Variabili strumentali

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="it" dir="ltr"> <head> <meta charset="UTF-8"> <title>Variabili strumentali - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )itwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","gennaio","febbraio","marzo","aprile","maggio","giugno","luglio","agosto","settembre","ottobre","novembre","dicembre"],"wgRequestId":"7c03a14c-fefc-4290-a32e-52208ef84f5b","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Variabili_strumentali","wgTitle":"Variabili strumentali","wgCurRevisionId":139816191,"wgRevisionId":139816191,"wgArticleId":148896,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Voci con modulo citazione e parametro pagine","Voci con modulo citazione e parametro coautori","Voci con codice Thesaurus BNCF","Voci con codice LCCN","Voci con codice BNF","Voci con codice J9U","Voci non biografiche con codici di controllo di autorità","Pagine che utilizzano collegamenti magici ISBN","Analisi di regressione"],"wgPageViewLanguage":"it","wgPageContentLanguage":"it","wgPageContentModel":"wikitext","wgRelevantPageName": "Variabili_strumentali","wgRelevantArticleId":148896,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"it","pageLanguageDir":"ltr","pageVariantFallbacks":"it"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1665305","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true, "wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.coloriDarkMode-default":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.MainPageWikiList","ext.gadget.stru-commonsupload","ext.gadget.HiddenCat", "ext.gadget.ReferenceTooltips","ext.gadget.TitoloErrato","ext.gadget.NewSection","ext.gadget.RichiediRevisioneBozza","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=it&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=it&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=it&modules=ext.gadget.coloriDarkMode-default&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=it&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Variabili strumentali - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//it.m.wikipedia.org/wiki/Variabili_strumentali"> <link rel="alternate" type="application/x-wiki" title="Modifica" href="/w/index.php?title=Variabili_strumentali&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (it)"> <link rel="EditURI" type="application/rsd+xml" href="//it.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://it.wikipedia.org/wiki/Variabili_strumentali"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it"> <link rel="alternate" type="application/atom+xml" title="Feed Atom di Wikipedia" href="/w/index.php?title=Speciale:UltimeModifiche&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Variabili_strumentali rootpage-Variabili_strumentali skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Vai al contenuto</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Sito"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menu principale" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menu principale</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menu principale</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">nascondi</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigazione </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Pagina_principale" title="Visita la pagina principale [z]" accesskey="z"><span>Pagina principale</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Speciale:UltimeModifiche" title="Elenco delle ultime modifiche del sito [r]" accesskey="r"><span>Ultime modifiche</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Speciale:PaginaCasuale" title="Mostra una pagina a caso [x]" accesskey="x"><span>Una voce a caso</span></a></li><li id="n-nearby-pages-title" class="mw-list-item"><a href="/wiki/Speciale:NelleVicinanze"><span>Nelle vicinanze</span></a></li><li id="n-vetrina" class="mw-list-item"><a href="/wiki/Wikipedia:Vetrina"><span>Vetrina</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Aiuto:Aiuto" title="Pagine di aiuto"><span>Aiuto</span></a></li><li id="n-Sportello-informazioni" class="mw-list-item"><a href="/wiki/Aiuto:Sportello_informazioni"><span>Sportello informazioni</span></a></li> </ul> </div> </div> <div id="p-Comunità" class="vector-menu mw-portlet mw-portlet-Comunità" > <div class="vector-menu-heading"> Comunità </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Portale:Comunit%C3%A0" title="Descrizione del progetto, cosa puoi fare, dove trovare le cose"><span>Portale Comunità</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikipedia:Bar"><span>Bar</span></a></li><li id="n-wikipediano" class="mw-list-item"><a href="/wiki/Wikipedia:Wikipediano"><span>Il Wikipediano</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Wikipedia:Contatti"><span>Contatti</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Pagina_principale" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="L'enciclopedia libera" src="/static/images/mobile/copyright/wikipedia-tagline-it.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Speciale:Ricerca" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cerca in Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Ricerca</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Cerca in Wikipedia" aria-label="Cerca in Wikipedia" autocapitalize="sentences" title="Cerca in Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Speciale:Ricerca"> </div> <button class="cdx-button cdx-search-input__end-button">Ricerca</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Strumenti personali"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Modifica la dimensione, la larghezza e il colore del testo" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aspetto" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aspetto</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_it.wikipedia.org&uselang=it" class=""><span>Fai una donazione</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speciale:CreaUtenza&returnto=Variabili+strumentali" title="Si consiglia di registrarsi e di effettuare l'accesso, anche se non è obbligatorio" class=""><span>registrati</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speciale:Entra&returnto=Variabili+strumentali" title="Si consiglia di effettuare l'accesso, anche se non è obbligatorio [o]" accesskey="o" class=""><span>entra</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Altre opzioni" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Strumenti personali" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Strumenti personali</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu utente" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_it.wikipedia.org&uselang=it"><span>Fai una donazione</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speciale:CreaUtenza&returnto=Variabili+strumentali" title="Si consiglia di registrarsi e di effettuare l'accesso, anche se non è obbligatorio"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>registrati</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speciale:Entra&returnto=Variabili+strumentali" title="Si consiglia di effettuare l'accesso, anche se non è obbligatorio [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>entra</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pagine per utenti anonimi <a href="/wiki/Aiuto:Benvenuto" aria-label="Ulteriori informazioni sulla contribuzione"><span>ulteriori informazioni</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Speciale:MieiContributi" title="Un elenco delle modifiche fatte da questo indirizzo IP [y]" accesskey="y"><span>contributi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speciale:MieDiscussioni" title="Discussioni sulle modifiche