UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)
UvA-DARE (Digital Academic Repository)
Testing Parameters in GMM without assuming that they are identified
Kleibergen, FrankPublication date 2002
Link to publication
Citation for published version (APA):
Kleibergen, F. (2002). Testing Parameters in GMM without assuming that they are identified. Unknown Publisher.
General rights
It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.
Discussion Paper: 2002/
23
Testing Parameters in GMM without assuming that
they are identified
Frank Kleibergen
www.fee.uva.nl/ke/UvA-Econometrics
Department of Quantitative Economics
Faculty of Economics and Econometrics Universiteit van Amsterdam
Roetersstraat 11
1018 WB AMSTERDAM The Netherlands
Whvwlqj Sdudphwhuv lq JPP zlwkrxw dvvxplqj wkdw
wkh| duh lghqwlhg
Iudqn Nohlehujhq
|Devwudfw
Zh sursrvh d Jhqhudol}hg Phwkrg ri Prphqwv +JPP, Odjudqjh pxowlsolhu vwdwlvwlf/ l=h= wkh N0vwdwlvwlf/ wkdw xvhv wkh Mdfreldq dw wkh hydoxdwhg sdudphwhu ydoxh lqvwhdg ri wkh h{shfwhg Mdfreldq1 Wr rewdlq lwv olplw ehkdylru/ zh xvh d qryho dvvxpswlrq wkdw eulqjv JPP forvhu wr pd{lpxp olnholkrrg dqg zklfk lv hdvlo| vdwlvhg1 Wkh xvxdo dv|pswrwlf "2glvwulexwlrq ri wkh N0vwdwlvwlf wkhq krogv xqghu d zlghu vhw ri flufxpvwdqfhv/ olnh zhdn
dqg pdq| lqvwuxphqw dv|pswrwlfv dqg frpelqdwlrqv wkhuhri/ wkdq wkh vwdqgdug ixoo udqn fdvh iru wkh Mdfreldq1 Wkh ehkdylru ri wkh N0vwdwlvwlf fdq eh vsxulrxv durxqg lq h{lrq srlqwv dqg wkh pd{lpxp ri wkh remhfwlyh ixqfwlrq vlqfh wkh prphqw frqglwlrqv duh wkhq qrw vdwlvhg1 Frpelqdwlrqv ri wkh N0vwdwlvwlf zlwk vwdwlvwlfv wkdw whvw wkh ydolglw| ri wkh prphqw htxdwlrqv ryhufrph wkh vsxulrxv ehkdylru1 Zh frqgxfw d srzhu frpsdulvrq wr whvw iru wkh ulvn dyhuvlrq sdudphwhu lq d vwrfkdvwlf glvfrxqw idfwru prgho dqg frqvwuxfw lwv frqghqfh vhw iru revhuyhg frqvxpswlrq jurzwk dqg dvvhw uhwxuq vhulhv1
4 Lqwurgxfwlrq
Wkh Jhqhudol}hg Phwkrg ri Prphqwv +JPP, ri Kdqvhq +4<;5, rhuv d h{leoh hvwlpdwlrq iudphzrun lq zklfk pdq| hfrqrphwulf prghov fdq eh fdvw1 Wklv doohyldwhv vwdwlvwlfdo lqihuhqfh lq wkhvh prghov ehfdxvh zh fdq xvh wkh hvwlpdwruv dqg whvw vwdwlvwlfv ghyhorshg iru JPP1 Wkh olplwlqj glvwulexwlrqv ri wkhvh duh frqvwuxfwhg xqghu d fhqwudo olplw wkhruhp iru wkh prphqwv dqg d ixoo udqn dvvxpswlrq ri wkh Mdfreldq1 Wkh udqn dvvxpswlrq lv riwhq eduho| vdwlvhg1 Vlpxodwlrq h{shulphqwv ri vxfk lqvwdqfhv vkrz wkdw wkh hpslulfdo glvwulexwlrqv ri JPP hvwlpdwruv dqg whvw vwdwlvwlfv duh wkhq txlwh glhuhqw iurp wkhlu qrupdo ru 2 olplwlqj
glvwulexwlrqv/ vhh e} Kdqvhq e| @, +4<<9,1 Zh frqvwuxfw wkh olplwlqj glvwulexwlrqv xqghu d glhuhqw dvvxpswlrq wkdw lv hdvlo| vdwlvhg dqg ohdyhv dvlgh wkh udqn dvvxpswlrq1 Wkh dvvxpswlrq lpsuryhv wkh dssur{lpdwlrq zklfk JPP lv ri pd{lpxp olnholkrrg1
Zh dqdo|}h wkh ghulydwlyh ri wkh remhfwlyh ixqfwlrq wr rewdlq d Odjudqjh pxowlsolhu +OP, ru vfruh vwdwlvwlf/ wr zklfk zh uhihu dv wkh N0vwdwlvwlf/ vhh Nohlehujhq +5335d,1 Lqvwhdg ri wkh h{shfwhg Mdfreldq/ wkh N0vwdwlvwlf xvhv wkh Mdfreldq dw wkh sdudphwhu ydoxh wkdw plqlpl}hv wkh remhfwlyh ixqfwlrq/ zklfk lv wkh frqwlqxrxv xsgdwlqj hvwlpdwru +FXH, ri Kdqvhq e| @, +4<<9,1
Ghsduwphqw ri Txdqwlwdwlyh Hfrqrplfv/ Xqlyhuvlw| ri Dpvwhugdp/ Urhwhuvvwuddw 44/ 434; ZE Dpvwhugdp/
Wkh Qhwkhuodqgv/ Hpdlo=nohlehujCihh1xyd1qo1 Krphsdjh= kwws=22zzz1ihh1xyd1qo2nh2nohlehujhq1kwp
|L wkdqn sduwlflsdqwv ri vhplqduv dw Eurzq/ Kduydug0PLW/ Qx!hog Froohjh/ Sulqfhwrq/ XVF/ XFVG/ Yl0
hqqd/ \doh/ XN Hfrqrphwulf Vwxg| Jurxs/ HF5 dw FRUH/ HVHPv dw Odxvdqqh dqg Yhqlfh dqg/ hvshfldoo|/
wkh hglwru dqg uhihuhhv iru khosixo frpphqwv dqg vxjjhvwlrqv1 Wklv uhvhdufk kdv ehhq ixqghg e| wkh QZR Yhuqlhxzlqjvlpsxov uhvhdufk judqw Hpslulfdo Frpsdulvrq ri Hfrqrplf Prghov1
