Consider the following model_y=Xβ+ ; ∼(0,σ2Ω)(10)whereXis ann×knon-singular matrix of regressors that are fixed in repeated sampling,βak×1 vector of unknown parameters, ann×1 vector error term, andΩann×npositive definitematrix.a) LetΩ=InwhereInis an identity matrix of ordern. Letˆβdenote the ordinary least squaresestimator ofβ. Consider any arbitrary linear estimator ̃β∗. Show thatˆβindeed attains theminimum sum of squares of residuals compared to any other ̃β∗.[40%]b) Let ̃βdenote the generalised least squares (GLS) estimator ofβ. There is no need to derive ̃β. Letˆβdenote the OLS estimator ofβin (10). Show that the covariance between ̃βandˆβ− ̃βis0.[60%

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WhereXis ann×knon-singular matrix of regressors that are fixed in repeated sampling,βak×1 vector of unknown parameters, ann×1 vector error term, andΩann×npositive definitematrix
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