Econometrics | Chapter 18 | SURE Models | Shalabh, IIT Kanpur - PDF document

Econometrics | CKaptr 18 | URE ModO | Shalabh, IIT Kanpur CKaptr 18 mLnJO UnrOatd RJrLon EquatLon ModO A aLc natur of tK muOtLpO rJrLon modO L tKat Lt dcrL tK KavLour of a partLcuOar tud varLaO ad on a t of [pOanator varLaO. WKn tK oM Econometrics | CKaptr 18 | URE ModO | Shalabh, IIT Kanpur aum tKat tK rror trm aocLatd ZLtK tK quatLon ma  contmporanouO corrOatd. TK quatLon ar apparntO or ³mLnJO´ unrOatd rJrLon ratKr tKan Lndpndnt rOatLonKLp. W conLdr Kr a modO comprLLnJ of muOtLpO rJrLon quatLon of tK form 12...;12...;12...titijijtii y xtTiM ZKr L tK orvatLon on tK dpndnt varLaO ZKLcK L to  [pOaLnd  tK rJrLon quatLon L tK orvatLon on [pOanator varLaO apparLnJ Ln tK quatLon L tK coffLcLnt aocLatd ZLtK tij at acK orvatLon and L tK vaOu of tK random rror componnt aocLatd ZLtK quatLon of tK modO. quatLon can  compactO [prd a 12...iiiiyXiM ZKr L vctor ZLtK Omnt XisTK matrL[ ZKo coOumn rprnt tK orvatLon on an [pOanator varLaO Ln tK quatLon; L a ik vctor ZLtK Omnt ; and L a vctor of dLturanc. TK quatLon can  furtKr [prd a 11112222MMMM §·§·§·§·¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸¨¸©¹©¹©¹©¹or ZKr tK ordr of1L TMXTMk L 1 L ii Econometrics | CKaptr 18 | URE ModO | Shalabh, IIT Kanpur Trat acK of tK quatLon a tK cOaLcaO rJrLon modO and maN convntLonaO aumptLon for 12... a L fL[d.rankXk OLmZKriiiiii X XQQL nonLnJuOar ZLtK fL[d and fLnLt Omnt. iiiiT uuI ZKr L tK varLanc of dLturanc Ln quatLon for acK orvatLon Ln tK ampO. ConLdrLnJ tK LntractLon tZn tK quatLon of tK modO Z aum OLmijij X ;12...ijijT uuIijMZKr L nonLnJuOar matrL[ ZLtK fL[d and fLnLt Omnt and L tK covarLanc tZn tK dLturanc of andthth quatLon for acK orvatLon Ln tK ampO. CompactO Z can ZrLt 1112121222TTMTTTMTMTMTMMTIIIIIIIII ZKr dnot tK .roncNr product oprator L M TMT matrL[ and isMMpoLtLv dfLnLt mmtrLc matrL[. TK dfLnLtn of avoLd tK poLLOLt of OLnar dpndncL amonJ tK contmporanou dLturanc Ln tK quatLon of tK modO. TK tructur E uuI LmpOL tKat varLanc of L contant for aOO contmporanou covarLanc tZn andtitj L contant for aOO LntrtmporaO covarLanc tZn andtitj ar ]ro for aOO Econometrics | CKaptr 18 | URE ModO | Shalabh, IIT Kanpur  uLnJ tK trmLnoOoJL ³contmporanou´ and ³LntrtmporaO´ covarLanc Z ar LmpOLcLtO aumLnJ tKat tK data ar avaLOaO Ln tLm rL form ut tKL L not rtrLctLv. TK ruOt can  ud for croctLon data aOo. TK contanc of tK contmporanou covarLanc acro ampO poLnt L a naturaO NdatLc dLturanc Ln a LnJO quatLon modO. ,t L cOar tKat tK quatLon ma appar to  