A Bartlett correction factor for tests on the cointegrating relations
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(2) All rights reserved. No part of this paper may be reproduced in any form without permission of the author.. c °1999 S¿ren Johansen Printed in Italy in March 1999 European University Institute Badia Fiesolana I-50016 San Domenico (FI) Italy.
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(4) 4. Lqwurgxfwlrq. Lq wklv sdshu zh ghulyh d Eduwohww fruuhfwlrq iru wkh whvw rq wkh frlqwhjudwlqj uhodwlrqv lq wkh yhfwru dxwruhjuhvvlyh prgho iru wkh ?glphqvlrqdo surfhvv f| jlyhq e| & [ 3. {f| ' kEq f| . . 3. n 4 (| n . K {f| n x_| n 0| c | ' c c A. +4,. 3. '. zkhuh 0| duh l1l1g1 ?Efc l c wkh lqlwldo ydoxhv duh {hg dqg _| E?_ dqg (| E?( duh ghwhuplqlvwlf whupv/ olnh frqvwdqw/ olqhdu whup hwf1 Wkh pdwulfhv k dqg q duh E? o dqg wkh pdwulfhv K c c K& c duh E? ?c x E? ?_ dqg 4 E?( Iru wkh dqdo|vlv ri wklv sdshu zh dvvxph wkdw f| lv UE D w|slfdo h{dpsoh ri wkh ghwhuplqlvwlf whupv lv wr ohw _| ' c dqg (| ' |c fruuhvsrqglqj wr kdylqj d olqhdu wuhqg lq wkh surfhvv dqg wuhqg vwdwlrqdu| frpelqdwlrqv q f| Wkh irupxodwlrq ehorz dovr doorzv/ zlwk plqru prgl fdwlrqv/ wkh srvvlelolw| wkdw ?( dqg ?_ duh }hur1 3. . Wkh olnholkrrg udwlr whvw iru k|srwkhvhv rq q kdv ehhq wuhdwhg lq Mrkdqvhq dqg Mxvholxv +4<<3, dqg Dkq dqg Uhlqvho +4<<3,/ dqg lw lv nqrzq wkdw lw lv dv|pswrwlfdoo| 2 glvwulexwhg/ ghvslwh wkh idfw wkdw wkh dv|pswrwlf glvwulexwlrq ri wkh hvwlpdwru lv pl{hg Jdxvvldq1 Wkh qlwh vdpsoh glvwulexwlrq/ krzhyhu/ lv qrw dozd|v zhoo dssur{lpdwhg e| wkh dv|pswrwlf glvwulexwlrq/ vhh iru lqvwdqfh Idfklq +4<<:,/ Mdfrevhq dqg Juhghqkr +4<<;,/ dqg Mdfrevrq/ Yuhglq/ dqg Zduqh +4<<;,/ Kdxj +4<<;,/ wr phqwlrq d ihz ri wkh pdq| vwxglhv ri wkh qlwh vdpsoh surshuwlhv ri wkh whvw ri uhvwulfwlrqv rq q Zh ghulyh khuh d fruuhfwlrq whup wr wkh olnholkrrg udwlr whvw vwdwlvwlf iru k|srwkhvhv rq q zlwk wkh sxusrvh ri lpsurylqj wkh dssur{lpdwlrq wr wkh dv|ps0 wrwlf 2 glvwulexwlrq1 Wkh fruuhfwlrq lv wkh vr0fdoohg Eduwohww fruuhfwlrq/ vhh Eduwohww +4<6:,1 Iru d uhfhqw vxuyh| ri wkh wkhru| ri wklv w|sh ri fruuhfwlrq vhh Fuledur0Qhwr dqg Frughlur +4<<9,1 Eulh | wkh phwkrg frqvlvwv ri fdofxodwlqj wkh h{shfwdwlrq ri wkh olnholkrrg udwlr +OU, vwdwlvwlf lq wkh irup 2uJ}u- iru d jlyhq sdudphwhu srlqw w xqghu wkh qxoo k|srwkhvlv1 Xvxdoo| lw lv qrw srvvleoh wr gr wklv h{solflwo| dqg rqh fdq lqvwhdg qg dq dssur{lpdwlrq ri wkh irup .d2 *L} u-o E n. Ew c A. zkhuh lv htxdo wr wkh ghjuhhv ri iuhhgrp iru wkh whvw dqg Ew c vkrzv krz wkh uhpdlqlqj sdudphwhuv xqghu wkh qxoo k|srwkhvlv glvwruwv wkh phdq dqg khqfh 4.