Dv uhfrjql}hg hovhzkhuh/ vhh e} Eurzq dqg Qhzh| +4<<;, dqg Grqdog dqg Qhzh| +5333,/ wklv Mdfreldq hvwlpdwru lv dv|pswrwlfdoo| xqfruuhodwhg dqg/ wkxv/ lqghshqghqw ri wkh dyhudjh prphqw yhfwru1 Xqghu rxu dvvxpswlrq/ wkh xvxdo dv|pswrwlf 2 glvwulexwlrq ri wkh N0vwdwlvwlf
krogv lq d zlghu vhw ri flufxpvwdqfhv wkdq wkh vwdqgdug ixoo udqn fdvh iru wkh Mdfreldq1 Wklv vhw lqfoxghv erwk wkh zhdn lqvwuxphqw dv|pswrwlfv ri Vwdljhu dqg Vwrfn +4<<;, dqg Vwrfn dqg Zuljkw +5333,/ wkh pdq| lqvwuxphqwv dv|pswrwlfv ri Ehnnhu +4<<7, dqg Grqdog dqg Qhzh| +5334, dqg frpelqdwlrqv ri wkhvh olplw vhtxhqfhv1
Wkh rxwolqh ri wkh sdshu lv dv iroorzv1 Lq wkh vhfrqg vhfwlrq/ zh glvfxvv JPP dqg pdnh wkh dvvxpswlrq wkdw eulqjv JPP forvhu wr pd{lpxp olnholkrrg1 Lq wkh wklug vhfwlrq/ zh rewdlq wkh N0vwdwlvwlf dqg frqvwuxfw lwv olplw ehkdylru xqghu vwdqgdug dqg zhdn lqvwuxphqw dv|pswrwlfv1 Lq vhfwlrq 7/ zh glvfxvv wkh olplw ehkdylru xqghu frpelqdwlrqv ri zhdn dqg pdq| lqvwuxphqw dv|pswrwlfv1 Wkh iwk vhfwlrq ghdov zlwk d vsxulrxv srzhu ghfolqh ri wkh N0vwdwlvwlf1 Ehfdxvh wkh N0vwdwlvwlf lv d txdgudwlf irup ri wkh remhfwlyh ixqfwlrq/ lw lv htxdo wr }hur erwk dw wkh plqlpxp/ pd{lpxp dqg lq h{lrq srlqwv ri wkh remhfwlyh ixqfwlrq1 Wkh odwwhu }hur ydoxhv uhvxow zkloh wkh prphqw frqglwlrqv duh qrw vdwlvhg vr wkh uhvxowv iurp wkh N0vwdwlvwlf duh wkhq vsxulrxv1 Wkh| fdq/ krzhyhu/ dhfw wkh glvfulplqdwru| srzhu ri wkh N0 vwdwlvwlf zkhq zh gr qrw dffrxqw iru wkhp dssursuldwho|1 Zh wkhuhiruh vxjjhvw frpelqdwlrqv ri wkh N0vwdwlvwlf zlwk vwdwlvwlfv wkdw whvw wkh prphqw htxdwlrq/ olnh/ iru h{dpsoh/ d JPP h{whqvlrq ri Pruhlud*v +5334, frqglwlrqdo olnholkrrg udwlr vwdwlvwlf1 Lq wkh vl{wk vhfwlrq/ zh glvfxvv wkh frqvwuxfwlrq ri frqghqfh vhwv1 Wkh vhyhqwk vhfwlrq frqgxfwv d srzhu frpsdulvrq wr whvw wkh ulvn dyhuvlrq sdudphwhu lq d vwrfkdvwlf glvfrxqw idfwru prgho dqg frqvwuxfw lwv frqghqfh vhw iru revhuyhg gdwd1 Ilqdoo|/ wkh hljkw vhfwlrq frqfoxghv1
Zh xvh wkh iroorzlqj qrwdwlrq wkurxjkrxw wkh sdshu= yhf+ vwdqgv iru wkh froxpq yhf0 wrul}dwlrq ri wkh A ? glphqvlrqdo pdwul{ c yhf+ ' E@
@? zkhq ' E@ @?c
gldj+@c c @? vwdqgv iru d eorfn0gldjrqdo pdwul{ zlwk wkh yhfwruv +pdwulfhv, @c c @? rq
wkh gldjrqdo/ ' E3 dqg ' UA iru d ixoo udqn pdwul{ dqg wkh A A
lghqwlw| pdwul{ UAc m@ vwdqgv iru hydoxdwhg lq @/ $R lqglfdwhv frqyhujhqfh lq suredelolw| dqg
$
_ lqglfdwhv frqyhujhqfh lq glvwulexwlrq1
5 Jhqhudol}hg Phwkrg ri Prphqwv
Zh frqvlghu wkh hvwlpdwlrq ri wkh 6 sdudphwhu yhfwru w ' Ew w6c zkrvh sdudphwhu
uhjlrq lv wkh U6c iru zklfk wkh , glphqvlrqdo prphqw htxdwlrq
. d)Ewfc t|mU|o ' f +4,
krogv1 Wkh h{shfwdwlrq/ lqglfdwhg e| .c lq +4, lv wdnhq zlwk uhvshfw wr wkh lqirupdwlrq vhw U| dw wlph2lqglylgxdo | Wkh gdwd yhfwru t| lv revhuyhg dw wlph2lqglylgxdo |1 Wkh ,
glphqvlrqdo yhfwru ixqfwlrq ) ri w lv qlwh iru qlwh ydoxhv ri wc frqwlqxrxv dqg wzlfh frqwlqxrxv glhuhqwldeoh1 Wkh vshflf wuxh ydoxh ri wc dw zklfk +4, krogv/ lv htxdo wr wf1 Wr hvwlpdwh
wkh sdudphwhu w lq +4,/ zh xvh Kdqvhq*v +4<;5, JPP iudphzrun1 Zh lqyroyh d &0glphqvlrqdo yhfwru ri lqvwuxphqwv f|wkdw lv vxfk wkdw &s E' &, h{fhhgv 6 Wkh lqvwuxphqwv vsdq wkdw sduw
ri wkh lqirupdwlrq vhw U| zklfk lv ri lpsruwdqfh iru wkh hvwlpdwlrq ri w dqg duh xqfruuhodwhg
zlwk )Ewfc t|c
. df|)Ewfc t|mU|o ' . df|)Ewfc t|o ' f +5,
Iru d gdwd0vhw Et|c f|c | ' c c A c wkh remhfwlyh ixqfwlrq lq wkh JPP iudphzrun uhdgv
'Ew ' sAEwc t TssEw3sAEwc t c +6,
zlwk sAEwc t 'SA|'s|Ewc
s|Ew ' yhf Ef|)Ewc t| ' E)Ewc t| f|c +7,
dqg TssEw lv wkh fryduldqfh pdwul{ ri sAEwc t zlwk 7s|Ew ' s|Ew .Es|Ewc
TssEw ' *4A <". q A SA |' SA 's7|Ew 7sEw r +8,
Dvvxpswlrq 41 Wkh &s glphqvlrqdo ghulydwlyh ri s|Ewf zlwk uhvshfw wr wc
Rc|Ewf ' YsYw|Ew mwf G &s c ' c c 6c +9,
lv vxfk wkdw
7Rc|Ewf ' 7^c|Ewf +:,
zlwk 7Rc|Ewf ' Rc|Ewf .ERc|Ewfc ^c|Ewf G & c 7^c|Ewf ' ^c|Ewf .E^c|Ewf dqg d
ghwhuplqlvwlf ixoo0udqn &s & glphqvlrqdo pdwul{/ & &s Wkh mrlqw olplwlqj ehkdylru ri wkh