fatte da questo indirizzo IP [n]" accesskey="n"><span>discussioni</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sito"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Indice" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Indice</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Definizione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definizione"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definizione</span> </div> </a> <ul id="toc-Definizione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Validità_degli_strumenti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Validità_degli_strumenti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Validità degli strumenti</span> </div> </a> <ul id="toc-Validità_degli_strumenti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Illustrazione_del_metodo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Illustrazione_del_metodo"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Illustrazione del metodo</span> </div> </a> <button aria-controls="toc-Illustrazione_del_metodo-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Illustrazione del metodo</span> </button> <ul id="toc-Illustrazione_del_metodo-sublist" class="vector-toc-list"> <li id="toc-Il_caso_multivariato" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Il_caso_multivariato"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Il caso multivariato</span> </div> </a> <ul id="toc-Il_caso_multivariato-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applicazione_del_metodo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applicazione_del_metodo"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applicazione del metodo</span> </div> </a> <ul id="toc-Applicazione_del_metodo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Variabili strumentali</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 15 lingue" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-15" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">15 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Instrumentvariablensch%C3%A4tzung" title="Instrumentvariablenschätzung - tedesco" lang="de" hreflang="de" data-title="Instrumentvariablenschätzung" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Instrumental_variables_estimation" title="Instrumental variables estimation - inglese" lang="en" hreflang="en" data-title="Instrumental variables estimation" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Variable_instrumental" title="Variable instrumental - spagnolo" lang="es" hreflang="es" data-title="Variable instrumental" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AA%D8%BA%DB%8C%D8%B1_%D8%A7%D8%A8%D8%B2%D8%A7%D8%B1%DB%8C" title="متغیر ابزاری - persiano" lang="fa" hreflang="fa" data-title="متغیر ابزاری" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Instrumenttimuuttuja" title="Instrumenttimuuttuja - finlandese" lang="fi" hreflang="fi" data-title="Instrumenttimuuttuja" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9thode_des_variables_instrumentales" title="Méthode des variables instrumentales - francese" lang="fr" hreflang="fr" data-title="Méthode des variables instrumentales" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%AA%D7%A0%D7%94_%D7%A2%D7%96%D7%A8" title="משתנה עזר - ebraico" lang="he" hreflang="he" data-title="משתנה עזר" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%93%8D%E4%BD%9C%E5%A4%89%E6%95%B0%E6%B3%95" title="操作変数法 - giapponese" lang="ja" hreflang="ja" data-title="操作変数法" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8F%84%EA%B5%AC_%EB%B3%80%EC%88%98" title="도구 변수 - coreano" lang="ko" hreflang="ko" data-title="도구 변수" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Metoda_zmiennych_instrumentalnych" title="Metoda zmiennych instrumentalnych - polacco" lang="pl" hreflang="pl" data-title="Metoda zmiennych instrumentalnych" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Vari%C3%A1veis_instrumentais" title="Variáveis instrumentais - portoghese" lang="pt" hreflang="pt" data-title="Variáveis instrumentais" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%B8%D0%BD%D1%81%D1%82%D1%80%D1%83%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D1%85_%D0%BF%D0%B5%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%BD%D1%8B%D1%85" title="Метод инструментальных переменных - russo" lang="ru" hreflang="ru" data-title="Метод инструментальных переменных" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Instrumentvariabel" title="Instrumentvariabel - svedese" lang="sv" hreflang="sv" data-title="Instrumentvariabel" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D1%96%D0%BD%D1%81%D1%82%D1%80%D1%83%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D1%85_%D0%B7%D0%BC%D1%96%D0%BD%D0%BD%D0%B8%D1%85" title="Метод інструментальних змінних - ucraino" lang="uk" hreflang="uk" data-title="Метод інструментальних змінних" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B7%A5%E5%85%B7%E5%8F%98%E9%87%8F" title="工具变量 - cinese" lang="zh" hreflang="zh" data-title="工具变量" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1665305#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Variabili_strumentali" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discussione:Variabili_strumentali&action=edit&redlink=1" rel="discussion" class="new" title="Vedi le discussioni relative a questa pagina (la pagina non esiste) [t]" accesskey="t"><span>Discussione</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambia versione linguistica" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">italiano</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Visite"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Variabili_strumentali"><span>Leggi</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&veaction=edit" title="Modifica questa pagina [v]" accesskey="v"><span>Modifica</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&action=history" title="Versioni precedenti di questa pagina [h]" accesskey="h"><span>Cronologia</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Strumenti" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Strumenti</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Strumenti</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">nascondi</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Altre opzioni" > <div class="vector-menu-heading"> Azioni </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Variabili_strumentali"><span>Leggi</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&veaction=edit" title="Modifica questa pagina [v]" accesskey="v"><span>Modifica</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&action=history"><span>Cronologia</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generale </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Variabili_strumentali" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j"><span>Puntano qui</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Variabili_strumentali" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k"><span>Modifiche correlate</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciale:PagineSpeciali" title="Elenco di tutte le pagine speciali [q]" accesskey="q"><span>Pagine speciali</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&oldid=139816191" title="Collegamento permanente a questa versione di questa pagina"><span>Link permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&action=info" title="Ulteriori informazioni su questa pagina"><span>Informazioni pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciale:Cita&page=Variabili_strumentali&id=139816191&wpFormIdentifier=titleform" title="Informazioni su come citare questa pagina"><span>Cita questa