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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">ARCH-Modelle</span></h1>

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<p><b>ARCH-Modelle</b> (ARCH, <a href="Akronym" title="Akronym">Akronym</a> für: <i><b>A</b>uto<b>R</b>egressive <b>C</b>onditional <b>H</b>eteroscedasticity</i>, <a href="Deutsche_Sprache" title="Deutsche Sprache">deutsch</a> <i>autoregressive bedingte <a href="Heteroskedastizit%C3%A4t" class="mw-redirect" title="Heteroskedastizität">Heteroskedastizität</a></i>) bzw. <b>autoregressive bedingt heteroskedastische Zeitreihenmodelle</b> sind <a href="Stochastik" title="Stochastik">stochastische</a> Modelle zur <a href="Zeitreihenanalyse" title="Zeitreihenanalyse">Zeitreihenanalyse</a>, mit deren Hilfe insbesondere <a href="Finanzmathematik" title="Finanzmathematik">finanzmathematische</a> Zeitreihen mit nicht konstanter <a href="Volatilit%C3%A4t" title="Volatilität">Volatilität</a> beschrieben werden können. Sie gehen von der Annahme aus, dass die <a href="Bedingte_Varianz" title="Bedingte Varianz">bedingte Varianz</a> der zufälligen <a href="Modellfehler" title="Modellfehler">Modellfehler</a> abhängig ist vom realisierten Zufallsfehler der Vorperiode, so dass große und kleine Fehler dazu tendieren, in Gruppen aufzutreten. ARCH-Modelle wurden von <a href="Robert_F._Engle" title="Robert F. Engle">Robert F. Engle</a> in den 1980er Jahren entwickelt. Im Jahr 2003 wurde ihm dafür der <a href="Nobelpreis_f%C3%BCr_Wirtschaftswissenschaften" class="mw-redirect" title="Nobelpreis für Wirtschaftswissenschaften">Nobelpreis für Wirtschaftswissenschaften</a> verliehen.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Eine Zeitreihe <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 (x_{t})_{t\in \mathbb {Z} }}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{t})_{t\in \mathbb {Z} }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1866e7bd1ece543a64e54bd7015700a60dfe5f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.983ex; height:2.843ex;" alt="{\displaystyle (x_{t})_{t\in \mathbb {Z} }}" loading="lazy"></span> heißt <i>ARCH(p)-Zeitreihe</i>, wenn sie rekursiv definiert ist durch<sup id="cite_ref-kreiss_1-0" class="reference"><a href="#cite_note-kreiss-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{t}&=\sigma _{t}\epsilon _{t}\\\sigma _{t}^{2}&=a_{0}+a_{1}x_{t-1}^{2}+\dotsb +a_{p}x_{t-p}^{2},\end{aligned}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88d34d2d7a60b574fdec13411c0c9e4bb0c86d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.5ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}x_{t}&=\sigma _{t}\epsilon _{t}\\\sigma _{t}^{2}&=a_{0}+a_{1}x_{t-1}^{2}+\dotsb +a_{p}x_{t-p}^{2},\end{aligned}}}" 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 a_{0},\dotsc ,a_{p}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0},\dotsc ,a_{p}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38fe6938ccdd3517adf6b5ee561d0be1c9703db8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.751ex; height:2.343ex;" alt="{\displaystyle a_{0},\dotsc ,a_{p}}" 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 a_{p}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle a_{p}\neq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/869c96f62d353edc9d7d7c9fee3f38c38f577ca1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.55ex; height:2.843ex;" alt="{\displaystyle a_{p}\neq 0}" loading="lazy"></span> reelle, nichtnegative Parameter sind, und der Prozess <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})_{t\in \mathbb {Z} }}">

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<annotation encoding="application/x-tex">{\displaystyle (\epsilon _{t})_{t\in \mathbb {Z} }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8af147c277259de91f4c37cafa79381cee5e0fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.598ex; height:2.843ex;" alt="{\displaystyle (\epsilon _{t})_{t\in \mathbb {Z} }}" loading="lazy"></span> aus <a href="Unabh%C3%A4ngig_identisch_verteilte_Zufallsvariablen" class="mw-redirect" title="Unabhängig identisch verteilte Zufallsvariablen">unabhängigen identisch verteilten Zufallsvariablen</a> 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 \operatorname {E} (\epsilon _{t})=0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\epsilon _{t})=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6181ca7c5a4c2402ee6126a83effa2b8f9f0a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.423ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (\epsilon _{t})=0}" 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 \operatorname {Var} (\epsilon _{t})=1}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (\epsilon _{t})=1}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<p>Für ARCH-Modelle gelten unter der Zusatzbedingung, dass <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 \sigma _{t}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{t}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5dd6db0238ac32f34c6feb604748e253e842356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle \sigma _{t}}" loading="lazy"></span> für alle <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\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {Z} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71d44797c8c9175ffba8896bddd5749b2aedfd38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {Z} }" loading="lazy"></span> bezüglich der durch <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 _{s})_{s\leq t-1}}">