(5) wkh glvwulexwlrq ri wkh whvw vwdwlvwlf1 Wkh lghd lv wkdw wkh txdqwlw| 2Orju n AEw . kdv d glvwulexwlrq wkdw lv forvhu wr wkh olplw glvwulexwlrq/ vhh Odzoh| +4<89, iru d surri ri wklv vwdwhphqw xqghu fodvvlfdo l1l1g1 dvvxpswlrqv/ wkdw duh qrw vdwlv hg lq wkh UE prgho1 Wkh prgho +4, lv fkdudfwhul}hg e| glphqvlrq +? / frlqwhjudwlqj udqn Eo / odj ohqjwk E& c wkh qxpehu ri ghwhuplqlvwlf whupv uhvwulfwhg wr wkh frlqwhjudwlqj vsdfh E?( c wkh qxpehu ri xquhvwulfwhg whupv E?_ dqg qdoo| ri frxuvh wkh ydoxh ri doo wkh sdudphwhuv dqg wkh vdpsoh vl}h EA Wkh pdlq uhvxow suhvhqwhg lq Vhfwlrq 7 lv wkdw wkh Eduwohww fruuhfwlrq lv d ixqfwlrq ri wkh sdudphwhuv wkurxjk rqo| wzr ixqfwlrqv/ dqg ydulrxv frpelqd0 wlrqv ri wkh deryh fkdudfwhulvwlf qxpehuv1 Zh qg iru lqvwdqfh iru wkh whvw wkdw q ' Mc EME? r wkdw ' oE? r c dqg iru 6 ' ? n r o n n 2?( Ew ' d 6 n ?_ n &?o n dE2E? o n 6 Ek n 2ESEk n S_ Ek o 2 o Wkh frh!flhqwv Ek c SEk c dqg S_ Ek duh jlyhq lq Wkhruhp 8 ehorz1 Wklv uhvxow lpsolhv wkdw rqh fdq vhh iru zklfk frpelqdwlrqv ri wkh sdudphwhuv wkh xvxdo 2 dssur{lpdwlrq euhdnv grzq/ dqg pruh frqvwuxfwlyho| zkhq lw lv xvh0 ixo1 Lq ehwzhhq wkhuh lv dq duhd zkhuh wkh Eduwohww fruuhfwlrq fdq vhuyh dv dq lpsuryhphqw wr wkh xvxdo dv|pswrwlf uhvxowv1 Wkh sodq ri wkh sdshu lv uvw wr hvwdeolvk lq Vhfwlrq 5/ wkdw d qxpehu ri k|srwkhvhv fdq eh jlyhq d jhqhudo irupxodwlrq dv whvwv lq d uhgxfhg udqn uhjuhvvlrq prgho1 Lq Vhfwlrq dq h{sdqvlrq lv jlyhq ri wkh hvwlpdwruv ri wklv uhgxfhg udqn uhjuhvvlrq/ dqg wkhq dq h{sdqvlrq lv jlyhq ri wkh orj olnholkrrg udwlr whvw vwdwlvwlf1 Lq Vhfwlrq 7 wkh pdlq uhvxow rq wkh Eduwohww fruuhfwlrqv duh jlyhq dqg wkh uhvxowv vshfldol}hg wr wkh prghov glvfxvvhg lq Vhfwlrq 5/ dqg qdoo| lq Vhfwlrq 8 vrph vlpxodwlrq h{shulphqwv duh frqgxfwhg zklfk vkrz wkdw wkh Eduwohww fruuhfwlrq lv d xvhixo dgglwlrq wr wkh xvxdo dv|pswrwlf dqdo|vlv1 Wkh yhu| orqj dqg whglrxv surriv duh jlyhq lq dq Dsshqgl{1. 5. Wkh prghov dqg wkh k|srwkhvhv. Zh gh qh lq wklv vhfwlrq wkuhh prghov e| uhvwulfwlrqv rq wkh frlqwhjudwlqj uh0 odwlrqv1 Doo prghov fdq eh dqdo|vhg e| uhgxfhg udqn uhjuhvvlrq/ vhh Mrkdqvhq 5.