voce</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speciale:UrlShortener&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FVariabili_strumentali"><span>Ottieni URL breve</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speciale:QrCode&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FVariabili_strumentali"><span>Scarica codice QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Stampa/esporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciale:Libro&bookcmd=book_creator&referer=Variabili+strumentali"><span>Crea un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciale:DownloadAsPdf&page=Variabili_strumentali&action=show-download-screen"><span>Scarica come PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Variabili_strumentali&printable=yes" title="Versione stampabile di questa pagina [p]" accesskey="p"><span>Versione stampabile</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In altri progetti </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1665305" title="Collegamento all'elemento connesso dell'archivio dati [g]" accesskey="g"><span>Elemento Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><p>In <a href="/wiki/Statistica" title="Statistica">statistica</a>, la stima con il <b>metodo delle variabili strumentali</b> è utilizzata nell'analisi di regressione lineare. Un'ipotesi standard del <a href="/wiki/Regressione_lineare" title="Regressione lineare">modello classico di regressione lineare</a> è che le variabili esplicative non siano <a href="/wiki/Correlazione_(statistica)" title="Correlazione (statistica)">correlate</a> con la componente non spiegata, o disturbo; laddove tale ipotesi viene meno, la <a href="/wiki/Regressione_lineare" title="Regressione lineare">regressione</a> con il consueto <a href="/wiki/Metodo_dei_minimi_quadrati" title="Metodo dei minimi quadrati">metodo dei minimi quadrati</a> non consentirà di ottenere <a href="/wiki/Stima" title="Stima">stime</a> <a href="/wiki/Consistenza_(statistica)" title="Consistenza (statistica)">consistenti</a> (cioè asintoticamente <a href="/wiki/Correttezza_(statistica)" class="mw-redirect" title="Correttezza (statistica)">corrette</a> e con <a href="/wiki/Varianza" title="Varianza">varianza</a> asintoticamente nulla). Se tuttavia è disponibile una <i>variabile strumentale</i>, è ancora possibile ottenere <a href="/wiki/Consistenza_(statistica)" title="Consistenza (statistica)">stime consistenti</a>. </p><p>Il metodo di stima di un modello lineare tramite variabili strumentali è anche noto come metodo dei <b>minimi quadrati a due stadi</b> (o 2SLS, dall'<a href="/wiki/Lingua_inglese" title="Lingua inglese">inglese</a> <i>Two-Stages Least Squares</i>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variabili_strumentali&veaction=edit&section=1" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variabili_strumentali&action=edit&section=1" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Illustrazione del modello generale di regressione con variabili strumentali e la sua terminologia:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+\cdots +\beta _{k}X_{ki}+\beta _{k+r}W{ri}+Z_{1i},Z_{2i}+u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mi>r</mi> </mrow> </msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>i</mi> </mrow> <mo>+</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+\cdots +\beta _{k}X_{ki}+\beta _{k+r}W{ri}+Z_{1i},Z_{2i}+u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff280bf67421b8cb25dc18be219ceafd547aee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:66.334ex; height:2.509ex;" alt="{\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+\cdots +\beta _{k}X_{ki}+\beta _{k+r}W{ri}+Z_{1i},Z_{2i}+u_{i}}"></span> </p> </div> <p>dove: </p> <dl><dd><i>i</i> varia tra le osservazioni, <i>i</i> = 1, ...., <i>n</i>;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}"></span> è <a href="/wiki/Variabile_dipendente" class="mw-redirect" title="Variabile dipendente">variabile dipendente</a>;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1i},X_{2i}+\cdots +X_{ki}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1i},X_{2i}+\cdots +X_{ki}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61bbb1567592d5a0368e28b13c895b85935d6834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.11ex; height:2.509ex;" alt="{\displaystyle X_{1i},X_{2i}+\cdots +X_{ki}}"></span> sono <i>k</i> <a href="/w/index.php?title=Regressori&action=edit&redlink=1" class="new" title="Regressori (la pagina non esiste)">regressori</a> <a href="/w/index.php?title=Endogeni&action=edit&redlink=1" class="new" title="Endogeni (la pagina non esiste)">endogeni</a> potenzialmente <a href="/wiki/Correlazione_(statistica)" title="Correlazione (statistica)">correlati</a> con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}"></span>;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1i},W_{2i}+\cdots +W_{ri}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1i},W_{2i}+\cdots +W_{ri}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5cf8bc1f46404432021e6a52fa4f65a132322b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.804ex; height:2.509ex;" alt="{\displaystyle W_{1i},W_{2i}+\cdots +W_{ri}}"></span> sono <i>r</i> <a href="/w/index.php?title=Regressori&action=edit&redlink=1" class="new" title="Regressori (la pagina non esiste)">regressori</a> <a href="/w/index.php?title=Esogeni&action=edit&redlink=1" class="new" title="Esogeni (la pagina non esiste)">esogeni</a> inclusi <a href="/wiki/Correlazione_(statistica)" title="Correlazione (statistica)">incorrelati</a> con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}"></span>;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{0}+\beta _{1}x_{1i}+\beta _{2}x_{2i}\cdots +\beta _{k}x_{ki}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>⋯</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0}+\beta _{1}x_{1i}+\beta _{2}x_{2i}\cdots +\beta _{k}x_{ki}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b66fcd3399c436bc496839e2679ef6b4315eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.034ex; height:2.509ex;" alt="{\displaystyle \beta _{0}+\beta _{1}x_{1i}+\beta _{2}x_{2i}\cdots +\beta _{k}x_{ki}}"></span> è la <i>retta di regressione</i>;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{0},\beta _{1}\cdots \beta {k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0},\beta _{1}\cdots \beta {k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01237f5c2f91a9f1c1f48175c03fe27039c7eef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.817ex; height:2.509ex;" alt="{\displaystyle \beta _{0},\beta _{1}\cdots \beta {k+1}}"></span> sono coefficienti di regressione ignoti;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{1i},Z_{2i}\cdots Z_{mi}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>⋯</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{1i},Z_{2i}\cdots Z_{mi}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9341a2b267f45ca0458f7c653af2a2c67b48c0bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.78ex; height:2.509ex;" alt="{\displaystyle Z_{1i},Z_{2i}\cdots Z_{mi}}"></span> sono <i>m</i> variabili strumentali;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}"></span> è l'<a href="/wiki/Errore_statistico" title="Errore statistico">errore statistico</a>.