vxpv ri wkh pduwlqjdoh glhuhqfh vhulhv 7s|Ewf E' s|Ewf dqg 7^|Ewf ' E7^|Ewf 7^6|Ewf
dffrugv zlwk wkh fhqwudo olplw wkhruhp
I A [A |' 7s|Ewf 7^|Ewf $ _ s w c +;,
zkhuh s G &s c w G &w c &w 'S6'&c
s w Efc T Ewc +<, dqg T Ew ' TssEw TswEw TwsEw TwwEw c +43,
zlwk TssEw G &s &sc TwsEw ' TswEw G &w &sc TwwEw G &w &wc dqg
T Ew ' *4A <". A SA |' SA
' 7s7^||EwEw 7s7^||EwEw
+44,
Wkh lqvwuxphqwv vsdq wkdw sduw ri wkh lqirupdwlrq vhw zklfk lv ri lpsruwdqfh iru wkh hvwlpdwlrq ri w vr . d^c|Ewo ' . d^c|EwmU|o Khqfh/ 7^|Ewc | ' c c Ac lv d pduwlqjdoh glhuhqfh
vhulhv dqg Dvvxpswlrq 4 lv d fhqwudo olplw wkhruhp iru pduwlqjdoh glhuhqfhv1 Lw lv wkhuhiruh vdwlvhg xqghu zhdn frqglwlrqv iru 7s|Ewf dqg 7^|Ewf Vx!flhqw frqglwlrqv wkdw hqvxuh vxfk
frqyhujhqfh duh wkdw= wkh o0wk prphqw ri wkh devroxwh ydoxh ri 7s|Ewf dqg 7^c|Ewfc '
c c 6c lv qlwh iru vrph o : 2c 2 T Ew lv zhoo0ghqhg dqg wkh dyhudjh ydoxh ri wkh rxwhu0surgxfw ri E 7s|Ewf 7^|Ewf frqyhujhv lq suredelolw| wr T Ew/ vhh e} Zklwh +4<;7,1
Dvvxpswlrq 4 lv dq h{whqvlrq ri wkh dvvxpswlrq wkdw lv xvxdoo| pdgh lq rughu wr rewdlq wkh olplwlqj glvwulexwlrqv ri JPP hvwlpdwruv1 Wudglwlrqdoo| rqo| d fhqwudo olplw wkhruhp iru s|Ewf lv dvvxphg wr krog/ vhh e} Kdqvhq +4<;5,/ Qhzh| dqg PfIdgghq +4<<7, dqg Vwrfn dqg
Zuljkw +5333,1 Dorqjvlgh wkdw dvvxpswlrq/ Dvvxpswlrq 4 dovr lpsrvhv d glvwulexwlrq rq wkh olplw ehkdylru ri 7^|Ewf Dvvxpswlrq 4 wkhuhiruh frpsohwho| fkdudfwhul}hv wkh olplw ehkdylru
ri wkh vwrfkdvwlf surfhvv iru 7s|Ew dqg 7^|Ew vr/ jlyhq wf dqg wkh lqvwuxphqwv/ zh fdq jhqhudwh
vwrfkdvwlf surfhvvhv wkdw dffrug zlwk Dvvxpswlrq 4 zlwkrxw pdnlqj dq| ixuwkhu dvvxpswlrqv1 6
Wkh wudglwlrqdo dvvxpswlrq pdgh lq JPP lv lqfrpsohwh vlqfh lw rqo| pdnhv dq dvvxpswlrq derxw wkh olplw ehkdylru ri wkh vwrfkdvwlf surfhvv iru s|Ewf zkloh 7^|Ew dovr uhvxowv iurp d
vwrfkdvwlf surfhvv1 Khqfh/ zh qhhg wr pdnh dgglwlrqdo dvvxpswlrqv/ e derxw 7^|Ewc zkhq zh
zdqw wr vlpxodwh iurp d vwrfkdvwlf surfhvv wkdw vdwlvhv wklv wudglwlrqdo dvvxpswlrq1 JPP lv ghyhorsshg dv d vwdwlvwlfdo surfhgxuh wkdw plplfv pd{lpxp olnholkrrg exw iru zklfk lw lv qrw qhfhvvdu| wr vshfli| wkh frpsohwh vwrfkdvwlf surfhvv1 Vlqfh Dvvxpswlrq 4 ixoo| vshflhv wkh mrlqw olplw ehkdylru ri erwk 7s|Ew dqg 7^|Ewc wkh JPP surfhgxuhv wkdw uhvxow iurp lw duh
pruh lq olqh zlwk pd{lpxp olnholkrrg wkdq wkh surfhgxuhv wkdw uhvxow iurp wkh wudglwlrqdo dvvxpswlrq1 Dvvxpswlrq 4 krogv xqghu plog frqglwlrqv vr qr sulfh lv sdlg iru wkhvh lpsuryhg JPP surfhgxuhv1
Dvvxpswlrq 4 lpsolhv d olplwlqj glvwulexwlrq iru wkh ghulydwlyh ri sAEwfc t Wkh olplwlqj
glvwulexwlrq rqo| krogv iru wkdw sduw ri wkh ghulydwlyh zlwk uhvshfw wr w zklfk olhv lq wkh vsdq
ri c E3
YsAEwfct
Yw
Lq wklv pdqqhu/ zh doorz iru d ghjhqhudwh olplwlqj glvwulexwlrq ri wkh ixoo ghulydwlyh ri sAEwfc t zlwk uhvshfw wr w Wklv ghjhqhudf| fdq/ iru h{dpsoh/ rffxu
zkhq wkh ghulydwlyh ri )Ewc t| zlwk uhvshfw wr w lv frpsohwho| vsdqqhg e| f| Wkh h{shfwhg
ydoxh ri wkh ghulydwlyh/ . dRc|Ewfo c lv wkhq htxdo wr wkh ghulydwlyh/ Rc|Ewfc vr 7Rc|Ewf ' f E|
vshfli|lqj wkh olplwlqj glvwulexwlrq rq 7^c|Ewfc zlwk 7Rc|Ewf ' 7^c|Ewf dqg dq dssursuldwho|
vshflhg &s & pdwul{ c zh dyrlg d ghjhqhudwh olplwlqj glvwulexwlrq1 Iru wklv h{dpsoh/ wkh
dssursuldwh vshflfdwlrq ri lv d }hur pdwul{ zlwk & ' f vr 7^c|Ewf grhv qrw h{lvw1 Zkhq
zh gluhfwo| vshfli| wkh olplwlqj glvwulexwlrq rq 7Rc|Ewf/ d ghjhqhudwh olplwlqj glvwulexwlrq dovr
rffxuv zkhq wkh ghulydwlyhv ri vhyhudo hohphqwv ri )Ewc t| zlwk uhvshfw wr w duh lghqwlfdo1 Zh
djdlq dyrlg wklv ghjhqhudf| e| vshfli|lqj d olplwlqj glvwulexwlrq rq 7^c|Ewf1 Wklv h{sodlqv zk|
zh xvh 7^|Ewf
6 Wkh Iluvw Rughu Ghulydwlyh dqg wkh N0vwdwlvwlf
614 Whvwlqj k|srwkhvhv rq doo sdudphwhuv
Lq rughu wr rewdlq dq hvwlpdwru iru wc lw lv fxvwrpdu| wr plqlpl}h wkh remhfwlyh ixqfwlrq 'Ew zlwk uhvshfw wr wc
w ' @h} 4?