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(7) Eo ' o n o2 / fruuhvsrqglqj wr suhvshfl hg frh!flhqwv q f dqg 4f lq vrph ri wkh frlqwhjudwlqj uhodwlrqv1 Wkh olnholkrrg udwlr whvw ri P2 lq Pf c vdwlv hg . 2 *L} u-EP2 mPf $ 2 Eo E? n ?( o Lw zrxog dovr eh uhohydqw wr irupxodwh khuh wkh uhvwulfwlrq wkdw rqo| q zdv sduwo| nqrzq1 Wklv prgho/ krzhyhu/ fdq qrw eh hvwlpdwhg e| uhgxfhg udqn uhjuhvvlrq dqg wkh dqdo|vlv jlyhq ehorz zrxog kdyh wr eh prgl hg1 Lq wkh iroorzlqj vhfwlrqv zh ghulyh d fruuhfwlrq idfwru iru whvw ri d vlpsoh k|srwkhvlv rq q dqg 4 lq hdfk ri wkh prghov Pf c P dqg P2 c dqg dsso| wkhvh wr ghulyh d fruuhfwlrq idfwru iru wkh whvw ri P lq Pf dqg P2 lq Pf xvlqj wkh iroorzlqj wulfn= Wr whvw P lq Pf c vd|/ zh wdnh d sdudphwhuv q f ' M f dqg 4f fruuh0 vsrqglqj wr d sdudphwhu srlqw lq P Zh gh qh wkh frqfhqwudwhg olnholkrrg ixqfwlrq u+qc 4 c dqg qg wkh olnholkrrg udwlr whvw 4@ q'Mc4 uEqc 4 4@ qc4 uEqc 4 4@ q'q f c4'4f uEqc 4 4@ q'q f c4'4f uEqc 4 * ' 4@ qc4 uEqc 4 4@ q 'Mc4 uEqc 4 . u-EP mPf '. ' u-Eq ' q f c 4 ' 4f mPf *u-Eq ' q f c 4 ' 4f mP c vxfk wkdw 2 *L} u-EP mPf ' 2 *L} u-Eq ' q f c 4 ' 4f mPf n 2 *L} u-Eq ' q f c 4 ' 4f mP Khqfh zh vhh wkdw wkh fruuhfwlrq iru wkh whvw zh duh uhdoo| lqwhuhvwhg lq/ qdpho| P lq Pf c fdq eh irxqg dv wkh gl huhqfh ri wkh fruuhfwlrqv wr wzr whvwv ri vlpsoh k|srwkhvhv rq q dqg 4 lq Pf dqg P2 1 Wkxv/ li zh fdq qg d jhqhudo uhvxow zklfk doorzv xv wr ghulyh d fruuhfwlrq iru d vlpsoh k|srwkhvlv rq q dqg 4 lq wkhvh ydulrxv prghov/ wkhq zh fdq ghulyh wkh fruuhfwlrqv e| vxewudfwlrq1. 514. Wkh ghwhuplqlvwlf whupv. Wkh fruuhfwlrq zloo ghshqg rq wkh ghwhuplqlvwlf whupv dqg lq rughu wr jhw uhd0 vrqdeo| vlpsoh h{suhvvlrqv zh dvvxph wkdw wkh| vdwlvi| wkh uhodwlrq _|n ' _| c ' c c fc c 7. +5,.
(8) iru vrph pdwul{ zlwk wkh surshuw| wkdw me}E m ' . +6,. {(| ' g _| c. +7,. Ixuwkhu zh dvvxph wkdw . iru vrph E?_ ?( pdwul{ g Ilqdoo| zh dvvxph wkdw E(| c _| A|' duh olqhduo| lqghshqghqw1 Wkxv zh doorz iru lqvwdqfh _| ' Ec |c |2 dqg (| ' |
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(16) Lw iroorzv wkdw .E{f| ' x_| n Eu Ek4 g n x{ _| S ' d x n E 'f Ek4 g n xE o_| ' g{ _| c . ". . . . 3. 3. 3 3. . vd|1 Wdnlqj h{shfwdwlrqv lq +4, zh qg & [ 3. g{ _| ' k.Eq f| . . . 3. n 4 (| n . '. K g{ _| n x_| c . 3. zklfk vkrzv wkh uhvxow iru .Eq f| n 4 (| 1 Qrwh wkdw wkh uhvxow wkdw jurzv dw prvw dv d sro|qrpldo lq c vhh Ohppd 43c vkrzv wkdw wkh vxpv duh frqyhujhqw/ vlqfh duh h{srqhqwldoo| ghfuhdvlqj1 . . 3. Zh qh{w vkrz krz wkh vlpsoh k|srwkhvhv rq q dqg 4 lq Pf c P c dqg P2 c jlyh ulvh wr uhjuhvvlrq htxdwlrqv zklfk fdq eh jlyhq wkh vdph irupxodwlrq1 Wklv doorzv xv wr ghulyh doo wkh uhvxowv iurp rqh jhqhudo uhgxfhg udqn uhjuhvvlrq htxdwlrq1. 515. D vlpsoh k|srwkhvlv rq q dqg 4 lq P3. Wkh prgho htxdwlrq lv jlyhq e| +4, dqg zh frqvlghu wkh k|srwkhvlv= q ' q f c 4 ' 4f c vxfk wkdw & [ 3. f . {f| ' kEq f|. . 3. f . n 4 (| n. K {f| n x_| n 0| c 3. '. zklfk lv hdvlo| hvwlpdwhg e| uhjuhvvlrq ri {f| rq q f f| n 4f (| c odjjhg gl hu0 hqfhv dqg _| . . Lw lv frqyhqlhqw iru wkh fdofxodwlrqv wr uhsdudphwul}h wkh prgho gh qlqj qhz sdudphwhuv dqg uhjuhvvruv zklfk lqyroyh wkh wuxh ydoxh1 Lq wkh iroorzlqj vxevhfwlrqv zh wkhuhiruh qhhg d qrwdwlrq iru wkh wuxh ydoxh ri wkh sdudphwhuv/ dv zhoo dv iru wkh sdudphwhuv ri wkh prgho1 Zh dovr qhhg d qrwdwlrq iru wkh hvwlpd0 wru xqghu wkh qxoo k|srwkhvlv dqg rqh iru wkh hvwlpdwru xqghu wkh dowhuqdwlyh1 Wkxv iru lqvwdqfh zh ohw k ghqrwh wkh sdudphwhu/ kf wkh wuxh ydoxh ri wkh sd0 udphwhu/ iru zklfk zh fdofxodwh wkh h{shfwdwlrqv/ k wkh uhgxfhg udqn hvwlpdwru lq wkh prgho dqg k wkh uhjuhvvlrq hvwlpdwru xqghu wkh qxoo k|srwkhvlv1 Zh xvh wkh qrwdwlrq & [ 3. [ ' EK c c K& c K ' U? 3. 9. '. K .
