Gegeben sei ein typisches multiples lineares Regressionsmodell
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{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}}
wahres Modell ), mit
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{\displaystyle {\boldsymbol {\beta }}}
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{\displaystyle (p\times 1)}
Regressionsparameter , der
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Versuchsplanmatrix
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{\displaystyle \mathbf {X} }
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{\displaystyle (n\times 1)}
abhängigen Variablen
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{\displaystyle \mathbf {y} }
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{\displaystyle (n\times 1)}
Störgrößen
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verallgemeinerte Kleinste-Quadrate-Schätzer (VKQ-Schätzer) kann auf unterschiedliche Art und Weise ausgedrückt werden. Jede dieser Ausdrucksweisen hat ihre eigene Interpretation. Eine bekannte Spezifikation ist die sogenannte zweistufige Kleinste-Quadrate-Schätzung , die von Henri Theil entwickelt wurde. Für die Herleitung des zweistufigen Kleinste-Quadrate-Schätzers lässt sich der verallgemeinerte Kleinste-Quadrate-Schätzer
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{\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}}
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{\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}=(\mathbf {Z} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Z} _{i})^{-1}\mathbf {Z} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}={\begin{pmatrix}\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}&\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {X} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}&\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {X} _{i}\\\\\end{pmatrix}}^{-1}\cdot {\begin{pmatrix}\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}\\\\\end{pmatrix}}}
Die reduzierte Form lautet
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{\displaystyle \mathbf {Y} =\mathbf {X} \mathbf {\Pi } +\mathbf {V} }
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{\displaystyle [\mathbf {y} _{i}\;\;\mathbf {Y} _{i}\;\;\mathbf {Y} _{i}^{*}]=\mathbf {X} [\pi _{i}\;\;\mathbf {\Pi } \;\;\mathbf {\Pi } _{i}^{*}]+[\mathbf {v} _{i}\;\;\mathbf {V} _{i}\;\;\mathbf {V} _{i}^{*}]}
wobei
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{\displaystyle (T\times 1)}
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{\displaystyle \mathbf {Y} _{i}^{*}}
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{\displaystyle \mathbf {Y} _{i}^{*}}
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{\displaystyle (T\times m_{i}^{*})}
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{\displaystyle [\pi _{i}\;\;\mathbf {\Pi } \;\;\mathbf {\Pi } _{i}^{*}]}
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{\displaystyle {\hat {\mathbf {\Pi } }}_{i}=(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}}
Prädiktionsmatrix
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{\displaystyle \mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}=\mathbf {X} {\hat {\mathbf {\Pi } }}_{i}={\hat {\mathbf {Y} }}_{i}=\mathbf {Y} _{i}-{\hat {\mathbf {V} }}_{i}}
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{\displaystyle (T\times (m_{i}-1))}
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{\displaystyle (\mathbf {X} ^{\top }\mathbf {X} )^{-1}(\mathbf {X} ^{\top }\mathbf {X} )=\mathbf {I} }
[ 1]
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{\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}={\begin{pmatrix}\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }(\mathbf {X} ^{\top }\mathbf {X} )^{-1}(\mathbf {X} ^{\top }\mathbf {X} )\mathbf {Y} _{i}&\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {X} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}&\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {X} _{i}\\\\\end{pmatrix}}^{-1}\cdot {\begin{pmatrix}\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}\\\\\end{pmatrix}}}
bzw.
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{\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}={\begin{pmatrix}{\hat {\mathbf {Y} }}_{i}^{\top }{\hat {\mathbf {Y} }}_{i}&{\hat {\mathbf {Y} }}_{i}^{\top }\mathbf {X} _{i}\\\\\mathbf {X} _{i}^{\top }{\hat {\mathbf {Y} }}_{i}&\mathbf {X} _{i}^{\top }\mathbf {X} _{i}\\\\\end{pmatrix}}^{-1}\cdot {\begin{pmatrix}{\hat {\mathbf {Y} }}_{i}^{\top }\mathbf {y} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {y} _{i}\\\\\end{pmatrix}}}
Wenn man
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{\displaystyle {\hat {\mathbf {Z} }}_{i}=[{\hat {\mathbf {Y} }}_{i}\;\;\mathbf {X} _{i}]}
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{\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}=({\hat {\mathbf {Z} }}_{i}^{\top }{\hat {\mathbf {Z} }}_{i})^{-1}{\hat {\mathbf {Z} }}_{i}^{\top }\mathbf {y} _{i}}
[ 2]
George G. Judge, R. Carter Hill, W. Griffiths, Helmut Lütkepohl , T. C. Lee. Introduction to the Theory and Practice of Econometrics. 2. Auflage. John Wiley & Sons, New York/ Chichester/ Brisbane/ Toronto/ Singapore 1988, ISBN 0-471-62414-4 .
↑ George G. Judge, R. Carter Hill, W. Griffiths, Helmut Lütkepohl , T. C. Lee. Introduction to the Theory and Practice of Econometrics. 2. Auflage. John Wiley & Sons, New York/ Chichester/ Brisbane/ Toronto/ Singapore 1988, ISBN 0-471-62414-4 , S. 645.
↑ George G. Judge, R. Carter Hill, W. Griffiths, Helmut Lütkepohl, T. C. Lee. Introduction to the Theory and Practice of Econometrics. 2. Auflage. John Wiley & Sons, New York/ Chichester/ Brisbane/ Toronto/ Singapore 1988, ISBN 0-471-62414-4 , S. 645.
![{\displaystyle [\mathbf {y} _{i}\;\;\mathbf {Y} _{i}\;\;\mathbf {Y} _{i}^{*}]=\mathbf {X} [\pi _{i}\;\;\mathbf {\Pi } \;\;\mathbf {\Pi } _{i}^{*}]+[\mathbf {v} _{i}\;\;\mathbf {V} _{i}\;\;\mathbf {V} _{i}^{*}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/523237bc09f7b88238b6b4569154f785bfcfa999)
![{\displaystyle [\pi _{i}\;\;\mathbf {\Pi } \;\;\mathbf {\Pi } _{i}^{*}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9002c68cc3ab13643ad34a095304865b59b04e)
![{\displaystyle {\hat {\mathbf {Z} }}_{i}=[{\hat {\mathbf {Y} }}_{i}\;\;\mathbf {X} _{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aae3456a7b7e9b22deb7a4a162de5b69e4dbbd1f)