</dd></dl> <p>Vengono stimati col <a href="/wiki/Minimi_Quadrati#Minimi_quadrati_a_due_stadi_-_2SLS_o_TSLS" class="mw-redirect" title="Minimi Quadrati">Metodo dei minimi quadrati a due stadi</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Validità_degli_strumenti"><span id="Validit.C3.A0_degli_strumenti"></span>Validità degli strumenti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variabili_strumentali&veaction=edit&section=2" title="Modifica la sezione Validità degli strumenti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variabili_strumentali&action=edit&section=2" title="Edit section's source code: Validità degli strumenti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un insieme di strumenti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{1i},Z_{2i}\cdots Z_{mi}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>⋯</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{1i},Z_{2i}\cdots Z_{mi}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9341a2b267f45ca0458f7c653af2a2c67b48c0bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.78ex; height:2.509ex;" alt="{\displaystyle Z_{1i},Z_{2i}\cdots Z_{mi}}"></span> deve soddisfare due condizioni per essere valido: </p> <ul><li><b>Rilevanza</b>: lo strumento è correlato con la X. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Cor[Z_{i},X_{i}]\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>o</mi> <mi>r</mi> <mo stretchy="false">[</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Cor[Z_{i},X_{i}]\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9294224a63b0e24fa0e443f9a8cf4b57fec798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.642ex; height:2.843ex;" alt="{\displaystyle Cor[Z_{i},X_{i}]\neq 0}"></span></li> <li><b>Esogeneità</b>: la parte della variazione di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span>, catturata dalla variabile strumentale è esogena. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[u_{i}|Z_{i}]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[u_{i}|Z_{i}]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156b0cb686321587fd64009078e8d6f08a173c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.494ex; height:2.843ex;" alt="{\displaystyle E[u_{i}|Z_{i}]=0}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Illustrazione_del_metodo">Illustrazione del metodo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variabili_strumentali&veaction=edit&section=3" title="Modifica la sezione Illustrazione del metodo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variabili_strumentali&action=edit&section=3" title="Edit section's source code: Illustrazione del metodo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si consideri il modello di <a href="/wiki/Regressione_lineare" title="Regressione lineare">regressione lineare</a>: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ y_{i}=x_{i}\beta +\varepsilon _{i},\ i=1,\ldots ,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>β</mi> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ y_{i}=x_{i}\beta +\varepsilon _{i},\ i=1,\ldots ,N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef7c2a24d39f6c712240ba239bb080c22b8438f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.723ex; height:2.509ex;" alt="{\displaystyle \ y_{i}=x_{i}\beta +\varepsilon _{i},\ i=1,\ldots ,N}"></span></dd></dl></dd></dl> <p>Nel modello classico di regressione si ipotizza che le variabili esplicative <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627df53731a09bc2fc34b492b0dfc639a405ec48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.009ex;" alt="{\displaystyle \ x_{i}}"></span> non presentino correlazione con i disturbi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \varepsilon _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \varepsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29f326af611e792fe74845d4f706ac536492c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle \ \varepsilon _{i}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mbox{E}}[x_{i}\varepsilon _{i}]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>E</mtext> </mstyle> </mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mbox{E}}[x_{i}\varepsilon _{i}]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d743e17ee82c4e967e0701c21bff499b7401cc08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.731ex; height:2.843ex;" alt="{\displaystyle \ {\mbox{E}}[x_{i}\varepsilon _{i}]=0}"></span>. Il <a href="/wiki/Metodo_dei_minimi_quadrati" title="Metodo dei minimi quadrati">metodo dei minimi quadrati</a> ottiene lo <a href="/wiki/Stimatore" title="Stimatore">stimatore</a> per il parametro <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d76b9bdff3e27343f8157af06a1008c9688a3f46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.913ex; height:2.509ex;" alt="{\displaystyle \ \beta }"></span> come soluzione all'equazione: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sum _{i}x_{i}(y_{i}-x_{i}\beta )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sum _{i}x_{i}(y_{i}-x_{i}\beta )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2de28a71658598650edf5471a722b0f3ed03b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.15ex; height:5.509ex;" alt="{\displaystyle \ \sum _{i}x_{i}(y_{i}-x_{i}\beta )=0}"></span></dd></dl></dd></dl> <p>Ciò conduce allo <a href="/wiki/Stimatore" title="Stimatore">stimatore</a> (dei <a href="/wiki/Metodo_dei_minimi_quadrati" title="Metodo dei minimi quadrati">minimi quadrati ordinari</a>, in <a href="/wiki/Lingua_inglese" title="Lingua inglese">inglese</a> <i>Ordinary Least Squares</i>, o OLS): </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\hat {\beta }}_{\mathrm {OLS} }={\frac {\sum _{i}x_{i}y_{i}}{\sum _{i}x_{i}^{2}}}={\frac {\sum _{i}x_{i}(x_{i}\beta +\varepsilon _{i})}{\sum _{i}x_{i}^{2}}}=\beta +{\frac {\sum _{i}x_{i}\varepsilon _{i}}{\sum _{i}x_{i}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>β</mi> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>β</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\hat {\beta }}_{\mathrm {OLS} }={\frac {\sum _{i}x_{i}y_{i}}{\sum _{i}x_{i}^{2}}}={\frac {\sum _{i}x_{i}(x_{i}\beta +\varepsilon _{i})}{\sum _{i}x_{i}^{2}}}=\beta +{\frac {\sum _{i}x_{i}\varepsilon _{i}}{\sum _{i}x_{i}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2454d26531ea8c008ee4448f8028ea44437655a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.586ex; height:6.843ex;" alt="{\displaystyle \ {\hat {\beta }}_{\mathrm {OLS} }={\frac {\sum _{i}x_{i}y_{i}}{\sum _{i}x_{i}^{2}}}={\frac {\sum _{i}x_{i}(x_{i}\beta +\varepsilon _{i})}{\sum _{i}x_{i}^{2}}}=\beta +{\frac {\sum _{i}x_{i}\varepsilon _{i}}{\sum _{i}x_{i}^{2}}}}"></span></dd></dl></dd></dl> <p>Poiché <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627df53731a09bc2fc34b492b0dfc639a405ec48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.009ex;" alt="{\displaystyle \ x_{i}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \varepsilon _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \varepsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29f326af611e792fe74845d4f706ac536492c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle \ \varepsilon _{i}}"></span> sono incorrelati, passando al limite per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ N\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ N\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670e40fdce11256456e8a4b90b30bd258078a74d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.582ex; height:2.176ex;" alt="{\displaystyle \ N\rightarrow \infty }"></span> il secondo termine nell'espressione sopra converge a zero in probabilità, così che la <a href="/wiki/Stima" title="Stima">stima</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\hat {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\hat {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c28b57f6275eed2ec21501f1707ef3c30a2fc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.032ex; height:3.176ex;" alt="{\displaystyle \ {\hat {\beta }}}"></span> è <a href="/wiki/Consistenza_(statistica)" title="Consistenza (statistica)">consistente</a>. </p><p>Quando l'ipotesi standard viene meno, tuttavia, lo stimatore