(17) Qrwh wkdw EU? K q ' EU? q Ek Kq k K q ' fc . z. vxfk wkdw iru q7 ' qEq q . 3. z. 3. . z. z. z. z. . 7 n q q7 ' EU? K qq 7 EU? K ' EU? K Eqq . . z. . z. Zh wkhuhiruh ghfrpsrvh wkh surfhvv lqwr vwdwlrqdu| dqg qrq vwdwlrqdu| frpsr0 qhqwv= 7 f q f f| n f Kf f| f| ' EU? f Kf q . Zh qg/ xvlqj wkh wuxh ydoxh ri wkh sdudphwhuv/ q f| . . 3. f ' q EU? f Kf q7 q f f| . . 3. . n q q f Ekf Kf q f kf Kf f| . . z. 3. +8,. . z. z. 3. z. Zh fkrrvh qhz sdudphwhuv ' q EU? Kf q7 B ' q q f B2 ' 4 4f . . . . . . f. Eo o Eo E? o Eo ?( . z. . . vxfk wkdw wkh rog sdudphwhuv lq whupv ri wkh qhz duh jlyhq e| q ' B Ekf Kf q f kf Kf n q f c 4 ' B2 n 4f . . . z. 3. . . z. . . . . . z. Wkh k|srwkhvlv q ' q f c 4 ' 4f lv h{suhvvhg lq wkh qhz sdudphwhuv dv B ' fc ' Uo Wkh prgho htxdwlrq +4, zlwk wkh qhz sdudphwhuv lv {f| ' k Eq f f| n 4f (| n kEB Ekf Kf q f kf Kf f| S n &' K {f| n x_| n 0| . . . . 3. . 3. z. z. 3. . z. . 3. n B 2 (| . +9,. 3. Qrwlfh wkdw wkh prgho lv ryhusdudphwul}hg vlqfh k# c # c B# . 3. 3. . jlyh wkh vdph suredelolw| phdvxuh dv Ekc c B iru dq| # Eo o ri ixoo udqn Zh fdq dfklhyh mxvw lghqwl fdwlrq e| fkrrvlqj ' Uo c wkdw lv/ e| devruelqj Eo o lqwr kE? o dqg dgmxvwlqj B dffruglqjo|1 Wkh k|srwkhvlv ri lqwhuhvw lv wkhq B ' f Lq wkh +uhgxfhg udqn, uhjuhvvlrq +9, zh xvh wkh uhvxow wkdw q f f| n 4f (| dqg {f| kdyh d phdq wkdw lv olqhdu lq _| c vhh Ohppd 4/ dqg wkdw x . 3. . :.