not rOatd Ln tK n tKat tKr L no LmuOtanLt tZn tK varLaO Ln tK tm and acK quatLon Ka Lt oZn [pOanator varLaO to [pOaLn tK tud varLaO. TK quatLon ar rOatd tocKatLcaOO tKrouJK tK dLturanc ZKLcK ar rLaOO corrOatd acro tK quatLon of tK modO. TKat L ZK tKL tm L rfrrd to a URE modO. TK URE modO L a partLcuOar ca of LmuOtanou quatLon modO LnvoOvLnJ tructuraO quatLon ZLtK MoLntO dpndnt varLaO and foraOOkki dLtLnct [oJnou varLaO and Ln ZKLcK nLtKr currnt nor OoJJd ndoJnou varLaO appar a [pOanator varLaO Ln an of tK tructuraO quatLon. TK URE modO dLffr from tK muOtLvarLat rJrLon modO onO Ln tK n tKat Lt taN account of prLor LnformatLon concrnLnJ tK anc of crtaLn [pOanator varLaO from crtaLn quatLon of tK modO. ucK [cOuLon ar KLJKO raOLtLc Ln man conomLc LtuatLon. O/ and */ tLmatLon: TK URE modO L 0.yXEVI Aum tKat L NnoZn. TK O/ tLmator of L bXXXyFurtKr 000 .VbEbbXXXXXXTK JnraOL]d Oat quar */ tLmator of Econometrics | CKaptr 18 | URE ModO | Shalabh, IIT Kanpur ˆˆˆXXXy IXXIyXIXDfLn GXXXXXX tKn 0andZ fLnd tKat VbVGGLnc L poLtLv dfLnLt o L atOat poLtLv mLdfLnLt and o */E L Ln JnraO mor ffLcLnt tKan O/E for tLmatLnJ . ,n fact uLnJ tK ruOt tKat */E t OLnar unLad tLmator of  o Z can concOud tKat L tK t OLnar unLad tLmator Ln tKL ca aOo. FaLO JnraOL]d Oat quar tLmatLon: L unNnoZn tKn */E of cannot  ud. TKn can  tLmatd and rpOacd  M matrL[ WLtK ucK rpOacmnt Z otaLn a faLO JnraOL]d Oat quar F*/ tLmator of a FTTXSIXXSIyAum tKat L a nonLnJuOar matrL[ and L om tLmator of EtLmatLon of TKr ar tZo poLO Za to tLmat Econometrics | CKaptr 18 | URE ModO | Shalabh, IIT Kanpur 1. U of unrtrLctd rLduaO  tK totaO numr of dLtLnct [pOanator varLaO out of ...kkk varLaO Ln tK fuOO modO 0yXEVI  a orvatLon matrL[ of tK varLaO. RJr acK of tK tud varLaO on tK coOumn of and otaLn T rLduaO vctor 12...iiiyZZZZyiMZKr . IZZZZTKn otaLn ijijiZj y H y and contruct tK matrL[ accordLnJO. Lnc X L a umatrL[ of Z ZKr L a K OctLon matrL[. TKn Z iii XXZZZZXXZJand tKu iZjiiiZjjjiZjyHyXHX  Econometrics | CKaptr 18 | URE ModO | Shalabh, IIT Kanpur +nc ijiZjijZijijEsEHtrH TKu an unLad tLmator of L JLvn  s 2. U of rtrLctd rLduaO ,n tKL approacK to fLnd an tLmator of tK rLduaO otaLnd  taNLnJ Lnto account tK rtrLctLon on tK coffLcLnt ZKLcK dLtLnJuLK tK URE modO from tK muOtLvarLat rJrLon modO ar ud a foOOoZ. X L.. rJr acK quatLon 12...  O/ and otaLn tK rLduaO vctor iiiiiiuIXXXX A conLtnt tLmator of L otaLnd a ijijiXXjsuuyHHyZKr Xiiii jjjj IXXXX IXXXXULnJ a conLtnt tLmator of can  contructd. ,f Ln L rpOacd  XijiiijjjjitrHHTkktrXXXXXXXXtKn L an unLad tLmator of