计量经济学（英文版）课件.ppt

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1、Chapter6MultipleLinearRegressionXi’AnInstituteofPost&TelecommunicationDeptofEconomic&ManagementProf.LongTwoExplanatoryVariablesyt=b1+b2xt2+b3xt3+etytxt2=b2xt3yt=b3xt‘saffectytseparatelyButleastsquaresestimationofb2nowdependsuponbothxt2andxt3.6.1Cor

2、relatedVariablesyt=outputxt2=capitalxt3=laborAlways5workerspermachine.Ifnumberofworkerspermachineisnevervaried,itbecomesimpossibletotellifthemachinesortheworkersareresponsibleforchangesinoutput.yt=b1+b2xt2+b3xt3+et6.2TheGeneralModelyt=b1+b2xt2+b3xt3+..

3、.+bKxtK+etTheparameterb1istheintercept(constant)term.The“variable”attachedtob1isxt1=1.Usually,thenumberofexplanatoryvariablesissaidtobeK-1(ignoringxt1=1),whilethenumberofparametersisK.(Namely:b1...bK).6.31.E(et)=02.var(et)=s23.cov(et,es)=0forts4.et~N(

4、0,s2)StatisticalPropertiesofet6.41.E(yt)=b1+b2xt2+...+bKxtK2.var(yt)=var(et)=s23.Cov(yt,ys)=Cov(et,es)=0forts4.yt~N(b1+b2xt2+...+bKxtK,s2)StatisticalPropertiesofyt6.5Assumptions1.yt=b1+b2xt2+...+bKxtK+et2.E(yt)=b1+b2xt2+...+bKxtK3.var(yt)=var(et)=s24.

5、cov(yt,ys)=cov(et,es)=0ts5.Thevaluesofxtkarenotrandom6.yt~N(b1+b2xt2+...+bKxtK,s2)6.6LeastSquaresEstimationyt=b1+b2xt2+b3xt3+etSºS(b1,b2,b3)=S(yt-b1-b2xt2-b3xt3)2t=1TDefine:yt=yt-y*xt2=xt2-x2*xt3=xt3-x3*6.7b1=y-b1-b2x2-b3x3b3=(Sytxt3)(Sxt2)-(Sytxt2)(S

6、xt3xt2)*******2(Sxt2)(Sxt3)-(Sxt2xt3)****222b2=(Sytxt2)(Sxt3)-(Sytxt3)(Sxt2xt3)*******2(Sxt2)(Sxt3)-(Sxt2xt3)****222LeastSquaresEstimators6.8DangersofExtrapolationStatisticalmodelsgenerallyaregoodonly“withintherelevantrange”.Thismeansthatextendingthemt

7、oextremedatavaluesoutsidetherangeoftheoriginaldataoftenleadstopoorandsometimesridiculousresults.Ifheightisnormallydistributedandthenormalrangesfromminusinfinitytoplusinfinity,pitythemanminusthreefeettall.6.9ErrorVarianceEstimations2^=n-Ket^2SUnbiasedes

8、timatoroftheerrorvariance:s2s2^(n-K)~n-KcTransformtoachi-squaredistribution:6.10Gauss-MarkovTheoremUndertheassumptionsofthemultipleregressionmodel,theordinaryleastsquaresestimatorshavethesmallestvarianceofalllinearandunbiasedestimators.