(18) hqwhuv xquhvwulfwhgo|/ wr uhsodfh wkh uhjuhvvruv q f f| gl huhqfhv zlwk wkh vwdwlrqdu| uhjuhvvruv . . 3. T|. . ' q f f|. ~|. . ' E{f|. 3. 3. . 3. . n 4f (| dqg wkh odjjhg . . .f Eq f f| c. . .f E{f| c c {f| &n .f E{f| &n . 3. +:,. . 3. . . 3. . 3. +;,. . 3. Zh dovr zdqw wr uhsodfh wkh uhjuhvvru Ekf Kf q f kf Kf f| e| vrphwklqj vlpsohu zlwkrxw fkdqjlqj wkh vwdwlvwlfdo prgho dqg khqfh wkh whvw wkdw B ' f1 Zh qg e| vxpplqj htxdwlrq +4, wkdw . z. & [ 3. k Ef| ff ' k . . z. z. E| vxewudfwlqj. S&. k Kf| z. 3. | [ K Ef| f n k E0 n x_ 3. z. '. . ' k ff n k. . 3. 3. ' kz K f| rq erwk vlghv dqg uhsodflqj | e| | zh jhw. & [ 3. . . z. . 3. '. 3. z. . . z. z. | [ K Ef| f| f n k E0 n x_ 3. . 3 3. '. 3. 3. z. '. Ehfdxvh zh duh fruuhfwlqj iru odjjhg gl huhqfhv lq wkh uhjuhvvlrq +9, zh fdq uhsodfh Ekf Kf q f kf Kf f| dqg (| e| wkh qrq vwdwlrqdu| uhjuhvvru jlyhq e| wkh frpprq wuhqgv S f n kf |' E0 n xf _ | ' c +<, (| . z. 3. z. . 3. z. 3. . z. 3. zkhuh f ghshqgv rq lqlwldo frqglwlrqv1 Wkh prgho htxdwlrq +9, lq wkh qhz yduldeohv dqg zlwk vxlwdeo| uhgh qhg sdudphwhuv ehfrphv {f| ' kT| E? . . 3. Eo . n kB | . 3 E?3o n?( . n [ ~|. 3 EE&3 ? . n x _| n 0| c E?_ . E? . +43,. zkhuh wkh glphqvlrqv duh lqglfdwhg ehorz hdfk yduldeoh1 Wkh hvwlpdwruv iru wkh sdudphwhuv Bc kc [c xc dqg l fdq eh irxqg e| uhgxfhg udqn uhjuhvvlrq ri {f| rq ET| c | fruuhfwhg iru ~| dqg _| Xqghu wkh k|srwkhvlv B ' f wkh sdudphwhuv fdq eh irxqg e| uhjuhvvlrq ri {f| rq ~| dqg _| . 3. . 3. 3. 3. Odwhu zh vkdoo fkrrvh | vxfk wkdw lw lv ruwkrjrqdo wr wkh ghwhuplqlvwlf whup _| zklfk vlpsol hv vrph qrwdwlrq1 Qrwh wkdw li (| ' f wkhq/ ri frxuvh/ zh gr qrw h{whqg wkh surfhvv/ dqg | lv gh qhg hqwluho| lq whupv ri wkh udqgrp zdonv/ dqg lqlwldo ydoxhv1 Qrwh dovr wkdw li _| frqwdlqv d frqvwdqw wkhq/ zkhq fruuhfwlqj iru _| c wkh lqlwldo ydoxhv glvdsshdu1 3. 3. ;.
(19) 516. D vlpsoh k|srwkhvlv rq q dqg 4 lq prgho P4. Lq prgho P wkh frlqwhjudwlqj yhfwruv q duh uhvwulfwhg dv q ' M +Er o dqg htxdwlrq +4, lv & [ 3. {f| ' kE M f| . . . 3. n 4 (| n . K {f| n x_| n 0| . +44,. 3. '. Zh frqvlghu djdlq wkh vlpsoh k|srwkhvlv q ' M f c 4 ' 4f c fruuhvsrqglqj wr d srlqw lq P Zh zdqw wr vkrz wkdw e| lqwurgxflqj wkh wuxh sdudphwhuv q f c 4f c dv ehiruh zh fdq uhirupxodwh wkh htxdwlrqv wr kdyh wkh irup +43,/ vxfk wkdw d whvw ri d vlpsoh k|srwkhvlv lv d whvw wkdw B ' f Zh ghfrpsrvh wkh surfhvv f| dqg qg. xvlqj wkh wuxh ydoxh ri wkh sdudphwhuv. . 3. q f| . f ' M EU? f Kf q7 q f f| . . 3. . . . 3. n M q f Ekf Kf q f kf Kf f| . . . z. 3. . z. z. 3. z. Lq wklv fdvh zh kdyh wkdw q f ' M f lpsolhv wkdw q f ' EM c MEM M f dqg khqfh . 3. z. z. z. q q f ' M q f ' f EfEr o E? r c Ur o . . . . z. z. 3. z. f. 3. 3. Zh lqwurgxfh wkh qhz sdudphwhuv f ' M EU? f Kf q7 Eo o B ' f Eo Er o B 2 ' 4 4f Eo ?( . . . . . z. . . . . vlqfh wkhq M f| n 4 (| ' Eq f f| n 4f (| n B EfEr o E? r c Ur o Ekf Kf q f kf Kf f| . . . 3. . . . . . 3. 3. f. 3. 3. 3. . z. z. . 3. z. n B 2 (| . Wkh k|srwkhvlv lv irupxodwhg dv B ' fc ' Uo Zh ohw T| dqg ~| eh gh qhg e| +:, dqg +;,/ dqg uhsodfh lq wklv fdvh wkh Er o glphqvlrqdo qrq vwdwlrqdu| uhjuhvvru 3. EfEr o E? r c Ur o Ekf Kf q f kf Kf f| . 3. f. 3. 3. z. <. 3. z. . z. . 3. 3.
