<!DOCTYPE html>
<html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-0 vector-toc-not-available 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-0 skin-theme-clientpref-day vector-sticky-header-enabled" lang="de" dir="ltr"><head>
<meta charset="UTF-8">
<title>Durbin-Watson-Test</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="icon" type="image/png" href="./_res_/favicon.png">
<link rel="canonical" href="https://de.wikipedia.org/wiki/Durbin-Watson-Test"> <link href="./_mw_/ext.cite.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.wikimediamessages.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link href="./_mw_/ext.gadget.citeRef.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.defaultPlainlinks.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonHide.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonLayout.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonStyle.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiDarkmode.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiResponsive.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.specialSearch.css" rel="stylesheet" type="text/css">
<link rel="stylesheet" type="text/css" href="./_mw_/site.styles.css">
<link rel="stylesheet" type="text/css" href="./_mw_/noscript.css">
<link rel="stylesheet" type="text/css" href="./_res_/footer.css">
<link rel="stylesheet" type="text/css" href="./_res_/vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Durbin-Watson-Test rootpage-Durbin-Watson-Test skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Durbin-Watson-Test</span></h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="contentSub">
<div id="mw-content-subtitle"></div>
</div>
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Durbin-Watson-Test</b> ist ein <a href="Statistischer_Test" title="Statistischer Test">statistischer Test</a>, mit dem man versucht zu überprüfen, ob eine <a href="Autokorrelation" title="Autokorrelation">Autokorrelation</a> 1. Ordnung vorliegt, d. h., ob die <a href="Korrelation" title="Korrelation">Korrelation</a> zwischen zwei aufeinanderfolgenden Residualgrößen bei einer <a href="Regressionsanalyse" title="Regressionsanalyse">Regressionsanalyse</a> ungleich null ist. Der Test wurde von dem britischen Statistiker <a href="James_Durbin" title="James Durbin">James Durbin</a> und dem Australier Geoffrey Watson entwickelt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Vorgehen">Vorgehen</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Hypothesen">Hypothesen</h3></div>
<p>Die Störterme werden bei Autokorrelation 1. Ordnung wie folgt modelliert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{t}=\rho _{1}\epsilon _{t-1}+\nu _{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>ν<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \epsilon _{t}=\rho _{1}\epsilon _{t-1}+\nu _{t}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bdab1768d1df5a4b757fbe83bb9c7855e7d1923.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.81ex; height:2.509ex;" alt="{\displaystyle \epsilon _{t}=\rho _{1}\epsilon _{t-1}+\nu _{t}}" loading="lazy"></span>. Beim Durbin-Watson-Test wird eine Nullhypothese, die besagt, dass keine Autokorrelation vorliegt (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{1}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b94e75d78df1b3ff7f0a7b7652d9139e0ba8150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.517ex; height:2.676ex;" alt="{\displaystyle \rho _{1}=0}" loading="lazy"></span>) und deren Gegenhypothese, welche aussagt, dass Autokorrelation vorliegt (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}\neq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{1}\neq 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90100751dda6291f290862db022500da7e80b1cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.517ex; height:2.676ex;" alt="{\displaystyle \rho _{1}\neq 0}" loading="lazy"></span>), aufgestellt.
</p>
<div class="mw-heading mw-heading3"><h3 id="Teststatistik">Teststatistik</h3></div>
<p>Die Teststatistik lautet:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d:={\frac {\sum _{t=2}^{T}(\epsilon _{t}-\epsilon _{t-1})^{2}}{\sum _{t=1}^{T}\epsilon _{t}^{2}}}\approx 2(1-{\hat {\rho }}_{1})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>:=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</munderover>
<msubsup>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d:={\frac {\sum _{t=2}^{T}(\epsilon _{t}-\epsilon _{t-1})^{2}}{\sum _{t=1}^{T}\epsilon _{t}^{2}}}\approx 2(1-{\hat {\rho }}_{1})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f32ed2f1dfe40dcb84062da42afbca5c17ab5b84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.025ex; height:7.509ex;" alt="{\displaystyle d:={\frac {\sum _{t=2}^{T}(\epsilon _{t}-\epsilon _{t-1})^{2}}{\sum _{t=1}^{T}\epsilon _{t}^{2}}}\approx 2(1-{\hat {\rho }}_{1})}" loading="lazy"></span></dd></dl>
<p>Hierbei bezeichnen die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \epsilon _{t}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1ff4ca7b5c076264ecd5bbc087863a23c4dfb9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.77ex; height:2.009ex;" alt="{\displaystyle \epsilon _{t}}" loading="lazy"></span> jeweils die Residuen der Regression in der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>-ten Periode. Wenn die Differenz zwischen den Residualgrößen sehr klein bzw. sehr groß ist, so liegt positive bzw. negative Autokorrelation vor. Dies führt dazu, dass der Durbin-Watson-Wert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> gegen den Wert null bzw. vier strebt.
