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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Korrelogramm</span></h1>
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<p>Ein <b>Korrelogramm</b> ist die graphische Darstellung der <a href="Autokorrelation" title="Autokorrelation">Autokorrelation</a> einer <a href="Zeitreihe" class="mw-disambig" title="Zeitreihe">Zeitreihe</a>. Dazu werden die geschätzten <a href="Korrelationskoeffizient" class="mw-redirect" title="Korrelationskoeffizient">Korrelationskoeffizienten</a> <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 }}_{l}}">
<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>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}_{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/238cc8c23c3d6340f2d9ab7a18f4e188aab61ec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.098ex; height:2.676ex;" alt="{\displaystyle {\hat {\rho }}_{l}}" loading="lazy"></span> gegen die Dauer der Zeitverschiebung <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 l}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
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<annotation encoding="application/x-tex">{\displaystyle l}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span> abgetragen.
</p><p>Mit dem Korrelogramm kann untersucht werden, ob eine signifikante Autokorrelation bei einer Zeitverschiebung vorliegt. Die <a href="Nullhypothese" class="mw-redirect" title="Nullhypothese">Nullhypothese</a> ist, dass <i>keine</i> Autokorrelation vorliegt.
</p><p>Dazu wird <a href="Konfidenzintervall" title="Konfidenzintervall">Konfidenzintervall</a> der geschätzten Korrelationskoeffizienten betrachtet:
</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 B=\pm t_{1-\alpha /2}\,SE({\hat {\rho }}_{l}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo>=</mo>
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<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>−<!-- − --></mo>
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<mi>S</mi>
<mi>E</mi>
<mo stretchy="false">(</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
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<mo stretchy="false">)</mo>
<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle B=\pm t_{1-\alpha /2}\,SE({\hat {\rho }}_{l}),}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/babc8e2975f58e75153fa500e35a79f649e65149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.755ex; height:3.176ex;" alt="{\displaystyle B=\pm t_{1-\alpha /2}\,SE({\hat {\rho }}_{l}),}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<ul><li>dem entsprechenden <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 \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> <a href="Perzentil" class="mw-disambig" title="Perzentil">Perzentil</a> der <a href="Students_t-Verteilung" class="mw-redirect" title="Students t-Verteilung">t-Verteilung</a> für die Irrtumswahrscheinlichkeit für den Fehler 1. Ordnung (Signifikanzniveau)</li>
<li>dem <a href="Standardfehler" title="Standardfehler">Standardfehler</a>&nbsp;SE</li></ul>
<p>Liegt <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 }}_{l}}">
<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>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}_{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/238cc8c23c3d6340f2d9ab7a18f4e188aab61ec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.098ex; height:2.676ex;" alt="{\displaystyle {\hat {\rho }}_{l}}" loading="lazy"></span> in dem Intervall <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 B}">
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<mi>B</mi>
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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>, so wird die Nullhypothese mit einer <a href="Irrtumswahrscheinlichkeit" title="Irrtumswahrscheinlichkeit">Irrtumswahrscheinlichkeit</a> von <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 \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> <i>abgelehnt</i>.
</p><p>Der <a href="Standardfehler" title="Standardfehler">Standardfehler</a>&nbsp;SE wird in diesem Zusammenhang meist anhand <a href="Maurice_Bartlett" title="Maurice Bartlett">Bartletts</a> Formel für <a href="ARMA-Modell#Moving-Average-_oder_MA-Modell" title="ARMA-Modell">MA(l)-Prozesse</a> berechnet (<i>moving average</i>, siehe dazu <a href="ARMA-Modell" title="ARMA-Modell">ARMA-Modell</a>):
</p>
<dl><dd><dl><dd>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 l=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fc60b21200ebf2b338c4fa71b103cb697b02bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=1}" loading="lazy"></span>: <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 SE({\hat {\rho }}_{1})={\frac {1}{T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mi>E</mi>
<mo stretchy="false">(</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ρ<!-- ρ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>T</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle SE({\hat {\rho }}_{1})={\frac {1}{T}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e6bc4750c3c54c093dfdaca51b7deeaa36413fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.085ex; height:5.176ex;" alt="{\displaystyle SE({\hat {\rho }}_{1})={\frac {1}{T}}}" loading="lazy"></span></dd>
<dd>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 l>1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>&gt;</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l&gt;1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1c4d657af24094218b2daf444fe15b6316ed825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l>1}" loading="lazy"></span>: <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 SE({\hat {\rho }}_{l})={\sqrt {\frac {1+2\sum _{i=1}^{l-1}{\hat {\rho }}_{i}^{2}}{T}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mi>E</mi>
<mo stretchy="false">(</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">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
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</munderover>
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>T</mi>
</mfrac>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle SE({\hat {\rho }}_{l})={\sqrt {\frac {1+2\sum _{i=1}^{l-1}{\hat {\rho }}_{i}^{2}}{T}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d52e38bf420def6df03bec7b2d9523b1f4c813e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.164ex; height:7.509ex;" alt="{\displaystyle SE({\hat {\rho }}_{l})={\sqrt {\frac {1+2\sum _{i=1}^{l-1}{\hat {\rho }}_{i}^{2}}{T}}}}" loading="lazy"></span></dd></dl></dd></dl>
<ul><li>der geschätzten Autokorrelation <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 }}_{l}}">
<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>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}_{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/238cc8c23c3d6340f2d9ab7a18f4e188aab61ec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.098ex; height:2.676ex;" alt="{\displaystyle {\hat {\rho }}_{l}}" loading="lazy"></span> zwischen Beobachtungen, 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 l}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
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<annotation encoding="application/x-tex">{\displaystyle l}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span> Perioden auseinanderliegen.</li></ul>
<p>Im Bild wird daher die Nullhypothese (dass keine Autokorrelation zwischen benachbarten Perioden besteht) bei den ersten vorliegenden Verzögerung verworfen, da dort die Null nicht innerhalb des Konfidenzintervalls liegt. Für diese ersten Verzögerungen gilt, dass direkt aufeinanderfolgender Zeitpunkte <a href="Statistische_Signifikanz" title="Statistische Signifikanz">signifikant</a> korreliert sind. Für die übrigen Verzögerungen kann die Nullhypothese fehlender Autokorrelation allerdings nicht abgelehnt werden.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Portmanteau-Test" title="Portmanteau-Test">Portmanteau-Test</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><span class="cite"><a rel="nofollow" class="external text" href="https://quantdare.com/bootstrapping-time-series-data/"><i>Bootstrapping time series data ⋆ Quantdare.</i></a> In: <i>Quantdare.</i> 28.&nbsp;Juni 2018,<span class="Abrufdatum"> abgerufen am 2.&nbsp;Oktober 2021</span> (britisches Englisch).</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&amp;rfr_id=info%3Asid%2Fde.wikipedia.org%3AKorrelogramm&amp;rft.title=Bootstrapping+time+series+data+%E2%8B%86+Quantdare&amp;rft.description=Bootstrapping+time+series+data+%E2%8B%86+Quantdare&amp;rft.identifier=https%3A%2F%2Fquantdare.com%2Fbootstrapping-time-series-data%2F&amp;rft.date=2018-06-28&amp;rft.language=en-GB">&nbsp;</span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>John E. Hanke, Arthur G. Reitsch, Dean W. Wichern: <i>Business forecasting.</i> 7. Aufl. Prentice Hall, Upper Saddle River, NJ 2001, ISBN 0-13-087810-3.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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