(20) zlwk r o olqhdu frpelqdwlrqv g ri wkh frpprq wuhqgv h{whqghg e| (| = S gf n g kf |' E0 n xf _ | ' c +45, (| 3. . z. 3. iru vrph pdwulfhv gf +Er o ghshqglqj rq lqlwldo frqglwlrqv dqg g EEr o E? o Htxdwlrq +44, wkhq ehfrphv. {f| ' kT|. 3. E? . Eo . . n kB | . n 3 Er3on?( . [ ~|. n 3 EE&3 ? . x _| n 0| c E?_ . +46,. E? . zkhuh lv devruehg lq k dqg wkh uhpdlqlqj sdudphwhuv duh dgmxvwhg dffrug0 lqjo| vxfk wkdw dovr ~| c dqg T| kdyh phdq }hur1 Wkh k|srwkhvlv ri lqwhuhvw lv B ' fc zklfk fruuhvsrqgv wr 4 ' 4f c q ' q f ' M f . 3. 3. Wklv htxdwlrq kdv wkh vdph vwuxfwxuh dv +43, h{fhsw wkdw wkh glphqvlrq ri | lv fkdqjhg wr r o n ?( 3. 517. D vlpsoh k|srwkhvlv rq q dqg 4 lq prgho P5. Djdlq zh lqyhvwljdwh d vlpsoh k|srwkhvlv rq q dqg 4 zklfk fdq eh irupxodwhg dv 2 ' q f2 c 42 ' 4f2 Wkh sdudphwhu k lv ghfrpsrvhg fruuhvsrqglqj wr wkh frlqwhjudwlqj sdudphwhu lqwr k ' Ek c k2 c vxfk wkdw kEq f| . . 3. n 4 (| ' k Eq f f| . . . . 3. n 4f (| n k2 E 2 f| . . 3. n 42 (| . . Lq wklv fdvh zh devrue lqwr k dqg lqfoxgh wkh uhjuhvvru q f f| zlwk wkh odjjhg gl huhqfhv ~| lqvwhdg ri zlwk T| Zh wkhq ghfrpsrvh wkh vhfrqg frpsrqhqw 2 f| n 42 (| ri wkh surfhvv dv . . 3. 3. . 3. . 3. 2 f| n 42 (| f ' E2 EU? f Kf q7 q f f| . . 3. . . . 3. n 2 q f Ekf Kf q f kf Kf f| . Qrz f f. . 7 f f . 2 EU? K q q f| . . z. . 3. ' E c 2 . . 3. q f f| q f2 f| . . . 3. . . z. z. ' q f f| . . n 42 (| . . n 2 q f2 f| c. 3. z. . 3. . 3. . . . 3. vr wkdw 2 f| . . 3. n 42 (| ' Eq f f| . . . . 3. n 4f (| n 2 Eq f2 f| . . . . 3. 43. n 4f2 (| n B | . . 3.
(21) Zh kdyh gh qhg wkh sdudphwhuv 7f E c 2 ' 2 EU? f Kf q B ' 2q f B2 ' 4 4f 2 4f2 . . Eo2 o c Eo2 o2 Eo2 E? o Eo2 ?( . . . . z. . . . . . . Wkh k|srwkhvlv ri lqwhuhvw lq wkh qhz sdudphwhuv lv B ' fc ' fc 2 ' Uo2 Wkh surfhvv | lv gh qhg e| +<, vxfk wkdw wkh htxdwlrq ehfrphv 3. {f| ' k2 T|. Eo2 3. E? . n k 2 B | . n 3 E?3on?( . [. ~|3 Eo nE&3 ? . n x _| n 0| c E?_ . E? . +47,. zkhuh T|. . ' q f2 f|. ~|. . '. +48,. . 3. Ef| 3 q f c {f| 3 c 3. 3. c {f| &n . +49,. . 3. erwk fruuhfwhg iru wkhlu phdq/ dqg zkhuh djdlq kc [c dqg x kdyh ehhq uhgh qh wr dffrpprgdwh wkh fkdqjh lq uhjuhvvruv1 Lw lv vhhq wkdw htxdwlrq +47, lv ri wkh irup +43,/ zlwk d fkdqjhg gh qlwlrq ri T| dqg ~| / vlqfh wkh dvvxphg vwd0 wlrqdu| frpelqdwlrqv q f f| .f Eq f f| duh pryhg wr wkh odjjhg gl huhqfhv1 Wkh k|srwkhvlv fdq eh whvwhg dv B ' fc dv wkh rwkhu uhvwulfwlrqv duh devruehg lq k dqg [ 3. . 3. . 3. 3. Wkxv lq d jhqhudo irupxodwlrq wkdw fryhuv doo wkh k|srwkhvhv zh duh lqwhu0 hvwhg lq/ zh qhhg wr doorz wkh glphqvlrqv ri wkh yduldeohv hqwhulqj wkh htxdwlrq wr eh gl huhqw iurp wkh wkrvh jlyhq lq +43,1 Exw zh vwloo qhhg wr suhvhuyh wkh surshuwlhv wkdw xqghu wkh qxoo k|srwkhvlv wkh surfhvv +T| c ~| lv d phdq }hur vwdwlrqdu| dxwruhjuhvvlyh surfhvv/ dqg wkdw T| dqg B | kdyh wkh idfwru k Eru k2 lq iurqw1 Doo prghov +43,/ +46,/ dqg +47, kdyh wkh surshuw| wkdw wkh| fdq eh vroyhg e| uhgxfhg udqn uhjuhvvlrq dqg wkdw xqghu wkh qxoo k|srwkhvlv/ B ' fc wkh prgho lv vroyhg e| vlpsoh uhjuhvvlrq1 . . 3. . 3. . 3. 6. 3. D jhqhudo uhgxfhg udqn htxdwlrq dqg dq h{0 sdqvlrq ri wkh hvwlpdwruv dqg wkh whvw vwdwlv0 wlf. Lq rughu wr fryhu wkh gl huhqw fdvhv frqvlghuhg lq Vhfwlrq 5/ zh glvfxvv wkh h{sdqvlrq dqg Eduwohww fruuhfwlrq ri wkh olnholkrrg udwlr whvw iru wkh k|srwkhvlv B ' f lq wkh htxdwlrq 44.
