheses. This paper proposes a novel MDH test for FTS by introducing functional auto-martingale difference divergence (FAMDD), a new metric capable of effectively detecting nonlinear dependence within dependent functional data. Compared with the existing approaches, our test can detect a wider range of functional sequences with nonlinear dependence. The asymptotic behaviors of the test statistics are established under some suitable conditions. 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