</p>
<div class="mw-heading mw-heading3"><h3 id="Testentscheidung">Testentscheidung</h3></div>
<table class="wikitable" style="border:0;">
<tbody><tr>
<th>Wert der Teststatistik
</th>
<th>Korrelation
</th>
<th>Bedeutung
</th></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=2\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d=2\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25086cbeded0a8c1eb5f83feb0b98ce19bc1a4a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.864ex; height:2.176ex;" alt="{\displaystyle d=2\,}" loading="lazy"></span>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><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 {\rho }}_{1}=0\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}_{1}=0\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42c9062e69d8e9d04fd7639ad0264595e5577117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.078ex; height:2.676ex;" alt="{\displaystyle {\hat {\rho }}_{1}=0\,}" loading="lazy"></span>
</td>
<td>keine Autokorrelation
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=0\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>=</mo>
<mn>0</mn>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d=0\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36557e090dd49fd8dc292a0816b0091a50d30523.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.864ex; height:2.176ex;" alt="{\displaystyle d=0\,}" loading="lazy"></span>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><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 {\rho }}_{1}=1\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}_{1}=1\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7606aacad441be6f45272bcf6019d64b7f37442a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.078ex; height:2.676ex;" alt="{\displaystyle {\hat {\rho }}_{1}=1\,}" loading="lazy"></span>
</td>
<td>perfekte positive Autokorrelation
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=4\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>=</mo>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d=4\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2e103d2915e79cc30e55541b8d0d3eff14f5bc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.864ex; height:2.176ex;" alt="{\displaystyle d=4\,}" loading="lazy"></span>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><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 {\rho }}_{1}=-1\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}_{1}=-1\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fba0fdfb6338241408a6d8a88d9ca72877c045d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.886ex; height:2.676ex;" alt="{\displaystyle {\hat {\rho }}_{1}=-1\,}" loading="lazy"></span>
</td>
<td>perfekte negative Autokorrelation
</td></tr></tbody></table>
<p>Die An- und Ablehnungsbereiche können tabellarisch ermittelt werden.<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> Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d<d_{u}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo><</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d<d_{u}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa595be7f0658229f511893e53132d58d183f053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.696ex; height:2.509ex;" alt="{\displaystyle d<d_{u}}" loading="lazy"></span> liegt positive Autokorrelation vor, für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d>4-d_{u}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>></mo>
<mn>4</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d>4-d_{u}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee141ac866d1b13c1434021b6cd1903ce8d27a37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.699ex; height:2.509ex;" alt="{\displaystyle d>4-d_{u}}" loading="lazy"></span> negative Autokorrelation, zwischen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{o}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>o</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{o}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db086c421821a5de9a49de753a122c28ad4233c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle d_{o}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4-d_{o}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>4</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>o</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 4-d_{o}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28e32fff4fb568cc2c6322c6117fa942250c6552.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.241ex; height:2.509ex;" alt="{\displaystyle 4-d_{o}}" loading="lazy"></span> liegt keine Autokorrelation vor. In den Intervallen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrack d_{u};d_{o}\rbrack }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">[</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<mo>;</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>o</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lbrack d_{u};d_{o}\rbrack }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5def0d3fb8759ad45376cf4a429fd9c5fda7935f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.948ex; height:2.843ex;" alt="{\displaystyle \lbrack d_{u};d_{o}\rbrack }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrack 4-d_{o};4-d_{u}\rbrack }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">[</mo>
<mn>4</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>o</mi>
</mrow>
</msub>
<mo>;</mo>
<mn>4</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lbrack 4-d_{o};4-d_{u}\rbrack }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b58d915fff31def82b7fd75ff29201f378d7868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.953ex; height:2.843ex;" alt="{\displaystyle \lbrack 4-d_{o};4-d_{u}\rbrack }" loading="lazy"></span> liegen Unschärfebereiche vor, in denen keine Aussagen getroffen werden können.
