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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Wahres Modell</span></h1>
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<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>In der <a href="Statistik" title="Statistik">Statistik</a> ist das zugrundeliegende <b>wahre Modell</b> das eigentliche Modell in der <a href="Grundgesamtheit" title="Grundgesamtheit">Grundgesamtheit</a>, welches die <a href="Einflussgr%C3%B6%C3%9Fe_und_Zielgr%C3%B6%C3%9Fe" class="mw-redirect" title="Einflussgröße und Zielgröße">Antwortvariable</a> und die relevanten unabhängigen Variablen in Beziehung zueinander setzt. Diese Beziehung wird durch eine additive <a href="St%C3%B6rgr%C3%B6%C3%9Fe_und_Residuum" title="Störgröße und Residuum">Störgröße</a> überlagert, für die angenommen wird, dass sie einen <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> von Null aufweist.<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><p>Die grundlegende Annahme des Modells ist, dass es <a href="Linearit%C3%A4t#Allgemeine_Definition" title="Linearität">linear</a> in den <a href="Parameter_(Statistik)" title="Parameter (Statistik)">Parametern</a> ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Multiple_lineare_Regression">Multiple lineare Regression</h2></div>
<p>Gegeben sei das folgende <a href="Multiple_lineare_Regression" title="Multiple lineare Regression">multiple lineare Regressionsmodell</a>:
</p>
<table class="centered" style="clear:both; margin-left: 1.5em; border-collapse:collapse;">

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<td rowspan="2" style="white-space: nowrap;"><div style="margin:0;"><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 y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}+\varepsilon _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i},\quad \operatorname {E} (\varepsilon _{i})=0}">
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<annotation encoding="application/x-tex">{\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}+\varepsilon _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i},\quad \operatorname {E} (\varepsilon _{i})=0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2e61767f296c38ae023a58f4440917b5be0a918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.419ex; height:3.176ex;" alt="{\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}+\varepsilon _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i},\quad \operatorname {E} (\varepsilon _{i})=0}" loading="lazy"></span>&nbsp;</div>
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<td><div style="font-size:1pt; margin:0;">&nbsp;</div>
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<td rowspan="2" style="white-space: nowrap; text-align:right;"><div style="margin:0;">&nbsp;<span style="">(1)</span></div>
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<td style="border-top:3px dotted #D5D5D5; width:98%;"><div style="font-size:1pt; margin:0;">&nbsp;</div>
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<p>Hierbei ist <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 k+1=:p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>+</mo>
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<mi>p</mi>
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<annotation encoding="application/x-tex">{\displaystyle k+1=:p}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/616a5f1e786f0b1c18873813ebd54437a5e85a5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.129ex; height:2.509ex;" alt="{\displaystyle k+1=:p}" loading="lazy"></span> die Anzahl der zu schätzenden unbekannten (<a href="Wahrer_Wert" title="Wahrer Wert">wahren</a>) Parameter <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 \beta _{0},\beta _{1},\beta _{2},\dotsc ,\beta _{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>β<!-- β --></mi>
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<mn>0</mn>
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<mi>β<!-- β --></mi>
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</msub>
<mo>,</mo>
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<mo>,</mo>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \beta _{0},\beta _{1},\beta _{2},\dotsc ,\beta _{k}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad1f10daa93c46ec15d24994c647de521cfee5e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.761ex; height:2.509ex;" alt="{\displaystyle \beta _{0},\beta _{1},\beta _{2},\dotsc ,\beta _{k}}" loading="lazy"></span>. Die <a href="Regressionsparameter" title="Regressionsparameter">Regressionsparameter</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 \beta _{0},\beta _{1},\beta _{2},\dotsc ,\beta _{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<mo>,</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>,</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
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<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{0},\beta _{1},\beta _{2},\dotsc ,\beta _{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad1f10daa93c46ec15d24994c647de521cfee5e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.761ex; height:2.509ex;" alt="{\displaystyle \beta _{0},\beta _{1},\beta _{2},\dotsc ,\beta _{k}}" loading="lazy"></span> sind unbekannte, konstante Parameter des Interesses (sie gilt es zu schätzen) 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 \varepsilon _{i}}">
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<mstyle displaystyle="true" scriptlevel="0">
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<mi>ε<!-- ε --></mi>
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<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00e1b6ad3cbad4af49bf21a3ad2dc379ff045079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{i}}" loading="lazy"></span> ist eine unbeobachtbare Zufallsvariable, die <a href="St%C3%B6rgr%C3%B6%C3%9Fe_und_Residuum" title="Störgröße und Residuum">Störgröße</a> oder Fehlerterm genannt wird. Selbst wenn man die wahre <a href="Stichproben-Regressionsfunktion" title="Stichproben-Regressionsfunktion">Regressionsfunktion der Grundgesamtheit</a> kennen würde, dann würde sich der beobachtete Wert der Zielgröße <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 y_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>y</mi>
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<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"></span> immer noch vom vorhergesagten 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 {\hat {y}}_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<mi>y</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}_{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5058f005e41a46a3c8d75449e20bac02bfe1609e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.102ex; height:2.676ex;" alt="{\displaystyle {\hat {y}}_{i}}" loading="lazy"></span> durch ein gewisses Ausmaß unterscheiden, was der Störgröße entspricht.
