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STATGR5205 Midterm - Fall 2017 - October 18Name: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.If students violate these guidelines, they will receive a zero on this exam and potentiallyface more severe consequences. Students must include all relevant work in the handwrittenproblems to receive full credit.Problem 1 [65 pts]Consider the simple linear regression model(1)Yi=β0+β1xi+✏i,ı = 1, . . . , n,✏iiid⇠N(0,σ2),and least squares estimatorsˆβ1=SxySxxandˆβ0=¯Y-ˆβ1¯x.For this problem, you can use the following results:(2)E[ˆβ0] =β0,E[ˆβ1] =β1,V ar[ˆβ0] =σ2✓1n+¯x2Sxx◆,V ar[ˆβ1] =σ2Sxx.For this exercise, use the scalar form of the simple linear regression model, i.e.,don’t use matrices.Part A (5 pts)Under model (1), prove thatˆβ0-ˆβ1is an unbiased estimator ofβ0-β1. Note that you candirectly use the relations from (2).1BruceBanner
Part B (20 pts)Under model (1), derive an expression forCov(ˆβ0,ˆβ1), whereˆβ0andˆβ1are the least squaresestimators.Simplify the result as much as possible.Note that you can directly use therelations from (2).Notethat§,=§,KiYi,Ki=×ij,÷Notethatundermodel11),)=Cov(±¥,Yi,?IkiYi•
Part C (10 pts)Under model (1), derive an expression forV ar[ˆβ0-ˆβ1], whereˆβ0andˆβ1are the leastsquares estimators. Simplify the result as much as possible. Note that you can directly usethe relations from (2).Note:if you cannot complete Part B, then express the solution to Part C interms ofCov(ˆβ0,ˆβ1).3