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  "body": "\n## Definition\n\nLet $X$ be a fixed real $n \\times p$ matrix of rank $p$, and let $y = X\\beta + \\varepsilon$ with $\\beta \\in \\R^p$ unknown.\nAssume $E[\\varepsilon] = 0$ and $\\operatorname{Cov}(\\varepsilon) = \\sigma^2 \\Omega$, where $\\sigma^2 \u003e 0$ and $\\Omega$ is a known symmetric [[positive definite|0KX66CZMBB]] $n \\times n$ matrix.\nThe *generalized least squares estimator* of $\\beta$ is the minimiser of\n$$(y - X b)^T \\Omega^{-1} (y - X b)$$\nover $b \\in \\R^p$, which is\n$$\\hat\\beta_{\\mathrm{GLS}} = (X^T \\Omega^{-1} X)^{-1} X^T \\Omega^{-1} y.$$\n\n## Remarks\n\nEquivalent form: let $P$ be an invertible matrix with $P^T P = \\Omega^{-1}$, for example $P = \\Omega^{-1/2}$.\nThen $P y = P X \\beta + P \\varepsilon$ with $\\operatorname{Cov}(P\\varepsilon) = \\sigma^2 I_n$.\nThe estimator $\\hat\\beta_{\\mathrm{GLS}}$ is the [[ordinary least squares|YM708KHV2W]] estimator of this transformed model.\nBy the [[Gauss–Markov theorem|N3YMNMWANM]] applied to the transformed model, it is the best linear unbiased estimator; this is Aitken's theorem.\nIts covariance matrix is $\\sigma^2 (X^T \\Omega^{-1} X)^{-1}$.\n\nExample: if $\\Omega = I_n$, the estimator reduces to ordinary least squares.\nIf $\\Omega$ is diagonal, it is the weighted least squares estimator with weights $1/\\Omega_{ii}$.\n\nNon-example: if $\\Omega$ is only positive semidefinite and singular, then $\\Omega^{-1}$ does not exist and the formula is not defined.\n\nWhen $\\Omega$ is unknown and is replaced by an estimate, the method is called *feasible* generalized least squares.\n",
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