wMU6'Ew +45,
Wkh rswlpdo ydoxh iru w lv wkhq rewdlqhg e| xvh ri wkh uvw rughu frqglwlrq +IRF,=
IRF= Y'EwYw mw ' f +46,
Lqvwhdg ri dqdo|}lqj wkh uvw rughu ghulydwlyh ri 'Ew lq wc zh dqdo|}h +
2 wlphv, lw lq wkh wuxh
ydoxh ri wc wf Wklv uvw rughu ghulydwlyh uhdgv
2Y'EwYw mwf ' sAEwfc t TssEwf3RAEwfc t
2 EsAEwfc t TssEwf3 sAEwfc t TssEwf3YyhfETYwss Ewmwfc
+47, zlwk RAEwfc t ' YsAYwEwct mwf 'SA|'R|Ewf dqg R|Ewf ' YsYw|Ew mwf Wkh wudglwlrqdo frqvwuxfwlrq
ri wkh olplwlqj glvwulexwlrq ri JPP hvwlpdwruv dqg whvw vwdwlvwlfv rqo| lqyroyhv wkh uvw hohphqw ri wkh uvw rughu ghulydwlyh +47,1 Wkh vhfrqg hohphqw lv ohiw dvlgh ehfdxvh lw/ xqghu wkh fxvwrpdu| dvvxpswlrq ri d {hg ixoo udqn ydoxh ri wkh Mdfreldq
awEwf ' *4A <"A SA|'.ER|EwfmU|c +48,
ydqlvkhv dv|pswrwlfdoo| zkhq zh vfdoh e| wkh dssursuldwh idfwru ri A/ vhh e} Kdqvhq +4<;5,/ Qhzh| dqg PfIdgghq +4<<7, dqg Vwrfn dqg Zuljkw +5333, Zkhq awEwf grhv qrw kdyh d {hg
ixoo udqn ydoxh/ wklv vhfrqg hohphqw grhv qrw ydqlvk dv|pswrwlfdoo| dqg lq xhqfhv wkh olplwlqj glvwulexwlrqv1 Lq rughu wr rewdlq d vwdwlvwlf zkrvh olplwlqj glvwulexwlrq lv lqvhqvlwlyh wr wkh ydoxh ri awEwfc zh wkhuhiruh xvh doo hohphqwv ri wkh uvw rughu ghulydwlyh +47,1
Zh rewdlq wkh uvw rughu ghulydwlyh +47, iurp wkh h{suhvvlrq iru TssEwf +8,/ vhh wkh Ds0
shqgl{ iru wkh frqvwuxfwlrq ri wkh ghulydwlyh/
2Y'EwYw mwf ' sAEwfc t TssEwf3(AEwfc t c +49,
zlwk
(AEwfc t ' RcAEwfc t TwscEwfTssEwf3sAEwfc t
R6cAEwfc t 6Twsc6EwfTssEwf3sAEwfc t c +4:,
dqg TwscEwf G & &sc ' c c 6c TwsEwf ' ETwscEwf Twsc6Ewfc RcAEwfc t G &s c
' c c 6c RAEwfc t ' ERcAEwfc t R6cAEwfc t Ohppd 41 Zkhq Dvvxpswlrq 4 krogv/ s A yhf A(AEwfc t awEw $ _ wsc I AsAEwfc t $_ sc +4;, zkhuh 'gldj+c c 6c ws ' w TwsEwfTssEwf3s dqg ws Efc TwwsEwfc s Efc TssEwfc +4<, zlwk TwwsEwf ' TwwEwf TwsEwfTssEwf3TswEwfc dqg ws lv lqghshqghqw ri s Surri1 vhh wkh Dsshqgl{1
Ohppd 4 vkrzv wkdw (AEwfc t lv dq hvwlpdwru ri wkh Mdfreldq awEwf zkrvh olplwlqj ehkdylru
lv lqghshqghqw ri wkh olplwlqj ehkdylru ri sAEwfc t Zh wkhuhiruh dqdo|}h wkh mrlqw olplwlqj
ehkdylru ri (AEwfc t dqg sAEwfc t iru glhuhqw olplw vhtxhqfhv ri awEwf
Ohppd 51 Xqghu Dvvxpswlrq 4/ wkh olplwlqj ehkdylru ri d vfdolqj idfwru wkdw ghshqgv rq A wlphv (AEwfc t lv lqghshqghqw iurp wkh olplwlqj ehkdylru ri IAsAEwfc t zkhq=
41 awEwf kdv d {hg ixoo udqn ydoxh/ udqnEawEwf ' 61
51 awEwf kdv d zhdn ydoxh vxfk wkdw awEwf ' awcAc awcA ' IAc G &s6 dqg udqnE ' 6
61 awEwf lv htxdo wr }hur1
Surri1 vhh wkh Dsshqgl{1
Xqghu fdvh 4 dqg 5 ri Ohppd 5/ wkh ydoxh ri wkh udqn ri awEwf dqg lv pd{lpdo dqg htxdo wr
61 Fdvh 6 ri Ohppd 5 vkrzv wkdw wkh olplwlqj ehkdylru ri +d vfdolqj idfwru wkdw ghshqgv rq A wlphv, (AEwfc t lv dovr lqghshqghqw ri wkh olplwlqj ehkdylru ri IAsAEwfc t zkhq awEwf
lv htxdo wr }hur1 Wkh ixoo udqn dvvxpswlrqv lq fdvhv 4 dqg 5 duh wkhuhiruh uhgxqgdqw dqg wkh lqghshqghqfh krogv iru dq| ydoxh ri awEwf dqg 1 Iru uhdvrqv ri euhylw| zh gr qrw h{solflwo|
ghulyh wkhvh uhvxowv1
Wkhruhp 4 Xqghu Dvvxpswlrq 4/ wkh olplwlqj ehkdylru ri wkh qrupdol}hg uvw rughu ghulyd0
wlyh ri 'Ew lq wf uhdgv 2IA Y'Ew Yw mwf E(AEwfc t TssEwf3(AEwfc t 3 2 $ _ '_wc +53, zkhuh '_w G 6 dqg '_w Efc U6 +54,
Wkh olplwlqj ehkdylru lq +53, lv lghqwlfdo xqghu fdvhv 406 iru awEwf iurp Ohppd 51
Surri1 vhh wkh Dsshqgl{1 Ghqlwlrq 41 Wkh N0vwdwlvwlf iru whvwlqj Kf G w ' wf uhdgv NEwf ' eA Y'Ew Yw mwf d(AEwfc t TssEwf3(AEwfc t o3 Y'Ew Yw mwf ' AsAEwfc t TssEwf3 2 TssEwf3 2(AEwfct TssEwf 3 2sAEwfc t +55,
dqg kdv xqghu Kf dqg dvvxpswlrq 4 d 2E6 olplwlqj glvwulexwlrq iru doo wkuhh fdvhv ri awEwf
iurp Ohppd 51
Wkh N0vwdwlvwlf iurp Ghqlwlrq 4 kdv dq lghqwlfdo ixqfwlrqdo irup dv wkh JPP OP ru vfruh vwdwlvwlf ri Qhzh| dqg Zhvw +4<;:e,1 Lghqwlfdo wr wkdw vwdwlvwlf/ lw lv lqyduldqw wr wkh vshflfdwlrq ri Kf Wkh rqo| glhuhqfh zlwk wkh JPP OP vwdwlvwlf frqfhuqv wkh Mdfreldq
hvwlpdwru Wkh JPP OP vwdwlvwlf xvhv wkh h{shfwhg Mdfreldq hvwlpdwru RAEwfc t zkloh wkh
N0vwdwlvwlf xvhv (AEwfc t Wkh olplwlqj ehkdylru ri RAEwfc t lv qrw lqghshqghqw ri wkh olplwlqj
ehkdylru ri sAEwfc t vr wkh olplwlqj glvwulexwlrq ri wkh JPP OP vwdwlvwlf ghshqgv rq qxlvdqfh