dei <a href="/wiki/Metodo_dei_minimi_quadrati" title="Metodo dei minimi quadrati">minimi quadrati</a> non è più <a href="/wiki/Consistenza_(statistica)" title="Consistenza (statistica)">consistente</a>. Può in tal caso essere utile considerare una <i>variabile strumentale</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4632271a1b9cafab7716bb515cdc56395d3b88d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.009ex;" alt="{\displaystyle \ z_{i}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ i=1,\ldots ,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ i=1,\ldots ,N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa0f1a5c849401dfed63bfe36523851a4b2cadb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.886ex; height:2.509ex;" alt="{\displaystyle \ i=1,\ldots ,N}"></span>, non correlata con il disturbo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \varepsilon _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \varepsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29f326af611e792fe74845d4f706ac536492c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle \ \varepsilon _{i}}"></span> (e, idealmente, correlata con la variabile esplicativa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627df53731a09bc2fc34b492b0dfc639a405ec48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.009ex;" alt="{\displaystyle \ x_{i}}"></span>). Grazie ad essa è possibile impostare uno <a href="/wiki/Stimatore" title="Stimatore">stimatore</a> tramite il <a href="/wiki/Metodo_dei_momenti_(statistica)" title="Metodo dei momenti (statistica)">metodo dei momenti</a>, tale da soddisfare la condizione: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sum _{i}z_{i}(y_{i}-x_{i}\beta )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sum _{i}z_{i}(y_{i}-x_{i}\beta )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db0a1c44afd2e6022e54e98d930137405ed137b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.902ex; height:5.509ex;" alt="{\displaystyle \ \sum _{i}z_{i}(y_{i}-x_{i}\beta )=0}"></span></dd></dl></dd></dl> <p>Dalla condizione sopra discende lo <a href="/wiki/Stimatore" title="Stimatore">stimatore</a> (delle variabili strumentali, in <a href="/wiki/Lingua_inglese" title="Lingua inglese">inglese</a> <i>Instrumental Variables</i>, o IV): </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }={\frac {\sum _{i}z_{i}y_{i}}{\sum _{i}z_{i}x_{i}}}={\frac {\sum _{i}z_{i}(x_{i}\beta +\varepsilon _{i})}{\sum _{i}z_{i}x_{i}}}=\beta +{\frac {\sum _{i}z_{i}\varepsilon _{i}}{\sum _{i}z_{i}x_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>β</mi> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>β</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }={\frac {\sum _{i}z_{i}y_{i}}{\sum _{i}z_{i}x_{i}}}={\frac {\sum _{i}z_{i}(x_{i}\beta +\varepsilon _{i})}{\sum _{i}z_{i}x_{i}}}=\beta +{\frac {\sum _{i}z_{i}\varepsilon _{i}}{\sum _{i}z_{i}x_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66430e38b0772894ea492bf8c3a6e4390867c32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.884ex; height:6.509ex;" alt="{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }={\frac {\sum _{i}z_{i}y_{i}}{\sum _{i}z_{i}x_{i}}}={\frac {\sum _{i}z_{i}(x_{i}\beta +\varepsilon _{i})}{\sum _{i}z_{i}x_{i}}}=\beta +{\frac {\sum _{i}z_{i}\varepsilon _{i}}{\sum _{i}z_{i}x_{i}}}}"></span></dd></dl></dd></dl> <p>Poiché <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4632271a1b9cafab7716bb515cdc56395d3b88d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.009ex;" alt="{\displaystyle \ z_{i}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \varepsilon _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \varepsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29f326af611e792fe74845d4f706ac536492c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle \ \varepsilon _{i}}"></span> non presentano correlazione, lo <a href="/wiki/Stimatore" title="Stimatore">stimatore</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37e385ba09b58a0971ff542bb211f228db751df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:3.343ex;" alt="{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }}"></span> godrà della proprietà di <a href="/wiki/Consistenza_(statistica)" title="Consistenza (statistica)">consistenza</a>. Può essere interessante osservare che tale stimatore costituisce un caso più generale di quello ottenuto con il <a href="/wiki/Metodo_dei_minimi_quadrati" title="Metodo dei minimi quadrati">metodo dei minimi quadrati</a>: in altre parole, tale metodo può essere pensato come una <a href="/wiki/Stima" title="Stima">stima</a> tramite variabili strumentali, in cui le stesse variabili esplicative (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627df53731a09bc2fc34b492b0dfc639a405ec48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.009ex;" alt="{\displaystyle \ x_{i}}"></span> nella notazione sopra adottata) sono utilizzate come variabili strumentali. </p> <div class="mw-heading mw-heading3"><h3 id="Il_caso_multivariato">Il caso multivariato</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variabili_strumentali&veaction=edit&section=4" title="Modifica la sezione Il caso multivariato" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variabili_strumentali&action=edit&section=4" title="Edit section's source code: Il caso multivariato"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La procedura descritta sopra è immediatamente adattabile al caso multivariato. Si consideri una matrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> di <i>N</i> osservazioni di <i>K</i> regressori, e una matrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> di <i>N</i> osservazioni di <i>P</i> variabili strumentali, tale che: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=X\beta +\varepsilon ,\quad {\mbox{E}}[X\varepsilon ]\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>X</mi> <mi>β</mi> <mo>+</mo> <mi>ε</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>E</mtext> </mstyle> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mi>ε</mi> <mo stretchy="false">]</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=X\beta +\varepsilon ,\quad {\mbox{E}}[X\varepsilon ]\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0a88e3004ba6239b7392aa69730ab64a019a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.047ex; height:2.843ex;" alt="{\displaystyle y=X\beta +\varepsilon ,\quad {\mbox{E}}[X\varepsilon ]\neq 0}"></span></dd></dl></dd></dl> <p>dove <i>I</i> denota la matrice identità di dimensione <i>N</i>, e: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X=Z\Gamma +u,\quad {\mbox{E}}[u]=0,\quad {\mbox{E}}[Z\varepsilon ]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mo>=</mo> <mi>Z</mi> <mi mathvariant="normal">Γ</mi> <mo>+</mo> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>E</mtext> </mstyle> </mrow> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>E</mtext> </mstyle> </mrow> <mo stretchy="false">[</mo> <mi>Z</mi> <mi>ε</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X=Z\Gamma +u,\quad {\mbox{E}}[u]=0,\quad {\mbox{E}}[Z\varepsilon ]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/924cab483731c368cd9fb17098fd0b91ebc84d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.044ex; height:2.843ex;" alt="{\displaystyle \ X=Z\Gamma +u,\quad {\mbox{E}}[u]=0,\quad {\mbox{E}}[Z\varepsilon ]=0}"></span></dd></dl></dd></dl> <p>Si può allora scrivere: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\Gamma }}=(Z'Z)^{-1}Z'X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mi>Z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\Gamma }}=(Z'Z)^{-1}Z'X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1dcaf17b050e45a2cdbeec1c50562d90789fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.14ex; height:3.343ex;" alt="{\displaystyle {\hat {\Gamma }}=(Z'Z)^{-1}Z'X}"></span> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {X}}=Z{\hat {\Gamma }}=Z(Z'Z)^{-1}Z'X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mi>Z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {X}}=Z{\hat {\Gamma }}=Z(Z'Z)^{-1}Z'X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77dc746b9f4c1394b0960f7d0b043209f2649778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.579ex; height:3.343ex;" alt="{\displaystyle {\hat {X}}=Z{\hat {\Gamma }}=Z(Z'Z)^{-1}Z'X}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\beta }}_{\mathrm {IV} }=({\hat {X}}'{\hat {X}})^{-1}{\hat {X}}'y=\beta +(X'Z(Z'Z)^{-1}Z'X)^{-1}X'Z(Z'Z)^{-1}Z'\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mi>y</mi> <mo>=</mo> <mi>β</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mi>Z</mi> <mo stretchy="false">(</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mi>Z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>X</mi> <mo>′</mo> </msup> <mi>Z</mi> <mo stretchy="false">(</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mi>Z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}_{\mathrm {IV} }=({\hat {X}}'{\hat {X}})^{-1}{\hat {X}}'y=\beta +(X'Z(Z'Z)^{-1}Z'X)^{-1}X'Z(Z'Z)^{-1}Z'\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/356d96785141e33e6e1f00c2f3a38019e5494918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.633ex; height:3.676ex;" alt="{\displaystyle {\hat {\beta }}_{\mathrm {IV} }=({\hat {X}}'{\hat {X}})^{-1}{\hat {X}}'y=\beta +(X'Z(Z'Z)^{-1}Z'X)^{-1}X'Z(Z'Z)^{-1}Z'\varepsilon }"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applicazione_del_metodo">Applicazione del metodo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variabili_strumentali&veaction=edit&section=5" title="Modifica la sezione Applicazione del metodo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variabili_strumentali&action=edit&section=5" title="Edit section's source code: Applicazione del metodo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Correlazione_(statistica)" title="Correlazione (statistica)">correlazione</a> tra regressori e disturbi in un modello di <a href="/wiki/Regressione_lineare" title="Regressione lineare">regressione lineare</a> può insorgere in una serie di circostanze. Alcuni casi notevoli, generalmente menzionati in letteratura, sono: </p> <ul><li><i>Omissione di variabili rilevanti</i>, se il modello di regressione (multivariata) non include tra i regressori una variabile, che pure avrebbe rilevante potere esplicativo nei confronti della variabile dipendente;</li> <li><i>Errore nelle variabili esplicative</i>, laddove i dati relativi a uno o più regressori sono affetti da un errore di misura, distinto dal disturbo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \varepsilon _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \varepsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29f326af611e792fe74845d4f706ac536492c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle \ \varepsilon _{i}}"></span>;</li> <li><i><a href="/w/index.php?title=Modello_di_equazioni_simultanee&action=edit&redlink=1" class="new" title="Modello di equazioni simultanee (la pagina non esiste)">Equazioni simultanee</a></i>, nei casi in cui il sistema oggetto di analisi mette insieme diversi modelli statistici che operano simultaneamente.</li></ul> <p>Il metodo delle variabili strumentali è spesso applicato con una procedura di stima con i minimi quadrati a due stadi (in <a href="/wiki/Lingua_inglese" title="Lingua inglese">inglese</a>, <i>Two-Stages Least Squares</i>, o 2SLS). Nell'approccio 2SLS, in un primo stadio di stima i regressori (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627df53731a09bc2fc34b492b0dfc639a405ec48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.009ex;" alt="{\displaystyle \ x_{i}}"></span> nella notazione sopra) sono regrediti sulle variabili strumentali (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4632271a1b9cafab7716bb515cdc56395d3b88d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.009ex;" alt="{\displaystyle \ z_{i}}"></span>), ottenendo dei valori di previsione di primo stadio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\hat {x}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\hat {x}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570a1fe3a4a89da11888f3276846ca40ec0dfc24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle \ {\hat {x}}_{i}}"></span>. Nel secondo stadio, la variabile dipendente (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0886e2e6eae3b6bfe41912c2047c3901bb310fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.52ex; height:2.009ex;" alt="{\displaystyle \ y_{i}}"></span>) è regredita sui valori di previsione di primo stadio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\hat {x}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\hat {x}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570a1fe3a4a89da11888f3276846ca40ec0dfc24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle \ {\hat {x}}_{i}}"></span>, ottenendo le stime <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37e385ba09b58a0971ff542bb211f228db751df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:3.343ex;" alt="{\displaystyle \ {\hat {\beta }}_{\mathrm {IV} }}"></span>. </p><p>Per le sue caratteristiche, il metodo delle variabili strumentali è soggetto a problemi legati alla scelta delle variabili strumentali stesse. Al di là dei requisiti formali per il funzionamento del metodo (assenza di correlazione con i disturbi), queste ultime potranno essere individuate mediante considerazioni strettamente relative al problema oggetto di analisi statistica. Modifiche esogene di una data politica (ad es., cancellazione di un programma di borse di studio), differenze geografiche nell'applicazione di dati standard (ad es., differenze di risultati necessari per il superamento di un dato esame in diversi stati), o mera casualità potranno definire delle opportune variabili strumentali. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variabili_strumentali&veaction=edit&section=6" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variabili_strumentali&action=edit&section=6" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> James Stock, Mark Watson, <span style="font-style:italic;">Introduzione all'econometria</span>, Milano, Pearson Education, 2005, p. 337, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-7192-267-6" title="Speciale:RicercaISBN/978-88-7192-267-6">978-88-7192-267-6</a>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variabili_strumentali&veaction=edit&section=7" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variabili_strumentali&action=edit&section=7" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Greene, W.H. (2000), <i>Econometric Analysis</i>, Prentice-Hall, <a href="/wiki/Speciale:RicercaISBN/0130132977" class="internal mw-magiclink-isbn">ISBN 0-13-013297-7</a>, analizza nel dettaglio il modello classico di regressione lineare nel caso multivariato, con particolare riferimento alle sue applicazioni nell'ambito dell'<a href="/wiki/Econometria" title="Econometria">econometria</a>, disciplina per cui rappresenta il testo di livello universitario/master di riferimento (in <a href="/wiki/Lingua_inglese" title="Lingua inglese">inglese</a>); il metodo delle variabili strumentali è trattato nel Capitolo 9.