5205_4205_Midterm_Fall_2018_KEY.pdf - STATGR5205 Midterm...

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STATGR5205 Midterm - Fall 2018 - October 17Name:UNI:The GU5205 midterm is closed notes and closed book. Calculators are allowed. Tablets,phones, computers and other equivalent forms of technology are strictly prohibited. Studentsare not allowed to communicate with anyone with the exception of the TA and the professor.Students must include all relevant work in the handwritten problems to receive full credit.The exam time is 75 minutes and once the exam time has expired, please do not discuss themidterm until after you leave the room (or log oof Zoom).By signing below, you are acknowledging that if you do not follow the guidelines, thenyou will receive a score of zero on the midterm and potentially face more severe conse-quences.Signature:Date:ProblemPointsStudent’s Score1102203104.i154.ii155.i105.ii20Total1001PeterParker
Problem 1 [10 pts]Consider thesimple linear regressionandregression through the originmodelsdefined respectively by(1)Yi=β0+β1xi+i,i= 1,2, . . . , n,iiidN(0,σ2),and(2)Yi=β1xi+i,i= 1,2, . . . , n,iiidN(0,σ2).Assume that the true data generating process comes from the simple linear regression modelmodel (1). However, suppose that you estimateβ1using the regression through the originleast squares estimatorˆβ1=Pxiyi/Px2i. Compute the bias ofˆβ1., i.e., computeE[ˆβ1]-β1.2
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Problem 2 [20 pts]Consider the least squares estimatedsimplelinear regression model(3)ˆYi=ˆβ0+ˆβ1xi=nXj=1hijYj,wherehijare the hat-values.Consider vectors1=(11· · ·1)T,x=(x1x2· · ·xn)Tande=(e1e2· · ·en)T.Note thateis the vector of residuals based on the least squares estimated modelˆYidefinedin (3). Using properties of the hat-values, prove that any vector in the span of1andxisorthogonal to the residual vectore.Note:You are not allowed to use the hat-matrix in this computation. The results must beproved using the scalar form.P

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