sdudphwhuv xqghu fdvhv 506 ri Ohppd 5/ vhh Vwrfn dqg Zuljkw +5333,1 Wkh Mdfreldq hvwlpdwru wkdw lv lqghshqghqw ri wkh prphqw htxdwlrqv kdv dovr ehhq sursrvhg lq Eurzq dqg Qhzh| +4<<;, wr lpsuryh wkh h!flhqf| ri vhpl0sdudphwulf hvwlpdwruv1
Wkh N0vwdwlvwlf lv htxdo wr }hur dw wkh hvwlpdwru iru w wkdw vdwlvhv wkh IRF +46,1 Wkh hvwlpdwru wkdw vdwlvhv wkh IRF lv wkh FXH ri Kdqvhq e| @, +4<<9,1 Khqfh/ lqihuhqfh wkdw uhvxowv iurp wkh N0vwdwlvwlf lv fhqwhuhg durxqg wkh FXH1 Rwkhu dsshdolqj ihdwxuhv ri wkh FXH kdyh ehhq phqwlrqhg hovhzkhuh1 Iru h{dpsoh/ Qhzh| dqg Vplwk +5334, vkrz wkdw wkh FXH kdv d vpdoohu eldv wkdq rwkhu JPP hvwlpdwruv1 Grqdog dqg Qhzh| +5333, vkrz wkdw wkh FXH lqyroyhv d Mdfreldq pdwul{ hvwlpdwru wkdw lv d 04 Mdfnnqlih hvwlpdwru1 Lw lv wkhuhiruh dv|pswrwlfdoo| lqghshqghqw ri wkh prphqw htxdwlrqv1
615 Whvwlqj k|srwkhvhv rq vxe0vhwv ri wkh sdudphwhuv
Wkh N0vwdwlvwlf iurp Ghqlwlrq 4 frqgxfwv d mrlqw whvw rq doo hohphqwv ri w Zkhq w frqwdlqv vhyhudo hohphqwv/ iru h{dpsoh/ w ' Ek qc zlwk k G 6
k dqg q G 6q c 6 ' 6kn 6qc
zh fdq dgdsw wkh N0vwdwlvwlf wr whvw d k|srwkhvlv vshflhg rq d vxe0vhw ri wkh sdudphwhuv/ KW
f G q ' qf Lq rughu wr frqvwuxfw wkh olplwlqj glvwulexwlrq iru wklv vwdwlvwlf/ zh pdnh dq
dgglwlrqdo dvvxpswlrq/ vhh dovr Nohlehujhq +5333,1
Dvvxpswlrq 51 Wkh &s 6k glphqvlrqdo Mdfreldq pdwul{
akEkc q ' *4A <"A SA|'. Ys|Ekcq Yk kcqmU| c +56, 9
lv d frqwlqxrxv ixqfwlrq ri Ekc q dqg kdv ixoo udqn 6k lq Ekfc qf
Xqghu Dvvxpswlrqv 405/ wkh hvwlpdwru wkdw vroyhv wkh IRF zlwk uhvshfw wr k jlyhq qfc
kEqfc lv d frqvlvwhqw hvwlpdwru ri kf dqgsAEkEqf kf kdv d qrupdo olplwlqj glvwulexwlrq/
vhh e} Vwrfn dqg Zuljkw +5333,1 Zh fdq wkhuhiruh dqdo|}h wkh olplwlqj ehkdylru ri wkh ghulydwlyh 'Ew zlwk uhvshfw wr w lq wf ' EkEqf qf wkdw lv htxdo wr
2Y'EwYwfmwf ' sAEwfc t TssEwf3(AEwfc t ' f sAEwfc t TssEwf32 TssEwf3 2(kcAEwfct TssEwf 3 2(qcAEwfc t +57,
zlwk (AEwfc t ' E(kcAEwfc t (qcAEwfc t c (kcAEwfc t G &s 6k dqg (qcAEwfc t G &s 6qc
dqg zklfk uhvxowv vlqfh sAEwfc t TssEwf3(kcAEwfc t ' f
Wkhruhp 5 Xqghu Dvvxpswlrqv 405/ wkh olplwlqj ehkdylru ri wkh qrupdol}hg uvw rughu ghuly0
dwlyh ri 'Ew zlwk uhvshfw wr q lq wf ' EkEqf qf uhdgv 2IA Y'Ew Yq mwf k (qcAEwfc t TssEwf32 TssEwf3 2(kcAEwfct TssEwf 3 2(qcAEwfc t l3 2 $ _ '_qc +58, zkhuh '_q G 6q dqg '_q fc U6q +59,
Wkh olplwlqj ehkdylru lq +58, lv lghqwlfdo xqghu fdvhv 406 iru
aqEwf ' *4A <"A SA|'. Ys|Ekcq Yq kEqfcqf mU| c +5:, iurp Ohppd 51
Surri1 uhvxowv gluhfwo| iurp Ohppdv 405 dqg Dvvxpswlrqv 4051 Ghqlwlrq 51 Wkh N0vwdwlvwlf iru whvwlqj KW f G q ' qf uhdgv NEqf ' eA Y'Ew Yw mwf d(AEwfc t TssEwf3(AEwfc t o3 Y'Ew Yw mwf c +5;,
zkhuh wf ' EkEqf qfc dqg kdv xqghu KWf dqg Dvvxpswlrqv 405 d 2E6q olplwlqj glvwulexwlrq
iru doo wkuhh fdvhv ri aqEwf iurp Ohppd 51
Frpsduhg wr N0vwdwlvwlf +55,/ wkh h{suhvvlrq ri wkh N0vwdwlvwlf uhpdlqv xqdowhuhg exw wkh olplwlqj glvwulexwlrq ri +5;, kdv ohvv ghjuhhv ri iuhhgrp1 Lw vkrzv wkdw zh fdq frqgxfw whvwv rq wkrvh vxe0vhwv ri wkh sdudphwhuv iru zklfk Dvvxpswlrq 5 krogv iru wkh uhpdlqlqj sdudphwhuv1 Zkhq Dvvxpswlrq 5 grhv qrw krog/ wkh olplwlqj glvwulexwlrq ri +5;, ghshqgv rq qxlvdqfh sdudphwhuv dqg zh vkrxog xvh +55, lqvwhdg1
Lw lv qrw dozd|v vwudljkwiruzdug wr ghwhuplqh wkh sdudphwhuv iru zklfk Dvvxpswlrq 5 lv vdwlvhg1 Dvvxpswlrq 5 lv dozd|v vdwlvhg iru wkrvh sdudphwhuv w iru zklfk 7RcAEwfc t htxdov
}hur1 Zh qrwh wkdw rqh kdv wr eh fduhixo zlwk frqgxfwlqj suh0whvwv iru wkh udqn ri akEkc q
ehfdxvh vxfk suh0whvwv fdq dhfw wkh vl}h ri wkh vwdwlvwlfv frpsxwhg frqvhfxwlyho|1
7 Pdq| lqvwuxphqwv dqg fryduldqfh pdwul{ hvwlpdwruv
Wkh ixqfwlrqdo h{suhvvlrqv ri N0vwdwlvwlfv +55, dqg +5;, ghshqg rq xqnqrzq fryduldqfh sd0 udphwhuv1 Zkhq zh xvh frqvlvwhqw hvwlpdwruv iru wkhvh fryduldqfh sdudphwhuv/ wkh olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf grhv qrw dowhu1 Wkhvh hvwlpdwruv fdq eh erwk sdudphwulf ru qrq0sdudphwulf1 Wkh frqyhujhqfh udwh ri wkh fryduldqfh pdwul{ hvwlpdwruv wr wkh wuxh ydoxh lv ri lpsruwdqfh zkhq zh dqdo|}h wkh olplwlqj ehkdylru ri wkh N0vwdwlvwlf lq d olplwlqj vhtxhqfh zkhuh wkh vdpsoh vl}h dqg qxpehu ri lqvwuxphqwv mrlqwo| frqyhujh wr lqqlw|1 Zh uhihu wr vxfk d olplwlqj vhtxhqfh dv pdq| lqvwuxphqwv dv|pswrwlfv/ vhh e} Ehnnhu +4<<7, dqg Grqdog dqg Qhzh| +5334,1 Wkhruhp 6 vwdwhv wkh uhodwlrqvkls ehwzhhq wkhvh frqyhujhqfh udwhv1