(22) Prgho Pf P P2. ? o o o2. ?@ ? o n ?( r o n ?( ? o n ?(. ?5 1 E& ? k E& ? k o n E& ? k2. Wdeoh 4= Wkh fkrlfh ri glphqvlrqv lq wkh jhqhudo uhjuhvvlrq prgho +4:, zklfk fruuhvsrqgv wr wkh k|srwkhvhv glvfxvvhg lq Vhfwlrq 41. {f| ' 1T|. E? 3. E? . n 1B | . 3 E?@ . n [~|. n x _| n 0| c. 3 E?5 . E?_ . +4:,. E? . zkhuh 0| duh l1l1g1 ?Efc l dqg wkh sdudphwhuv E1c Bc [c xc l ydu| iuhho|1 Wklv qrwdwlrq fryhuv wkh gl huhqw vlwxdwlrqv frqvlghuhg iru vxlwdeoh fkrlfhv ri wkh uhjuhvvruv T| c | c dqg ~| c dqg wkhlu glphqvlrqv/ vhh Wdeoh 41 3. 3. 3. Lq doo fdvhv wkh yduldeohv T| dqg ~| duh/ xqghu wkh k|srwkhvlv B ' S fc vwdwlrqdu| zlwk phdq }hur dqg | lv d olqhdu ixqfwlrq ri 1 f |' E0 n xf _ c zlwk 1 f ' kf ru kf2 Qrwh wkdw wkh vwdfnhg Eo n E& ? glphqvlrqdo surfhvv t| ' ET| c ~| lv wkh vdph iru doo fdvhv dqg frqwdlqv q f| dqg wkh odjjhg gl huhqfhv fruuhfwhg iru wkhlu phdq1 3. 3. . 3. . . z. . . Wkh yhu| ghwdlohg fdofxodwlrqv lq wklv sdshu frqwlqxhv wkh zrun lq Mrkdqvhq +4<<<, zkhuh wkh fruuhfwlrq zdv irxqg iru wkh prgho zkhuh vs+1 lv nqrzq1 Wkh uhvxow ghulyhg wkhuh surylgh wkh pdlq whup ri wkh Eduwohww fruuhfwlrq lq wkh vlwxdwlrq zkhuh 1 lv xqnqrzq dqg zh wkhuhiruh eulh | glvfxvv wklv vlwxdwlrq lq wkh qh{w vxevhfwlrq1 Zh wkhq jlyh dq h{sdqvlrq ri wkh uhgxfhg udqn hvwlpdwru durxqg wkh uhjuhvvlrq hvwlpdwru ydolg xqghu wkh qxoo k|srwkhvlv dqg qdoo| zh ghulyh dq h{sdqvlrq ri d vlpsoh k|srwkhvlv iru B ' f1. 614. Wkh dqdo|vlv iru {hg 1 ' 1 3. Qrwh wkdw li 1 ' 1 f c zkhuh 1 f lv nqrzq wkh prgho htxdwlrq lv {f| ' 1 f T| . . 3. n 1 f B | . . 3. n [~|. . n x_| n 0| c. . f f n 71 x_| n 71 0|. 3. f zklfk lpsolhv/ iru 71 ' 1 f E1 f 1 f c wkdw . 71 f {f| ' T| . . 3. 3. n B | . . 1 f {f| ' 1 f [~| . . z. z. . 3. . 3. f . n 71 [~|. 3. n 1 f x_| n 1 f 0| . . z. z. 45. . .