</li> <li>Wooldridge, J.M. (2002), <i>Econometric Analysis of Cross Section and Panel Data</i>, MIT Press, <a href="/wiki/Speciale:RicercaISBN/0262232197" class="internal mw-magiclink-isbn">ISBN 0-262-23219-7</a>, tratta in maniera più approfondita la stima tramite variabili strumentali, nel caso di modelli ad una singola equazione (capitolo 5) e di modelli di equazioni simultanee (capitolo 8); si tratta di un testo di riferimento per studi a livello di dottorato (in <a href="/wiki/Lingua_inglese" title="Lingua inglese">inglese</a>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variabili_strumentali&veaction=edit&section=8" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variabili_strumentali&action=edit&section=8" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Regressione_lineare" title="Regressione lineare">Regressione lineare</a></li> <li><a href="/wiki/Metodo_dei_momenti_(statistica)" title="Metodo dei momenti (statistica)">Metodo dei momenti (statistica)</a></li> <li><a href="/wiki/Metodo_generalizzato_dei_momenti" title="Metodo generalizzato dei momenti">Metodo generalizzato dei momenti</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Statistica"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Statistica" title="Template:Statistica"><span title="Vai alla pagina del template">V</span></a> · <a href="/wiki/Discussioni_template:Statistica" title="Discussioni template:Statistica"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Statistica&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Statistica" title="Statistica">Statistica</a></span></th></tr><tr><th colspan="1" class="navbox_group">Teoria statistica</th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox"><tbody><tr><th class="subnavbox_group" style="width:200px;text-align:left;"><a href="/wiki/Statistica_descrittiva" title="Statistica descrittiva">Statistica descrittiva</a></th><td colspan="1"><a href="/wiki/Media_(statistica)" title="Media (statistica)">Media</a> (<a href="/wiki/Media_(statistica)#Media_aritmetica" title="Media (statistica)">aritmetica</a><b> ·</b> <a href="/wiki/Media_(statistica)#Media_geometrica" title="Media (statistica)">geometrica</a><b> ·</b> <a href="/wiki/Media_armonica" title="Media armonica">armonica</a><b> ·</b> <a href="/wiki/Media_(statistica)#Media_di_potenza" title="Media (statistica)">di potenza</a><b> ·</b> <a href="/wiki/Media_(statistica)#Media_aritmetico-geometrica" title="Media (statistica)">aritmetico-geometrica</a><b> ·</b> <a href="/wiki/Media_(statistica)#Media_integrale" title="Media (statistica)">integrale</a>)<b> ·</b> <a href="/wiki/Mediana_(statistica)" title="Mediana (statistica)">Mediana</a><b> ·</b> <a href="/wiki/Moda_(statistica)" title="Moda (statistica)">Moda</a><b> ·</b> <a href="/wiki/Campo_di_variazione" title="Campo di variazione">Intervallo di variazione</a><b> ·</b> <a href="/wiki/Varianza" title="Varianza">Varianza</a><b> ·</b> <a href="/wiki/Scarto_quadratico_medio" title="Scarto quadratico medio">Deviazione standard</a><b> ·</b> <a href="/wiki/Scarto_medio_assoluto" title="Scarto medio assoluto">Scarto medio assoluto</a><b> ·</b> <a href="/wiki/Simmetria_(statistica)" title="Simmetria (statistica)">Simmetria</a><b> ·</b> <a href="/wiki/Differenza_media" class="mw-redirect" title="Differenza media">Differenza media</a> (<a href="/wiki/Differenza_media_assoluta" title="Differenza media assoluta">assoluta</a><b> ·</b> <a href="/wiki/Differenza_media_logaritmica" title="Differenza media logaritmica">logaritmica</a>)<b> ·</b> <a href="/wiki/Curtosi" title="Curtosi">Curtosi</a></td></tr><tr><th class="subnavbox_group" style="width:200px;text-align:left;"><a href="/wiki/Inferenza_statistica" title="Inferenza statistica">Inferenza statistica</a></th><td colspan="1"><a href="/wiki/Test_di_verifica_d%27ipotesi" title="Test di verifica d'ipotesi">Test di verifica d'ipotesi</a><b> ·</b> <a href="/wiki/Significativit%C3%A0" title="Significatività">Significatività</a><b> ·</b> <a href="/wiki/Ipotesi_nulla" title="Ipotesi nulla">Ipotesi nulla/alternativa</a><b> ·</b> <a href="/wiki/Errore_di_primo_tipo" class="mw-redirect" title="Errore di primo tipo">Errore del I</a> e <a href="/wiki/Errore_di_secondo_tipo" class="mw-redirect" title="Errore di secondo tipo">del II tipo</a><b> ·</b> <a href="/wiki/Test_Q" title="Test Q">Test Q</a><b> ·</b> <a href="/wiki/Test_U" class="mw-redirect" title="Test U">Test U</a><b> ·</b> <a href="/wiki/Test_t" title="Test t">Test t</a><b> ·</b> <a href="/wiki/Test_Z" title="Test Z">Test Z</a><b> ·</b> <a href="/wiki/Metodo_della_massima_verosimiglianza" title="Metodo della massima verosimiglianza">Massima verosimiglianza</a><b> ·</b> <a href="/wiki/Standardizzazione_(statistica)" title="Standardizzazione (statistica)">Standardizzazione</a><b> ·</b> <a href="/wiki/Valore_p" title="Valore p">Valore p</a><b> ·</b> <a href="/wiki/Analisi_della_varianza" title="Analisi della varianza">Analisi della varianza</a></td></tr><tr><th class="subnavbox_group" style="width:200px;text-align:left;"><a href="/wiki/Analisi_di_sopravvivenza" title="Analisi di sopravvivenza">Analisi di sopravvivenza</a></th><td colspan="1"><a href="/wiki/Tasso_di_guasto" title="Tasso di guasto">Tasso di guasto</a><b> ·</b> <a href="/wiki/Stimatore_di_Kaplan-Meier" title="Stimatore di Kaplan-Meier">Stimatore di Kaplan-Meier</a><b> ·</b> <a href="/wiki/Test_dei_ranghi_logaritmici" title="Test dei ranghi logaritmici">Test dei ranghi logaritmici</a></td></tr><tr><th class="subnavbox_group" style="width:200px;text-align:left;"><a href="/wiki/Analisi_della_regressione" title="Analisi della regressione">Analisi della regressione</a></th><td colspan="1"><a href="/wiki/Regressione_lineare" title="Regressione lineare">Regressione lineare</a><b> ·</b> <a href="/wiki/Regressione_nonlineare" title="Regressione nonlineare">Regressione nonlineare</a><b> ·</b> <a class="mw-selflink selflink">Variabili strumentali</a><b> ·</b> <a href="/wiki/Metodo_generalizzato_dei_momenti" title="Metodo generalizzato dei momenti">Metodo generalizzato dei momenti</a><b> ·</b> <a href="/wiki/Regressione_logistica" class="mw-redirect" title="Regressione logistica">Regressione logistica</a><b> ·</b> <a href="/wiki/Modello_probit" title="Modello probit">Modello probit</a><b> ·</b> <a href="/wiki/Modello_logit" title="Modello logit">Modello logit</a></td></tr></tbody></table></td><td rowspan="2" class="navbox_image"><span typeof="mw:File"><a href="/wiki/File:Composition_of_the_European_Parliament.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Composition_of_the_European_Parliament.svg/100px-Composition_of_the_European_Parliament.svg.png" decoding="async" width="100" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Composition_of_the_European_Parliament.svg/150px-Composition_of_the_European_Parliament.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Composition_of_the_European_Parliament.svg/200px-Composition_of_the_European_Parliament.svg.png 2x" data-file-width="195" data-file-height="179" /></a></span></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Statistica_economica" title="Statistica economica">Statistica economica</a></th><td colspan="1" class="navbox_list navbox_even"><table class="subnavbox"><tbody><tr><th class="subnavbox_group" style="width:200px;text-align:left;">Istituti statistici</th><td colspan="1"><a href="/wiki/Istituto_nazionale_di_statistica" title="Istituto nazionale di statistica">ISTAT</a><b> ·</b> <a href="/wiki/Eurostat" title="Eurostat">EuroSTAT</a><b> ·</b> <a href="/wiki/Royal_Statistical_Society" title="Royal Statistical Society">Royal Statistical Society</a><b> ·</b> <a href="/wiki/Ufficio_del_censimento_degli_Stati_Uniti_d%27America" title="Ufficio del censimento degli Stati Uniti d'America">U.S. Census Bureau</a><b> ·</b> <a href="/wiki/Istituto_internazionale_di_statistica" title="Istituto internazionale di statistica">ISI</a><b> ·</b> <a href="/wiki/Institut_national_de_la_statistique_et_des_%C3%A9tudes_%C3%A9conomiques" title="Institut