</p>
<div class="mw-heading mw-heading2"><h2 id="Durbin_h-Statistik">Durbin h-Statistik</h2></div>
<p>Bei <a href="ARMA-Modell" title="ARMA-Modell">autoregressiven Modellen</a> ist diese Teststatistik zum Wert zwei hin verzerrt, sodass die Autokorrelation unterschätzt wird. Allerdings lässt sich aus der obigen Statistik leicht die bei großen Stichproben <a href="Normalverteilung" title="Normalverteilung">standardnormalverteilte</a> und unverzerrte Durbin h-Statistik herleiten:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=\left(1-{\frac {1}{2}}d\right){\sqrt {\frac {T}{1-T\cdot {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mi>T</mi>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">V</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">r</mi>
</mrow>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>β<!-- β --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h=\left(1-{\frac {1}{2}}d\right){\sqrt {\frac {T}{1-T\cdot {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb47cf707ce357d961e07a643caae62a5e4b73dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:34.502ex; height:8.509ex;" alt="{\displaystyle h=\left(1-{\frac {1}{2}}d\right){\sqrt {\frac {T}{1-T\cdot {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)}}}}" loading="lazy"></span>,</dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">V</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">r</mi>
</mrow>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>β<!-- β --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89aafa7a4a6a81ec21cd248e2f716870395bf56b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.431ex; height:4.843ex;" alt="{\displaystyle {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)}" loading="lazy"></span> die geschätzte Varianz des Regressionskoeffizienten der <a href="Exogene_und_endogene_Variable#Verzögerte_exogene_und_endogene_Variable" title="Exogene und endogene Variable">zeitlich verzögerten endogenen Variable</a> ist und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\cdot {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)<1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">V</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">r</mi>
</mrow>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>β<!-- β --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T\cdot {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)<1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af25ea33e4dfea63eb8c4e1a42f706f16ea4b00d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.007ex; height:4.843ex;" alt="{\displaystyle T\cdot {\widehat {\mathrm {Var} }}\left({\hat {\beta }}\right)<1}" loading="lazy"></span> sein muss.
</p>
<div class="mw-heading mw-heading2"><h2 id="Durbin-Watson-Test_für_Paneldaten"><span id="Durbin-Watson-Test_f.C3.BCr_Paneldaten"></span>Durbin-Watson-Test für Paneldaten</h2></div>
<p>Für <a href="Paneldaten" title="Paneldaten">Paneldaten</a> lässt sich die obige Teststatistik wie folgt verallgemeinern:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{pd}={\frac {\sum _{i=1}^{N}\sum _{t=2}^{T}(\epsilon _{it}-\epsilon _{i,t-1})^{2}}{\sum _{i=1}^{N}\sum _{t=1}^{T}\epsilon _{it}^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
<mi>d</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</munderover>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>t</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</munderover>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</munderover>
<msubsup>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>t</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{pd}={\frac {\sum _{i=1}^{N}\sum _{t=2}^{T}(\epsilon _{it}-\epsilon _{i,t-1})^{2}}{\sum _{i=1}^{N}\sum _{t=1}^{T}\epsilon _{it}^{2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ecda333ed42388ee7d77b70ec19d4bbecb9bbee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.121ex; height:7.676ex;" alt="{\displaystyle d_{pd}={\frac {\sum _{i=1}^{N}\sum _{t=2}^{T}(\epsilon _{it}-\epsilon _{i,t-1})^{2}}{\sum _{i=1}^{N}\sum _{t=1}^{T}\epsilon _{it}^{2}}}}" loading="lazy"></span>, mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{it}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \epsilon _{it}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28d403a812876e3b33b51807a608fd7845b2f503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.338ex; height:2.009ex;" alt="{\displaystyle \epsilon _{it}}" loading="lazy"></span> = Residuen der Within-Regression</dd></dl>
<p>Diese Teststatistik wird dann mit den in Abhängigkeit von <i>T</i> (Länge des <a href="Paneldaten#Einteilung_nach_Paneldesign" title="Paneldaten">balancierten Paneldatensatzes</a>), <i>K</i> (Zahl der Regressoren) und <i>N</i> (Zahl der beobachteten Individuen) tabellierten Annahme- und Ablehnungsbereiche verglichen [siehe hierzu bspw. Bhargava et al. (1982), Seite 537]. Eine Variante dieser Statistik für unbalancierte Paneldaten wurde von Baltagi und Wu (1999) entwickelt.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Gujarati, Damodar N. (1995): Basic Econometrics, 3. Aufl., New York et al.: McGraw-Hill, 1995, Seite 605f.</li>
<li>Eckey, Hans-Friedrich/Kosfeld, Reinhold/Dreger, Christian (2004): Ökonometrie, 3., überarb. und erw. Aufl., Wiesbaden: Gabler, 2004, Seite 114ff.</li>
<li>Verbeek, Marno (2004): A Guide to Modern Econometrics, 2. Aufl., Chichester: John Wiley & Sons, 2004, Seite 102f.</li>
<li>Bhargava, A./Franzini, L./Narendranathan, W. (1982): Serial Correlation and the Fixed Effects Models, in: Review of Economic Studies, Vol. 49 Iss. 158, 1982, Seite 533–549.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Vergleiche zu diesem Absatz: Eckey, Hans-Friedrich/Kosfeld, Reinhold/Dreger, Christian (2004): Ökonometrie, 3., überarb. und erw. Aufl., Wiesbaden: Gabler, 2004.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">d1 in Formel 16 in Baltagi/Wu (1999), Unequally spaced panel data regressions with AR(1) disturbances. Econometric Theory, 15(6), S. 814–823.</span>
</li>
</ol></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-12-07" href="https://de.wikipedia.org/wiki/?title=Durbin-Watson-Test&oldid=251046106">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
<script src="./_webp_/webpHandler.js"></script>
</body></html>