清华经管计量经济学习题2

ProblemSet2SuggestedSolutions1.Givenanrandomsample{(xi,yi)}Ni=1andasimplelinearregressionmodely=β0+β1x+u,wecalculatetheOLSestimators,denotedbyˆβ0andˆβ1.Nowsuppose˜x=axand˜y=by,andconsiderthefollowingregressionmode˜y=˜β0+˜β1˜x+ε.Letˆ˜β0andˆ˜β1betheOLSestimatorsof˜β0and˜β1calculatedfromthesample{(˜xi,˜yi)}Ni=1where˜xi=axiand˜yi=byi.Findˆ˜β0andˆ˜β1asfunctionsofˆβ0andˆβ1.SOLUTIONS:Firstnoticethat¯˜x=1NNXi=1˜xi(1)=1NNXi=1axi(2)=a·1NNXi=1xi(3)=a¯x.(4)Similarly,¯˜y=b¯y.(5)Therefore,ˆ˜β1=P˜xi−¯˜x
˜yi−¯˜y
P˜xi−¯˜xi
2(6)=P(axi−a¯x)(byi−b¯y)P(axi−a¯x)2(7)=Pab(xi−¯x)(yi−¯y)Pa2(xi−¯x)2(8)=abP(xi−¯x)(yi−¯y)a2P(xi−¯x)2(9)=ba·P(xi−¯x)(yi−¯y)P(xi−¯x)2(10)=baˆβ1.(11)Then,ˆ˜β0=¯˜y−ˆ˜β1¯˜x(12)=b¯y−baˆβ1·a¯x(13)=b¯y−ˆβ1¯x(14)=bˆβ0.(15)12.Wooldridge(4thEd)Ex2.2.ThefollowingtablecontainstheACTscoresandtheGPA(gradepointaverage)foreightcollegestudents.Gradepointaverageisbasedon...SOLUTIONS:23.Wooldridge(4thEd)Ex2.5.Considerthesavingfunctionsav=β0+β1inc+u,u=√inc·ewhere...SOLUTIONS:4.Wooldridge(4thEd)Ex2.6.Letˆβ0andˆβ1betheOLSinterceptandslopeestimators,respec-tively,andlet¯ubethesampleaverageoftheerrors(nottheresiduals!)...SOLUTIONS:35.Wooldridge(4thEd)Ex2.8(ii)-(iv).(ii)Now,let˜β0and˜β1befromtheregressionof(c1+yi)and(c2+xi)(withnorestrictiononc1orc2).Showthat...SOLUTIONS:45