Wkhruhp 6 Zkhq wkh frqyhujhqfh udwh ri wkh fryduldqfh pdwul{ hvwlpdwruv lqyroyhg lq wkh
N0vwdwlvwlf +55, lv htxdo wr A> dqg wkh qxpehu ri lqvwuxphqwv & dqg wkh vdpsoh vl}h A mrlqwo|
frqyhujh wr lqqlw|/ wkh N0vwdwlvwlf +55, kdv d 2E6 olplwlqj glvwulexwlrq xqghu doo fdvhv ri
Ohppd 5 zkhq wkh mrlqw olplwlqj vhtxhqfh ri & dqg A lv vxfk wkdw &
A2> $ f
Surri1 vhh wkh Dsshqgl{1
Wkhruhp 6 vkrzv wkh urexvwqhvv ri wkh olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf wr odujh qxpehuv ri lqvwuxphqwv1 Wkh olplwlqj glvwulexwlrqv ri vwdqgdug JPP Zdog/ OP dqg olnholkrrg udwlr vwdwlvwlfv ghshqg rq qxlvdqfh sdudphwhuv zkhq wkh qxpehu ri lqvwuxphqwv frqyhujhv wr lqqlw|/ vhh e} Ehnnhu +4<<7,1 Iru wkh N0vwdwlvwlf/ Wkhruhp 6 vkrzv wkdw lq fdvh ri d sdudphwulf fryduldqfh pdwul{ hvwlpdwru wkdw kdv d ydoxh ri > htxdo wr
2c wkh qxpehu ri
lqvwuxphqwv dqg wkh vdpsoh vl}h vkrxog frqyhujh lq vxfk d pdqqhu wkdw &
A $ f Wklv frqglwlrq
lv lghqwlfdo wr wkh rqh lq Grqdog dqg Qhzh| +5334,/ zklfk wkh| xvh wr rewdlq dq h{suhvvlrq iru wkh phdq vtxduhg huuru ri wkh olplwhg lqirupdwlrq pd{lpxp olnholkrrg hvwlpdwru lq wkh olqhdu lqvwuxphqwdo yduldeohv uhjuhvvlrq prgho1 Ehnnhu dqg Nohlehujhq +5335, dqdo|}h wkh olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf lq d olqhdu lqvwuxphqwdo yduldeohv uhjuhvvlrq prgho xvlqj d sdudphwulf fryduldqfh pdwul{ hvwlpdwru dqg d olplwlqj vhtxhqfh zkhuh &
A $ Bc zlwk f B
Wkh| vkrz wkdw wkh olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf wkhq glhuv ryhu wkh vhsdudwh fdvhv ri Ohppd 51 Lq fdvh 4/ wkh N0vwdwlvwlf frqyhujhv wr d 2E6 udqgrp yduldeoh zkloh lq fdvh 6/ lw
frqyhujhv wr
3B wlphv d 2E6 udqgrp yduldeoh1 Wkh olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf
lq wkhvh wzr fdvhv surylgh erxqgv rq wkh olplwlqj glvwulexwlrq lq doo lqwhuphgldwh fdvhv1 Qrq0sdudphwulf fryduldqfh pdwul{ hvwlpdwruv kdyh frqyhujhqfh udwhv wkdw duh vxfk wkdw > lv vpdoohu wkdq
2c vhh e} Dqguhzv +4<<4,/ Qhzh| dqg Zhvw +4<;:d, dqg Zklwh +4<;3,1 Wklv
vkrzv wkh lpsruwdqfh ri wkh frqyhujhqfh udwh ri wkh fryduldqfh pdwul{ hvwlpdwru xqghu pdq| lqvwuxphqwv dv|pswrwlfv1
8 Lpsurylqj srzhu wrzdugv luuhohydqw dowhuqdwlyhv
Wkh N0vwdwlvwlf htxdov d txdgudwlf irup ri wkh uvw rughu ghulydwlyh ri wkh JPP remhfwlyh ixqfwlrq1 Zkhq frqvlghuhg dv d ixqfwlrq ri uhdol}hg gdwd/ lw lv wkhuhiruh htxdo wr }hur dw wkrvh ydoxhv ri w zkhuh wkh JPP remhfwlyh ixqfwlrq dwwdlqv lwv plqlpxp/ pd{lpxp ru kdv dq lq h{lrq srlqw1 Wkh FXH jlyhv wkh ydoxh ri w zkhuh wkh remhfwlyh ixqfwlrq lv plqlpdo1 Wkh N0vwdwlvwlf lv wkhuhiruh htxdo wr }hur dw wkh FXH1 Wkh }hur ydoxhv ri wkh N0vwdwlvwlf dw wkh pd{lpdo ydoxh ri wkh remhfwlyh ixqfwlrq dqg lq h{lrq srlqwv duh hvvhqwldoo| luuhohydqw ehfdxvh wkh prphqw frqglwlrqv duh ylrodwhg dw wkhvh ydoxhv ri w Wkh| fdq/ krzhyhu/ glvwruw glvfulplqdwru| srzhu dqg frqghqfh vhwv1 Zh xvh wkh pdsslqj iurp wkh prphqw htxdwlrq hvwlpdwru sAEwc t rq wkh N0vwdwlvwlf wr dqdo|}h wklv1
Zh ghfrpsrvh wkh qrupdol}hg prphqw htxdwlrq hvwlpdwru dv
TssEw32sAEwc t ' TssEw32(AEwc t }AEwc t n TssEw2(AczEwc t AEwc t c +5<,
zlwk (AczEwc t G &s E&s 6c (AczEwc t (AEwc t fc (AczEwc t (AczEwc t U&s36 dqg
}AEwc t dqg AEwc t duh 6 dqg E&s 6 yhfwruv vr
}AEwc t ' E(AEwc t TssEw3(AEwc t 3(AEwc t TssEw3sAEwc t
AEwc t ' E(AczEwc t TssEw(AczEwc t 3(AczEwc t sAEwc t +63,
Wkh txdgudwlf irup ri wkh qrupdol}hg prphqw htxdwlrq hvwlpdwru +5<, frqvwlwxwhv Vwrfn dqg Zuljkwv +5333, V0vwdwlvwlf/
AsAEwc t TssEw3sAEwc t ' A}AEwc t (AEwc t TssEw3(AEwc t }AEwc t
n
AAEwc t (AczEwc t TssEw(AczEwc t AEwc t /
VEw ' NEw n MEwc +64,
vlqfh wkh N0vwdwlvwlf lv d txdgudwlf irup ri }AEwc t dqg wkh M0vwdwlvwlf4/
MEw '
AsAEwc t TssEw3
2
TssEw3 2(AEwct TssEw
3
2sAEwc t c +65,
lv d txdgudwlf irup ri AEwc t Ehfdxvh ri wkh ruwkrjrqdolw| ri (AEwc t dqg (AczEwc t c
xqghu Kf G w ' wf dqg Dvvxpswlrq 4/ wkh N dqg M vwdwlvwlfv frqyhujh wr lqghshqghqw 2E6
dqg 2E&
s 6 glvwulexwhg udqgrp yduldeohv1 Wkh V0vwdwlvwlf wkhq frqyhujhv wr d 2E&s
glvwulexwhg udqgrp yduldeoh1
