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Problem % (16 point<. ) Considkr thc Tcgreasion nuxkl Y = X8+ : AII : stochastic Ad i` such tat EcT") #0 Howerr thcre mnattix of suinbkz 2 ~uchthat E(Z'...

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Problem % (16 point<. ) Considkr thc Tcgreasion nuxkl Y = X8+ : AII : stochastic Ad i` such tat EcT") #0 Howerr thcre mnattix of suinbkz 2 ~uchthat E(Z':) Vn E(Z4+0 Thc ,ittension ofthe HatIir . iT k(Tithe mtkr ofolerations Ad kithe nutber of regreAOrs) wlcteas the: dimension of the mtrix Z is T x with 9 > k(D) (2 points) [s B from [cgrexsion of Y ou . consisteut for B:(2 point-) Rgte` the matrix 4 o tle matrix Z (ie F #anit [ regrexn Gch coltn of . on thc matrix 2) Exprexs thc f

Problem % (16 point<. ) Considkr thc Tcgreasion nuxkl Y = X8+ : AII : stochastic Ad i` such tat EcT") #0 Howerr thcre mnattix of suinbkz 2 ~uchthat E(Z':) Vn E(Z4+0 Thc ,ittension ofthe HatIir . iT k(Tithe mtkr ofolerations Ad kithe nutber of regreAOrs) wlcteas the: dimension of the mtrix Z is T x with 9 > k (D) (2 points) [s B from [cgrexsion of Y ou . consisteut for B: (2 point-) Rgte` the matrix 4 o tle matrix Z (ie F #anit [ regrexn Gch coltn of . on thc matrix 2) Exprexs thc fittevl sluer cotupctly i fuuction o uz (2 poiuts) Regr` the obtnvations } on the fittevl valut > fxom tlie previou ["gTtsion ( T utrix). Expre` compctk the "n" € tiuuor fuuction o42 Jd ! (Note: Jou could u Tery ~pxific idempotent Hatrix here). (3 poiuts) I the new etimator colsistent for B2 (3 poiuts) Asun k =% Docr thc for of tle &timantor simplify? poilta) Interpret all of Vour previous Fezults from # applicsl ~tand point . Wy ae the useful?



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$[\mathbf{M}]$ In parts $(\mathrm{a})-(\mathrm{d}),$ let $U$ be the matrix formed by normalizing each column of the matrix $A$ in Exercise $35 .$
a. Compute $U^{T} U$ and $U U^{T} .$ How do they differ?
b. Generate a random vector $\mathbf{y}$ in $\mathbb{R}^{8},$ and compute $\mathbf{p}=U U^{T} \mathbf{y}$ and $\mathbf{z}=\mathbf{y}-\mathbf{p} .$ Explain why $\mathbf{p}$ is in $\mathrm{Col} A$ Verify that $\mathbf{z}$ is orthogonal to $\mathbf{p} .$
c. Verify that $\mathbf{z}$ is orthogonal to each column of $U$
d. Notice that $\mathbf{y}=\mathbf{p}+\mathbf{z},$ with $\mathbf{p}$ in $\mathrm{Col} A .$ Explain why $\mathbf{z}$ is in $(\mathrm{Col} A)^{\perp} .$ (The significance of this decomposition of y will be explained in the next section.)


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