OLS_Variance_Unbiased_Estimate.pdf

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1、UnivariateOLS:UnbiasedEstimatorofErrorVarianceConsidertheunivariatelinearregressionmodelyt=+xt+ut;t=1;:::;T:wheretheregressorsarenon-stochastic(fixed)andthedisturbanceshavezeromeanandareuncorrelatedandhomoscedasticwithvarianceequalto2,i.e.,E[u]=0;tC[u;u]=0,t6=s,andV[u]=E[u2]=2:Wewillshowthattstt1X

2、T22s=u^tT2t=1isanunbiasedestimatorofthedisturbancevariance2;whereu^=yy^,y^=^+x^;tttttand^and^aretheOLSestimatorsofand,respectively.TheOLSestimates^and^aregivenbyPT^=t=1P(yty)(xtx);^=y^x;T2t=1(xtx)whichcanalsobeexpressedasXTXT^=wtyt;^=qtyt;t=1t=1wherextx1wt=PT2;qt=x
wt:t=1(xtx)TThe

3、followingpropertiesareeasilyverified:XTXTXTXTwt=0;wtxt=1;qt=1;qtxt=0;t=1t=1t=1t=1andXTXT1PT2XT212Tt=1xtxwt=PT2;qt=PT2;qtwt=PT2:t=1t=1(xtx)t=1t=1(xtx)t=1t=1(xtx)1Usingtheseproperties,weobservethatXTXTXT^=wtyt=wt(+xt+ut)=+wtut;t=1t=1t=1andXTXTXT^=qtyt=qt(+xt+ut)=+qtut:t=1t=1t=1Notethattheresi

4、dualscanbeexpressedasu^t=yty^t=yt^+x^t=(+xt+ut)^+x^t;orequivalentlyhiu^t=(^)+^xt+ut:Hence,hi2hiu^2=(^)+^x+u22(^)+^xutttttwhere!!XTXT(^)+^xt=qsus+wsusxt:s=1s=1ItfollowsthathhiiE(^)+^xtut""!!##XTXT=Eqsus+wsusxtuts=1s=12=(qt+wtxt);andsoXThhiiXTE(^)+^xu=(q+wx)2=(1+1

5、)2=22:tttttt=1t=12Moreover,hi2E(^)+^xt2"!!#23XTXT=E4qsus+wsusxt5s=1s=12!232!23"!!#XTXTXTXT=E4qu5+E4wu5
x2+2
Equwu
xsssstssssts=1s=1s=1s=1"!!!#XTXTXT2222=qs+ws
xt+2
qsws
xts=1s=1s=1"P#1Tx2x2x
x=2Tt=1t+t2t;PT2PT2PT2t=1(xtx)t=1(xtx)t=1(xtx)andsoXThi2E(^)+^xtt=1"PP#Tx2Tx2T
x2=2

6、t=1t+t=1t2;PT2PT2PT2t=1(xtx)t=1(xtx)t=1(xtx)"P#1Tx2x2=22Tt=1t1PT2Tt=1(xtx)2=2:PT22Finally,t=1E[ut]=TandsoXT22222Eu^t=2
+T
2
2
=(T2)
t=1whichyields"#1XT22Eu^t=:T2t=13