Wkh uhodwlrqvkls ehwzhhq wkh V/ N dqg M vwdwlvwlfv vkrzv wkdw wkh M0vwdwlvwlf lv htxdo wr wkh V0vwdwlvwlf zkhq wkh N0vwdwlvwlf lv htxdo wr }hur1 Wkh N0vwdwlvwlf vxhuv iurp d ghfolqh lq srzhu durxqg ydoxhv ri w zkhuh wkh remhfwlyh ixqfwlrq/ e wkh V0vwdwlvwlf/ lv pd{lpdo ru kdv dq lq h{lrq srlqw1 Iru wkhvh ydoxhv ri wc wkh M0vwdwlvwlf lv +dssur{lpdwho|, htxdo wr wkh V0vwdwlvwlf vr lw kdv glvfulplqdwru| srzhu1 Wkh M0vwdwlvwlf kdv glvfulplqdwru| srzhu ehfdxvh lw whvwv wkh ydolglw| ri wkh prphqw htxdwlrqv zkloh wkh N0vwdwlvwlf whvwv Kf G w ' wf jlyhq
wkdw wkh prphqw htxdwlrqv krog/ vhh Nohlehujhq +5335e,1 Durxqg wkh ydoxhv ri w zkhuh wkh remhfwlyh ixqfwlrq lv pd{lpdo ru kdv dq lq h{lrq srlqw/ wkh prphqw htxdwlrqv gr qrw krog vr wkh uhvxowv iurp wkh N0vwdwlvwlf duh vsxulrxv1 Zh sursrvh wzr pdqqhuv lq zklfk wkh M0vwdwlvwlf fdq eh frpelqhg zlwk wkh N0vwdwlvwlf wr ryhufrph vxfk vsxulrxv rxwfrphv ri wkh N0vwdwlvwlf1
Wkh olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf lv ydolg xqghu Dvvxpswlrq 4 dqg Kf G w ' wf
Dvvxpswlrq 4 lpsolhv wkdw wkh prphqw htxdwlrq +5, krogv1 Wr fkhfn zkhwkhu wkh prphqw htxdwlrq krogv/ zh frqgxfw d suh0whvw xvlqj wkh M0vwdwlvwlf +65,1 Vlqfh wkh M0vwdwlvwlf lv dv|ps0 wrwlfdoo| lqghshqghqw ri wkh N0vwdwlvwlf/ wkh ryhudoo vl}h k ri whvwlqj wkh prphqw htxdwlrqv dqg Kf htxdov wkh vxp ri wkh vl}hv wkdw zh xvh wr whvw wkh prphqw htxdwlrqv xvlqj wkh M0vwdwlvwlf/
kMc dqg Kf xvlqj wkh N0vwdwlvwlf/ kNc vr k ' kMn kN E| fkrrvlqj kM dqg kN dssursuldwho|/
zh hpskdvl}h whvwv ri wkh prphqw htxdwlrqv ru Kf Iru h{dpsoh/ zkhq k ' ffDc kM ' ff
dqg kN ' ffe lpsolhv wkdw zh irfxv rq Kf1 Wkh suh0whvw xvlqj wkh M0vwdwlvwlf uhvwulfwv wkh
sdudphwhu uhjlrq iru w wr ydoxhv wkdw vdwlvi| wkh prphqw htxdwlrqv ru vwdwhg glhuhqwo| lw sxwv dq xsshuerxqg rq wkh ydoxh ri wkh remhfwlyh ixqfwlrq1
Wkh vsxulrxv srzhu ghfolqh ri wkh N0vwdwlvwlf lv dovr h{solfdeoh xvlqj vshflfdwlrq +5<,1 Wklv lv d uhjuhvvlrq prgho lq zklfk (AEwc t dqg (AczEwc t duh uhjuhvvhg rq sAEwc t D
4Zh qrwh wkdw wklv M0vwdwlvwlf lv qrw wkh M0vwdwlvwlf ri Kdqvhq +4<;5, wkdw lv sursruwlrqdo wr wkh remhfwlyh
ixqfwlrq hydoxdwhg dw d JPP hvwlpdwru iru =
ydoxh ri (AEwc t wkdw lv forvh wr d uhgxfhg udqn ydoxh fdxvhv d pxowl0froolqhdulw| sureohp1
]hur ydoxhv ri wkh N0vwdwlvwlf dw vxfk ydoxhv ri (AEwc t duh wkhuhiruh fdxvhg e| wkh ydoxh
ri (AEwc t lqvwhdg ri wkh prphqw htxdwlrq sAEwc t Wklv h{sodlqv wkh }hur ydoxhv ri wkh
N0vwdwlvwlf dw ydoxhv ri w glhuhqw iurp wkh FXH1 Khqfh/ iru ydoxhv ri (AEwc t forvh wr
uhgxfhg udqn/ wkh N0vwdwlvwlf grhv qrw frqwdlq pxfk lqirupdwlrq rq zkhwkhu sAEwc t olhv lq
wkh gluhfwlrq ri (AEwc t dqg lw lv pruh dssursuldwh wr whvw sAEwc t xvlqj wkh V0vwdwlvwlf1 Wkh
+frqglwlrqdo, olplwlqj glvwulexwlrq ri wkh olnholkrrg udwlr vwdwlvwlf iru whvwlqj Kf lq d olqhdu
lqvwuxphqwdo yduldeohv uhjuhvvlrq prgho zlwk rqh lqfoxghg hqgrjhqrxv yduldeoh rshudwhv lq wklv pdqqhu1 Pruhlud +5334, frqvwuxfwv wkh olplwlqj glvwulexwlrq ri wkh OU vwdwlvwlf zklfk lv frqglwlrqdo rq (AEwc t TwwsEw3(AEwc t / zklfk lv d vfdodu zkhq 6 ' 1 Iru odujh ydoxhv
ri (AEwc t TwwsEw3(AEwc t c wkh olplwlqj glvwulexwlrq ri wkh OU vwdwlvwlf lv lghqwlfdo wr wkh
olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf zkloh lw lv lghqwlfdo wr wkh olplwlqj glvwulexwlrq ri wkh V0vwdwlvwlf iru }hur ydoxhv ri (AEwc t TwwsEw3(AEwc t 1
Nohlehujhq +5335e, dqdo|}hv wkh olplwlqj glvwulexwlrq ri wkh OU vwdwlvwlf iru whvwlqj Kf
lq d olqhdu lqvwuxphqwdo yduldeohv uhjuhvvlrq prgho zlwk pxowlsoh hqgrjhqrxv yduldeohv dqg d Nurqhfnhu vshflfdwlrq iru T Ewc T Ew ' ElEw 'ff zlwk lEw G , ,c 'ff G & & Wklv
olplwlqj glvwulexwlrq lv frqglwlrqdo rq wkh 6 hljhqydoxhv ri lwwsEw32(AEwc t '3ff(AEwc t
lwwsEw32c zkhuh l lv ghfrpsrvhg lq dq lghqwlfdo pdqqhu dv T Zkhq wkh vpdoohvw hljhqydoxh
frqyhujhv wr lqqlw|/ wkh olplwlqj glvwulexwlrq ri wkh OU vwdwlvwlf ehfrphv lghqwlfdo wr wkh 2E6
olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf zkloh lw lv lghqwlfdo wr wkh 2E&
s olplwlqj glvwulexwlrq
ri wkh V0vwdwlvwlf zkhq wkh vpdoohvw hljhqydoxh htxdov }hur1 Khqfh/ wkh vpdoohvw hljhqydoxh ri lwwsEw32(AEwc t '3ff(AEwc t lwwsEw32 jryhuqv wkh ehkdylru ri wkh olplwlqj glvwulexwlrq
