AMS572 - Stony Brook



AMS570.01 Regression Quiz Spring, 2018Name: _____________________________ ID: ___________________________ Signature:__________________This is a close book exam. You are allowed one-piece two-sided 8x11 formula sheet. No cellphone or calculator or computer of any kind is allowed. Cheating will result in a grade of F. Please provide complete solutions for full credit. The exam goes 9:20-9:50am. Please turn in this cover page along with solutions. Good luck! (1)) Please prove that: Total sum of squares = Regression sum of squares + Error sum of squares; that is,SST = SSR + SSE(2) Please derive the method of moment estimators of the regression model parametersSolution:(1) SST=∑j=1NYj-Y2=∑j=1NYj-Yj+Yj-Y2=∑j=1N[Yj-Yj2+(Yj-Y)2+2Yj-YjYj-Y]SST=∑j=1NYj-Yj2+∑j=1N(Yj-Y)2+2∑j=1NYj-YjYj-Y=∑j=1NYj-Yj2+∑j=1N(Yj-Y)2=SSE+SSRSince we have,∑j=1NYj-YjYj-Y=∑j=1NYjYj-Yj-YYj-Yj=∑j=1NYjej-Y∑j=1Nej=0(2)(simple regression) yi=α+βxi+εi, where εi~N(0,σ2)Moment Conditions:(a) E[εi]=E[yi-α-βxi]=0(b) Exiεi=Exiyi-α-βxi=0(c) Eεi2=σ2Then we have, εi=yi-α-βxi(a) 1Ni=1N(yi-α-βxi)=0(b) 1Ni=1Nxi(yi-α-βxi)=0(c) 1Ni=1Nεi2=σ2By solving the three equations above, we can derive the MOMEs as shown below,α=y-βxβ=i=1N(xi-x)(yi-y)i=1N(xi-x)2=SXYSXXσ2=1Ni=1N[yi-α-βxi]2 ................
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