{"paper_id":"009cdeff-406c-4bf3-a409-15a07092ff53","body_text":"Renormalized Maximum Likelihood for Spatial Lag Models | 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PDF Research Article Renormalized Maximum Likelihood for Spatial Lag Models Saïd Maanan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7178189/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 12 Nov, 2025 Read the published version in Networks and Spatial Economics → Version 1 posted 7 You are reading this latest preprint version Abstract Model selection is critical in spatial econometrics, particularly when specifying the degree of spatial dependence in regression models. Traditional criteria such as Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), and the Hannan-Quinn Criterion (HQC) are widely applied but do not adaptively account for the complexity introduced by spatial autocorrelation. This paper evaluates the performance of the Renormalized Maximum Likelihood (RNML) criterion, which incorporates a data-driven penalty derived from the Fisher Information Matrix to balance model fit and complexity. Through Monte Carlo simulations, we assess RNML’s capacity to accurately recover the true spatial dependence structure under varying degrees of spatial autocorrelation and sample sizes. We complement these simulations with three empirical applications: the spatial distribution of GDP per capita across European NUTS-2 regions, the spread of COVID-19 incidence across Italian provinces, and the share of foreign-born residents across Germany's NUTS-3 regions. These examples span diverse spatial scales and policy domains, economic performance, public health, and migration, providing a comprehensive evaluation of RNML in practice. Our findings consistently show that RNML tends to select higher spatial autoregressive coefficients compared to classical criteria, especially in settings with pronounced spatial dependence or higher spatial resolution. This adaptiveness enhances RNML’s reliability for robust inference in spatial modeling, particularly where spatial spillovers and feedback effects are substantively important. Model Selection Renormalized Maximum Likelihood Spatial Lag Models Information Criteria Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 12 Nov, 2025 Read the published version in Networks and Spatial Economics → Version 1 posted Editorial decision: Revision requested 18 Sep, 2025 Reviews received at journal 18 Sep, 2025 Reviewers agreed at journal 22 Aug, 2025 Reviewers invited by journal 18 Aug, 2025 Editor assigned by journal 06 Aug, 2025 Submission checks completed at journal 28 Jul, 2025 First submitted to journal 21 Jul, 2025 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-7178189\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":504366045,\"identity\":\"0a41af9c-7a83-4f95-8138-00b58407e8ac\",\"order_by\":0,\"name\":\"Saïd Maanan\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVRIiWNgGAWjYFADZgbDB0CKh48ULcYGIC1spNhjJgEiCWrRbT/87MHPHQz2/O3M2yq/5tjJsDEwP3x0A5/JZ9LMDXvPMCTOOMxWdlt2WzLQYWzGxjn4tBxIMJPgbWNIYDjMY3ZbchszUAsPmzReLeeff5P828ZgLw/UUiy5rZ4ILTdyzKSBtjBuAGph/LjtMDFa3pRJy7ZJJG48zFYszbjtOA8bMyG/nE/fJvm2zcZe7vzhjR9/bqu252dvfvgYnxYoAMcIAzMPmCSsHAEYf5CiehSMglEwCkYMAAB/LEEAno9DAgAAAABJRU5ErkJggg==\",\"orcid\":\"\",\"institution\":\"Mohammed V University of Rabat\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Saïd\",\"middleName\":\"\",\"lastName\":\"Maanan\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2025-07-21 13:53:17\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-7178189/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-7178189/v1\",\"draftVersion\":[],\"editorialEvents\":[{\"content\":\"https://doi.org/10.1007/s11067-025-09715-w\",\"type\":\"published\",\"date\":\"2025-11-12T15:57:56+00:00\"}],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":96105217,\"identity\":\"1bffd0ca-4743-4f64-ac22-9500787d0992\",\"added_by\":\"auto\",\"created_at\":\"2025-11-17 16:10:10\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":2099601,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"manuscript.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7178189/v1_covered_b2a236dd-3b21-48e2-943b-9abfe55e5b54.pdf\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"Renormalized Maximum Likelihood for Spatial Lag Models\",\"fulltext\":[],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":false,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":false,\"hideJournal\":false,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":true,\"isAuthorSuppliedPdf\":true,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":true,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"networks-and-spatial-economics\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":false,\"externalIdentity\":\"nets\",\"sideBox\":\"Learn more about [Networks and Spatial Economics](http://link.springer.com/journal/11067)\",\"snPcode\":\"11067\",\"submissionUrl\":\"https://submission.nature.com/new-submission/11067/3\",\"title\":\"Networks and Spatial Economics\",\"twitterHandle\":\"\",\"acdcEnabled\":true,\"dfaEnabled\":true,\"editorialSystem\":\"em\",\"reportingPortfolio\":\"Springer Hybrid\",\"inReviewEnabled\":true,\"inReviewRevisionsEnabled\":false},\"keywords\":\"Model Selection, Renormalized Maximum Likelihood, Spatial Lag Models, Information Criteria\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-7178189/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-7178189/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eModel selection is critical in spatial econometrics, particularly when specifying the degree of spatial dependence in regression models. Traditional criteria such as Akaike\\u0026rsquo;s Information Criterion (AIC), Bayesian Information Criterion (BIC), and the Hannan-Quinn Criterion (HQC) are widely applied but do not adaptively account for the complexity introduced by spatial autocorrelation. This paper evaluates the performance of the Renormalized Maximum Likelihood (RNML) criterion, which incorporates a data-driven penalty derived from the Fisher Information Matrix to balance model fit and complexity. Through Monte Carlo simulations, we assess RNML\\u0026rsquo;s capacity to accurately recover the true spatial dependence structure under varying degrees of spatial autocorrelation and sample sizes. We complement these simulations with three empirical applications: the spatial distribution of GDP per capita across European NUTS-2 regions, the spread of COVID-19 incidence across Italian provinces, and the share of foreign-born residents across Germany's NUTS-3 regions. 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