Projection matrix: Difference between revisions

Revision as of 02:24, 9 December 2021

Revision as of 06:21, 9 December 2021

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=== Weighted and generalized least squares ===

{{further|Weighted least squares|Generalized least squares}}

The above may be generalized to the cases where the weights are not identical and/or the errors are correlated. Suppose that the [[covariance matrix]] of the errors is Ψ'''Σ'''. Then since

: <math>

\hat{\mathbf\beta}_{\text{GLS}}= \left( \mathbf{X}^\textsf{T} \mathbf{\PsiSigma}^{-1} \mathbf{X} \right)^{-1} \mathbf{X}^\textsf{T} \mathbf{\PsiSigma}^{-1}\mathbf{y}

</math>.

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: <math>

\mathbf{H} = \mathbf{X}\left( \mathbf{X}^\textsf{T} \mathbf{\PsiSigma}^{-1} \mathbf{X} \right)^{-1} \mathbf{X}^\textsf{T} \mathbf{\PsiSigma}^{-1}

</math>