national de la statistique et des études économiques">INSEE</a></td></tr><tr><th class="subnavbox_group" style="width:200px;text-align:left;"><a href="/wiki/Sito_web" title="Sito web">Siti web</a> statistici</th><td colspan="1"><a href="/wiki/Our_World_in_Data" title="Our World in Data">Our World in Data</a><b> ·</b> <a href="/wiki/Statista_(sito_web)" title="Statista (sito web)">Statista</a><b> ·</b> <a href="/wiki/Bloomberg_(azienda)" title="Bloomberg (azienda)">Bloomberg Terminal</a><b> ·</b> <a href="/wiki/Google_Public_Data_Explorer" title="Google Public Data Explorer">Google Public Data Explorer</a><b> ·</b> <a href="/wiki/World_Inequality_Database" title="World Inequality Database">World Inequality Database</a><b> ·</b> <a href="/wiki/TradingEconomics" title="TradingEconomics">TradingEconomics</a><b> ·</b> <a href="/wiki/Armed_Conflict_Location_and_Event_Data_Project" title="Armed Conflict Location and Event Data Project">ACLED</a></td></tr><tr><th class="subnavbox_group" style="width:200px;text-align:left;"><a href="/wiki/Software" title="Software">Software</a> <a href="/wiki/Econometria" title="Econometria">econometrici</a></th><td colspan="1"><a href="/wiki/Gretl" title="Gretl">gretl</a><b> ·</b> <a href="/wiki/EViews" title="EViews">EViews</a></td></tr></tbody></table></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Nuovo_soggettario" title="Nuovo soggettario">Thesaurus BNCF</a> <span class="uid"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=57217">57217</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85066780">sh85066780</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb122988792">cb122988792</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb122988792">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007553253805171">987007553253805171</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-basis: calc( 100% / 2 - 8px / 2 );"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_kchart.svg" class="mw-file-description" title="Economia"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Nuvola_apps_kchart.svg/25px-Nuvola_apps_kchart.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Nuvola_apps_kchart.svg/38px-Nuvola_apps_kchart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Nuvola_apps_kchart.svg/50px-Nuvola_apps_kchart.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <b><a href="/wiki/Portale:Economia" title="Portale:Economia">Portale Economia</a></b></div></div><div style="flex-basis: calc( 100% / 2 - 8px / 2 );"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r140555418"><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Greek_uc_sigma.svg" class="mw-file-description" title="Statistica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Greek_uc_sigma.svg/25px-Greek_uc_sigma.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Greek_uc_sigma.svg/38px-Greek_uc_sigma.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Greek_uc_sigma.svg/50px-Greek_uc_sigma.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span> <b><a href="/wiki/Portale:Statistica" title="Portale:Statistica">Portale Statistica</a></b></div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Estratto da "<a dir="ltr" href="https://it.wikipedia.org/w/index.php?title=Variabili_strumentali&oldid=139816191">https://it.wikipedia.org/w/index.php?title=Variabili_strumentali&oldid=139816191</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categoria:Categorie" title="Categoria:Categorie">Categoria</a>: <ul><li><a href="/wiki/Categoria:Analisi_di_regressione" title="Categoria:Analisi di regressione">Analisi di regressione</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie nascoste: <ul><li><a href="/wiki/Categoria:Voci_con_modulo_citazione_e_parametro_pagine" title="Categoria:Voci con modulo citazione e parametro pagine">Voci con modulo citazione e parametro pagine</a></li><li><a href="/wiki/Categoria:Voci_con_modulo_citazione_e_parametro_coautori" title="Categoria:Voci con modulo citazione e parametro coautori">Voci con modulo citazione e parametro coautori</a></li><li><a href="/wiki/Categoria:Voci_con_codice_Thesaurus_BNCF" title="Categoria:Voci con codice Thesaurus BNCF">Voci con codice Thesaurus BNCF</a></li><li><a href="/wiki/Categoria:Voci_con_codice_LCCN" title="Categoria:Voci con codice LCCN">Voci con codice LCCN</a></li><li><a href="/wiki/Categoria:Voci_con_codice_BNF" title="Categoria:Voci con codice BNF">Voci con codice BNF</a></li><li><a href="/wiki/Categoria:Voci_con_codice_J9U" title="Categoria:Voci con codice J9U">Voci con codice J9U</a></li><li><a href="/wiki/Categoria:Voci_non_biografiche_con_codici_di_controllo_di_autorit%C3%A0" title="Categoria:Voci non biografiche con codici di controllo di autorità">Voci non biografiche con codici di controllo di autorità</a></li><li><a href="/wiki/Categoria:Pagine_che_utilizzano_collegamenti_magici_ISBN" title="Categoria:Pagine che utilizzano collegamenti magici ISBN">Pagine che utilizzano collegamenti magici ISBN</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Questa pagina è stata modificata per l'ultima volta il 18 giu 2024 alle 16:14.</li> <li id="footer-info-copyright">Il testo è disponibile secondo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it">licenza Creative Commons Attribuzione-Condividi allo stesso modo</a>; possono applicarsi condizioni ulteriori. Vedi le <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use/it">condizioni d'uso</a> per i dettagli.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/it">Informativa sulla privacy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Sala_stampa/Wikipedia">Informazioni su Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Avvertenze_generali">Avvertenze</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Codice di condotta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Sviluppatori</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/it.wikipedia.org">Statistiche</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Dichiarazione sui cookie</a></li> <li id="footer-places-mobileview"><a href="//it.m.wikipedia.org/w/index.php?title=Variabili_strumentali&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Versione mobile</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-d6wxr","wgBackendResponseTime":206,"wgPageParseReport":{"limitreport":{"cputime":"0.224","walltime":"0.359","ppvisitednodes":{"value":3096,"limit":1000000},"postexpandincludesize":{"value":27947,"limit":2097152},"templateargumentsize":{"value":175,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":7613,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 182.574 1 -total"," 36.05% 65.815 1 Template:Statistica"," 34.08% 62.227 1 Template:Navbox"," 25.76% 47.031 1 Template:Controllo_di_autorità"," 21.18% 38.661 1 Template:Portale"," 15.62% 28.512 1 Template:Cita_libro"," 12.03% 21.969 2 Template:Icona_argomento"," 9.58% 17.484 2 Template:Navbox_subgroup"," 2.26% 4.133 46 Template:·"]},"scribunto":{"limitreport-timeusage":{"value":"0.108","limit":"10.000"},"limitreport-memusage":{"value":1983606,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-84d8f4b96-mw6fk","timestamp":"20241117173237","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Variabili strumentali","url":"https:\/\/it.wikipedia.org\/wiki\/Variabili_strumentali","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1665305","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1665305","author":{"@type":"Organization","name":"Contributori ai progetti Wikimedia"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-08-24T10:18:08Z","dateModified":"2024-06-18T15:14:20Z"}</script> </body> </html>

CINXE.COM

Variabili strumentali - Wikipedia