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<annotation encoding="application/x-tex">{\displaystyle (\epsilon _{s})_{s\leq t-1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92f0ffc8c16e381d50087a812bcb1f3c1896dfea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.733ex; height:2.843ex;" alt="{\displaystyle (\epsilon _{s})_{s\leq t-1}}" loading="lazy"></span> erzeugten σ-Algebra <a href="Messbare_Funktion" title="Messbare Funktion">messbar</a> ist, die folgenden Aussagen:<sup id="cite_ref-kreiss_1-1" class="reference"><a href="#cite_note-kreiss-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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>

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<ul><li>Die auf die Vergangenheit <a href="Bedingter_Erwartungswert" title="Bedingter Erwartungswert">bedingten Erwartungswerte</a> und <a href="Bedingte_Varianz" title="Bedingte Varianz">bedingten Varianzen</a> sind:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (x_{t}\mid x_{t-1},x_{t-2},\dotsc )=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d861971e6cb83e92665399482817da08b66c2be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.049ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (x_{t}\mid x_{t-1},x_{t-2},\dotsc )=0}" loading="lazy"></span> und</dd>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (x_{t}\mid x_{t-1},x_{t-2},\dotsc )=\sigma _{t}^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd1602c158d493e39a3f22f32c08c3f1d87d38f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.506ex; height:3.176ex;" alt="{\displaystyle \operatorname {Var} (x_{t}\mid x_{t-1},x_{t-2},\dotsc )=\sigma _{t}^{2}}" loading="lazy"></span>.</dd></dl></dd></dl>

<ul><li>Eine ARCH(<i>p</i>)-Zeitreihe <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 (x_{t})}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5aeaece4a584f41f264efce53dab9ddbad9d7eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.965ex; height:2.843ex;" alt="{\displaystyle (x_{t})}" loading="lazy"></span> ist genau dann <a href="Station%C3%A4rer_stochastischer_Prozess" title="Stationärer stochastischer Prozess">(schwach) stationär</a>, wenn alle Nullstellen des <i>charakteristischen Polynoms</i></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle P(z)=1-a_{1}z-\dotsb -a_{p}z^{p}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/826266508e84d493c4d18c12efee92fc4c95deb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.959ex; height:3.009ex;" alt="{\displaystyle P(z)=1-a_{1}z-\dotsb -a_{p}z^{p}}" loading="lazy"></span></dd></dl></dd>

<dd>außerhalb des <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen</a> Einheitskreises liegen.</dd></dl>

<ul><li>Eine stationäre ARCH(<i>p</i>)-Zeitreihe hat den stationären Erwartungswert <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 \operatorname {E} (x_{t})=0}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0429e1de3ead3773bddfeff686db4fce2a43930d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.809ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (x_{t})=0}" loading="lazy"></span> und ihre <a href="Autokorrelation" title="Autokorrelation">Autokorrelation</a> verschwindet: <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 \operatorname {Cov} (x_{t},x_{t+h})=0}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f92cafe6996040a1aa31f17abed850a1cfca0c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.709ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cov} (x_{t},x_{t+h})=0}" loading="lazy"></span> 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 h>0}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbddb7a5cca6170575e4e73e769fbb434c2a3d71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h>0}" loading="lazy"></span>. Für ihre stationäre Varianz gilt die Formel</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (x_{t})={\frac {a_{0}}{1-\sum _{k=1}^{p}a_{k}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dad1014c1ab4edb4952ff251dc6a87b23b291e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.068ex; height:5.843ex;" alt="{\displaystyle \operatorname {Var} (x_{t})={\frac {a_{0}}{1-\sum _{k=1}^{p}a_{k}}}}" loading="lazy"></span>.</dd></dl></dd></dl>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccd7a31bca9bfec463d8bb756cfea77078aeed3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.193ex; height:3.176ex;" alt="{\displaystyle (x_{t}^{2})}" loading="lazy"></span> eine <a href="ARMA-Modell#AR-Modell" title="ARMA-Modell">AR-Zeitreihe</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerungen">Verallgemeinerungen</h2></div>

<p>Die Idee des ARCH-Modells wurde in verschiedener Weise weiterentwickelt und gehört heute ganz selbstverständlich zu den fortgeschrittenen Methoden der <a href="%C3%96konometrie" title="Ökonometrie">Ökonometrie</a>.