</p><p>Formal handelt es sich bei der obigen Gleichung um das Modell in der <i>Grundgesamtheit</i> bzw. das <i>Populationsmodell</i>. Dies wird manchmal <i>wahres Modell</i> genannt, da mit der Annahme eines wahren Modells sichergestellt wird, dass man ein Modell schätzt, was sich von (1) unterscheidet.<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><p>Man könnte beispielsweise redundante unabhängige Variablen hinzufügen. Allerdings muss das Einbeziehen von redundanten unabhängigen Variablen nicht immer ein <a href="Spezifikation_(Statistik)" title="Spezifikation (Statistik)">Spezifikationsfehler</a> darstellen (von einem Spezifikationsfehler spricht man, wenn die Annahme, dass der Erwartungswert der Störgröße gleich Null ist verletzt ist). Beispielsweise könnte das zugrundeliegende wahre Modell gegeben sein 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 y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\varepsilon _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo>=</mo>
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<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mi>β<!-- β --></mi>
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<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\varepsilon _{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/934a67b9b75d6d6305e285606b183e5d031bb222.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.454ex; height:2.509ex;" alt="{\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\varepsilon _{i}}" loading="lazy"></span>. Das gewählte (spezifizierte) Modell (mit der irrelevanten unabhängigen Variablen <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_{i3}}">
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<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle x_{i3}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ced2c970fc3dca6c8c0d8bc8e3850685f59e4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.951ex; height:2.009ex;" alt="{\displaystyle x_{i3}}" loading="lazy"></span>) könnte folgendes Modell sein: <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 y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+x_{i3}\beta _{3}+\varepsilon _{i}^{*}}">
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<mn>1</mn>
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</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mn>3</mn>
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</msub>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msub>
<mo>+</mo>
<msubsup>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+x_{i3}\beta _{3}+\varepsilon _{i}^{*}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11451643f2cf9c1929fcaff0f9567f4a22460a9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.871ex; height:2.843ex;" alt="{\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+x_{i3}\beta _{3}+\varepsilon _{i}^{*}}" loading="lazy"></span>. Dass die Variable als irrelevant angenommen wird bedeutet, dass der wahre Wert 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 \beta _{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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<annotation encoding="application/x-tex">{\displaystyle \beta _{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b1001bcd6968314a965726f5dd193d1b11ada59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{3}}" loading="lazy"></span> gleich Null ist (<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 \beta _{3}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{3}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6e09addeff8ad7092fc6f4ce9b13020a2f4d34b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.631ex; height:2.509ex;" alt="{\displaystyle \beta _{3}=0}" loading="lazy"></span>). Aus diesem Grund gilt: <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} (\varepsilon _{i}^{*})=\operatorname {E} (\varepsilon _{i}-x_{i3}\beta _{3})=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msubsup>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
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</msubsup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mn>3</mn>
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<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\varepsilon _{i}^{*})=\operatorname {E} (\varepsilon _{i}-x_{i3}\beta _{3})=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2d2df35d432773764ba7fae23eb012b53e75cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.326ex; height:3.009ex;" alt="{\displaystyle \operatorname {E} (\varepsilon _{i}^{*})=\operatorname {E} (\varepsilon _{i}-x_{i3}\beta _{3})=0}" loading="lazy"></span> . In diesem Fall sind die <a href="Methode_der_kleinsten_Quadrate" title="Methode der kleinsten Quadrate">KQ-Schätzer</a> immer noch <a href="Erwartungstreu" class="mw-redirect" title="Erwartungstreu">erwartungstreu</a> für die wahren Werte und es liegt kein Spezifikationsfehler vor.
</p>
<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"><a href="Jeffrey_Wooldridge" title="Jeffrey Wooldridge">Jeffrey Marc Wooldridge</a>: <i>Introductory econometrics: A modern approach.</i> 5. Auflage. Nelson Education, 2015, S.&nbsp;859.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Jeffrey Marc Wooldridge: <i>Introductory econometrics: A modern approach.</i> 5. Auflage. Nelson Education, 2015, S.&nbsp;83.</span>
</li>
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