(23) f Khqfh wkh prgho iru 71 {f jlyhq 1 f {f| dqg wkh sdvw lv . . z. 71 f {f| ' /| n T| . . . 3. n B | . 3. . f n E71 /1 f [~| . . . 3. z. f n E71 /1 f x_| n 0| c +4;, . . z. iru f . | ' 1 f {f| c / ' 71 l1 f E1 f l1 f c . z. . . z. z. 3. z. dqg f . 0| ' 71 0| /1 f 0| ' E1 f l 1 f 1 f l 0| . . 3. . 3. 3. z. Zh gh qh wkh qrupdol}hg huuru L| ' E1 f lf 1 f . 3. 3. f 2. 1 lf 0| c 3. vxfk wkdw iru wkh wuxh ydoxh ri wkh sdudphwhuv 0| ' E1 f lf 1 f . 3. 3. 2. L| . Zh gh qh wkh surgxfw prphqw pdwulfhv iru wkh yduldeohv {f| c | c 0| c L| c dqg _| dw wlph | exw T| c | c dqg ~| odjjhg rqh shulrg1 Wkxv iru lqvwdqfh 3. A [. 3. 3. 43 4 3 4 {f| {f| ff f f0 C T| D C T| D ' C f 0 D 0| 0| 0f 0 00 . 3. |'. 3. 3. Zh dovr xvh wkh qrwdwlrq iru dq| wkuhh surfhvv fc Lc dqg T % ' % %% % c 3. dqg lq sduwlfxodu zh xvh d qrwdwlrq iru wkh prphqw pdwulfhv fruuhfwhg iru wkh odjjhg gl huhqfhv ~| dqg _| / vlqfh pdq| uhvxowv orrn d elw vlpsohu wklv zd| 3. 5_ 7 ' 5c_ ' _ __ _ 5_ 55_ 3. 3. Wkhvh prphqw pdwulfhv duh qdwxudo zkhq wkh olnholkrrg ixqfwlrq lv frqfhqwudwhg zlwk uhvshfw wr [ dqg x1 Wkh pd{lpxp olnholkrrg hvwlpdwru ri B +iru nqrzq 1 ' 1 f lv irxqg e| uhjuhvvlrq lq +4;, BE1 f ' 7f 7f @@c5cKc_ @fc5cKc_ 1 ' 7@@cK 7@fcK 1 ' B n 7@@cK 7@0cKl 1 f E1 f l 1 f 3. 3. 3. . 3. 46. 3. 3.
(24) Wkh whvw iru wkh k|srwkhvlv B ' fc +vwloo iru nqrzq 1 f lv xqghu wkh k|srwkhvlv/ B ' fc htxdo wr u- 2*A EB ' fm1 f nqrzq f f f f ' m71 7ff@ccK71 m*m71 7ffcK71 m ' m7@ccK m*m7cKm ' mU? 7cK 7@cK 7@@cK 7@cK m ' mU? A 'mc. +4<,. |oi'2 j 2A. +53,. 3. . . 3. 3. 3. vd|/ vxfk wkdw . 2 *L} u-EB ' fm1 f nqrzq ' |oi'j n _. Zh xvh khuh wkh qrwdwlrq ' wr lqglfdwh wkdw zh kdyh nhsw whupv ri ru0 ghu A _ Dq dssur{lpdwlrq wr wkh h{shfwdwlrq ri 2 *L} u- jlyhq e| +53, zdv ghulyhg lq Mrkdqvhq +4<<<, dqg wxuqv rxw wr jlyh wkh pdlq frqwulexwlrq wr wkh h{shfwdwlrq ghulyhg lq wklv sdshu1 3. 615. Wkh uvw rughu frqglwlrqv iru wkh hvwlpdwlrq ri 1c Bc dqg l. Lq wkh uhvw ri wkh sdshu zh uhihu wr wkh wuxh ydoxh/ wkh rqh iru zklfk zh fdofxodwh wkh h{shfwdwlrq/ zlwkrxw wkh vxshuvfulsw vlqfh wkdw vlpsol hv wkh qrwdwlrq1 Zh h{suhvv wkh uhvxowv ehorz lq wkh qrwdwlrq iru wkh frqfhqwudwhg prgho/ zkhuh wkh sdudphwhuv [ dqg x kdyh ehhq holplqdwhg/ wkdw lv/ zh xvh wkh prphqw pdwulfhv 7 udwkhu wkdq Wkh pd{lpxp olnholkrrg hvwlpdwruv edvhg xsrq +4:, zloo eh ghqrwhg e| Bc 1c dqg l Wkh uvw rughu frqglwlrqv iru wkh hvwlpdwruv lq prgho +4:, fdq eh vroyhg iru hdfk ri wkh yduldeohv dv 1 ' E7f n 7f@B E7 n B 7@ n 7@B n B 7@@B . . +54,. E7ff 1E7 n B 7@ n 7@B n B 7@@B 1 . +55,. . 'A l. . 3. . . . B ' 7 E7@f l 1E1 l 1 @@ 3. . 3. 47. 3. 3. . 7@ . 3. . +56,.
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