</p><p>Eine Verallgemeinerung sind die <a href="GARCH-Modelle" title="GARCH-Modelle">GARCH-Modelle</a> (<i><b>g</b>eneralized <b>a</b>uto<b>r</b>egressive <b>c</b>onditional <b>h</b>eteroscedasticity</i>), die 1986 von <a href="Tim_Bollerslev" title="Tim Bollerslev">Tim Bollerslev</a> entwickelt wurden. Hierbei hängt die bedingte <a href="Varianz_(Stochastik)" title="Varianz (Stochastik)">Varianz</a> nicht nur von der Historie der Zeitreihe ab, sondern auch von ihrer eigenen Vergangenheit. Zeitstetige Analoga, sogenannte <a href="GARCH-Modell#COGARCH-Modell" class="mw-redirect" title="GARCH-Modell">COGARCH-Modelle</a> (<i><b>co</b>ntinuous-time <b>GARCH</b></i>), wurden von Feike C. <a href="Drost" title="Drost">Drost</a> und Bas J. C. Werker sowie <a href="Claudia_Kl%C3%BCppelberg" title="Claudia Klüppelberg">Claudia Klüppelberg</a>, Alexander Lindner und Ross Maller vorgestellt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Robert F. Engle: <i>Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of UK. Inflation.</i> In: <i>Econometrica.</i> Vol.: 50, pp. 987–1008, 1982. <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/1912773">1912773</a></li>

<li>Tim Bollerslev: <i>Generalized Autoregressive Conditional Heteroskedasticity.</i> In: <i>Journal of Econometrics.</i> Vol.: 31 No.: 3, pp. 307–327, 1986. <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1016/0304-4076%2886%2990063-1">10.1016/0304-4076(86)90063-1</a></span></li>

<li>Jürgen Franke, <a href="Wolfgang_H%C3%A4rdle" title="Wolfgang Härdle">Wolfgang Härdle</a>, Christian Matthias Hafner: <i>Statistics of Financial Markets: An Introduction.</i> 3. Auflage Springer, Berlin/Heidelberg/New York 2011, ISBN 978-3-642-16520-7, Kapitel 13, S. 283–342.</li>

<li>Christian Gouriéroux: <i>ARCH Models and Financial Applications.</i> Springer, New York 1997, ISBN 0-387-94876-7.</li>

<li>Feike C. Drost, F.C., Bas J. C. Werker: <i>Closing the GARCH gap: continuous GARCH modelling.</i> In: <i>Journal of Econometrics.</i> Vol.: 74, No.: 1, pp. 31–57, 1996. <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1016/0304-4076%2895%2901750-X">10.1016/0304-4076(95)01750-X</a></span></li>

<li>Claudia Klüppelberg, Alexander Lindner, Ross Maller: <i>A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour.</i> In: <i>Journal of Applied Probability.</i> Vol.: 41 No.: 3, pp. 601–622, 2004. <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1239/jap%2F1091543413">10.1239/jap/1091543413</a></span> <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/4141341">4141341</a></li>

<li>Evdokia Xekalaki, Stavros Degiannakis: <i>ARCH Models for Financial Applications.</i> Wiley, New York 2010, ISBN 978-0-470-06630-0.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-kreiss-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-kreiss_1-0">a</a></sup> <sup><a href="#cite_ref-kreiss_1-1">b</a></sup></span> <span class="reference-text">Jens-Peter Kreiß, Georg Neuhaus: <i>Einführung in die Zeitreihenanalyse.</i> Springer-Verlag, Berlin / Heidelberg 2006, ISBN 3-540-25628-8, S. 298f.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="Rainer_Schlittgen" title="Rainer Schlittgen">Rainer Schlittgen</a>, Bernd H. J. Streitberg: <i>Zeitreihenanalyse.</i> 9. Auflage. Oldenbourg Verlag, München/Wien 2001, ISBN 3-486-25725-0, S. 450 f.</span>

</li>

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