MIT-SCIENCE-Lectures-ps5(9)

5thproblemset,

Supposeε1,...,εn～i.i.d.N(0,σ2),andfori=1,...,nwehaveYi=β0+β1xi+εi.(Capital“Y”andlower-case“x”;theformerisrandom;thelatterisconstant.)Thismodelcanbewrittenas

Y=Xβ+ε,ε～Nn(0,σ2In)

oras

Y～Nn(Xβ,σ2In).

1.FindtheentriesinthematrixX.

2.LetH=X(X X) 1X .ShowthatifuisinthecolumnspaceofX,thenHu=u,andifuisorthogonal(i.e.,atrightangles)tothecolumnspaceofX,thenHu=0.AHintisinafootnote1.

3.Fillintheblanks:

(X X) 1X Y～N?(?,?).

Whatis(X X) 1X Yanunbiasedestimatorof?(XandYare“observable.”)

1oftheslopeβetheresultof#3to ndan unbiasedestimatorβ

estimatorintheform

1=[somerowvector(specifyandsimplify!)]Y.β

Fillintheblanks: 1～N?(?,?). β

etheresultof#4to nd a90%con denceintervalfortheslopeβ1,assuming(unrealistically)thatσisknown.I.e.,wewant

Pr(somestatistic<β1<somestatistic)=0.9.

6.Firstsomenotation:

ε =Y Xβ=(unobservable)vectorof“errors,”

=(observable)vectorof“residuals.”ε =Y Xβ

Showthatε =(I H)etheresult,andtheidentity

cov(AU,BV)=A(cov(U,V))B (A,Bareconstant)

1).Fromtheresult,andthejointnormalityofthesetworandomto ndcov(ε, β

variables,drawaconclusionaboutthenaturedependencebetweenthem.

Continued →

1AvectoruisinthecolumnspaceofXifandonlyifu=Xaforsomevectora.

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