ri wkh OU vwdwlvwlf1 D xsshuerxqg rq wkh glvwulexwlrq ri wkh OU vwdwlvwlf lv wkhuhiruh surylghg zkhq zh frqvlghu doo hljhqydoxhv wr eh htxdo wr wkh vpdoohvw rqh1 Wklv xsshuerxqg lv hvvhqwldoo| txlwh wljkw/ vhh Nohlehujhq +5335e,1 Zkhq doo hljhqydoxhv duh lghqwlfdo wr wkh vpdoohvw rqh/ wkh h{suhvvlrq iru wkh vpdoohvw urrw ri wkh sro|qrpldo iurp zklfk wkh OU vwdwlvwlf uhvxowv lv lghqwlfdo wr wkh h{suhvvlrq iru wkh rqh hqgrjhqrxv yduldeoh fdvh1 Khqfh/ wkh h{suhvvlrq ri wkh frqglwlrqdo OU vwdwlvwlf iru wkh rqh hqgrjhqrxv yduldeoh fdvh iurp Pruhlud +5334, surylghv d wljkw xsshuerxqg rq wkh OU vwdwlvwlf lq wkh pxowlsoh hqgrjhqrxv yduldeoh fdvh1 Lw lv wkhuhiruh frqyhqlhqw wr dsso| Pruhlud*v frqglwlrqdo OU vwdwlvwlf dovr lq wkh pxowlsoh hqgrjhqrxv yduldeoh fdvh iru zklfk qr forvhg irup h{suhvvlrq ri wkh OU vwdwlvwlf h{lvwv1
Wkh vpdoohvw hljhqydoxh frqvwlwxwhv wkh fdqrqlfdo fruuhodwlrq udqn vwdwlvwlf ri Dqghuvrq +4<84, wkdw whvwv iru d uhgxfhg udqn ri awEw Zh h{whqg wkh OU vwdwlvwlf wrzdugv d JPP vhw0
wlqj/ vxfk wkdw zh doorz iru d qrq0Nurqhfnhu T Ew dqg d fxvwrpdu| vshflhg prphqw htxdwlrq sAEwc t c e| xvlqj udqn vwdwlvwlfv wkdw dsso| pruh jhqhudo wkdq wkh Dqghuvrq fdqrqlfdo fruuh0
odwlrq udqn vwdwlvwlf1 Zh wkhuhiruh pdnh wkh JPP0OU vwdwlvwlf ghshqghqw rq udqn vwdwlvwlfv wkdw dsso| lq JPP1 Zh lqvhuw wkhvh udqn vwdwlvwlfv lq wkh h{suhvvlrq ri Pruhlud*v frqglwlrqdo OU vwdwlvwlf zkrvh olplwlqj glvwulexwlrq surylghv d wljkw xsshuerxqg rq wkh olplwlqj glvwule0 xwlrq ri wkh OU vwdwlvwlf lq wkh olqhdu lqvwuxphqwdo yduldeohv uhjuhvvlrq prgho zlwk pxowlsoh hqgrjhqrxv yduldeohv=
JPP0P+w '
2dNEw n M+w un+(AEwc t n
t
ENEw n M+w n un+(AEwc t 2 eM+wun+(AEwc t
+66,
Wkh vwdwlvwlf un+(AEwc t whvwv KoG udqn ri awEw htxdov 6 xvlqj (AEwc t dqg &s6n
lv wkh ghjuhhv ri iuhhgrp sdudphwhu ri wkh 2 olplwlqj glvwulexwlrq ri un+(
AEwc t Ghshqglqj
rq wkh vshflfdwlrq ri wkh fryduldqfh pdwul{/ ydulrxv udqn vwdwlvwlfv zlwk 2E&
s 6 n
olplwlqj glvwulexwlrqv fdq eh xvhg1 Zkhq wkh fryduldqfh pdwul{ lv lq Nurqhfnhu surgxfw irup/ zh fdq xvh wkh Dqghuvrq +4<84, fdqrqlfdo fruuhodwlrq udqn vwdwlvwlf1 Li wkh fryduldqfh pdwul{ grhv qrw kdyh d Nurqhfnhu surgxfw irup/ zh fdq/ iru h{dpsoh/ xvh wkh vwdwlvwlfv iurp Fudjj dqg Grqdog +4<<9/4<<:, ru Nohlehujhq dqg Sdds +5335,1
Xqghu Kf dqg Dvvxpswlrq 4/ wkh olplwlqj glvwulexwlrq ri JPP0P+w lv frqglwlrqdo rq
un+(AEwc t Zkhq un+(AEwc t lv odujh/ wkh olplwlqj glvwulexwlrq ri wkh JPP0P vwdwlvwlf
lv lghqwlfdo wr wkh 2E6 olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf zkloh lw lv lghqwlfdo wr wkh
2E&
s olplwlqj glvwulexwlrq ri wkh V0vwdwlvwlf zkhq un+(AEwc t htxdov }hur1 Ohppd 4 lpsolhv
wkdw wkh olplwlqj glvwulexwlrq ri (AEwc t lv lqghshqghqw ri wkh olplwlqj glvwulexwlrqv ri wkh
M dqg N vwdwlvwlfv1 Wkh olplwlqj glvwulexwlrq ri un+(AEwc t lv wkhuhiruh dovr lqghshqghqw
ri wkh olplwlqj glvwulexwlrqv ri wkh M dqg N vwdwlvwlfv1 Khqfh/ zh fdq vlpxodwh wkh olplwlqj glvwulexwlrq ri JPP0P+wc xqghu Kf dqg Dvvxpswlrq 4/ jlyhq un+(AEwc t
9 Frqghqfh Vhwv
Wr rewdlq d frqghqfh vhw iru wc zh vshfli| vhtxhqfhv ri ? lqfuhdvlqj ydoxhv iru hyhu| hohphqw ri w Zh wkhq kdyh d 60glphqvlrqdo julg wkdw frqwdlqv ?6 glhuhqw ydoxhv ri w
f Zh frpsxwh wkh
vwdwlvwlf ri lqwhuhvw/ e wkh M/ N/ JPP0P/ ru V vwdwlvwlf/ iru hdfk ri wkhvh ?6 glhuhqw ydoxhv
ri wf Doo hohphqwv lq wkh vshflhg julg iru zklfk wkh dv|pswrwlf R0ydoxh ri wkh vwdwlvwlf ri
lqwhuhvw h{fhhgv k duh lq wkh EkffI frqghqfh vhw ri w1 Xvlqj dq dssursuldwh vshflfdwlrq ri wkh julg/ zh rewdlq wkh E kffI dv|pswrwlf frqghqfh vhw1 Iru wkh frpelqhg M0N whvw/ wkh E kffI frqghqfh vhw wkdw xvhv kN dqg kM lv rewdlqhg dv wkh vhw ri ydoxhv iru zklfk
wkh dv|pswrwlf R0ydoxhv ri wkh M dqg N vwdwlvwlfv h{fhhgv kM dqg kN uhvs11
Dorqjvlgh lqqlwh dqg qlwh frqyh{ frqghqfh vhwv/ wkh frqghqfh vhwv wkdw uhvxow iurp wkh sursrvhg whvwv fdq dovr eh qrq0frqyh{/ vhh e} Gxirxu +4<<:,1 Wkh frqghqfh vhwv wkdw uhvxow iurp wkh V/ M dqg frpelqhg M0N vwdwlvwlfv fdq eh hpsw| zkloh wkh frqghqfh vhwv wkdw uhvxow iurp wkh N dqg JPP0P vwdwlvwlfv duh qhyhu hpsw| vlqfh wkhvh vwdwlvwlfv duh htxdo wr }hur dw wkh FXH1
: Vwrfkdvwlf Glvfrxqw Idfwruv
Zh whvw wkh ulvn dyhuvlrq sdudphwhu lq d vwrfkdvwlf glvfrxqw idfwru +VGI, prgho1 Zh frpsxwh srzhu fxuyhv iru vrph vlpxodwhg gdwd0vhwv dqg frqvwuxfw frqghqfh vhwv xvlqj revhuyhg dvvhw uhwxuqv dqg frqvxpswlrq jurzwk1
Vwrfkdvwlf Glvfrxqw Idfwru prgho
Wkh vshflfdwlrq ri )Ewc t| lq wkh Hxohu htxdwlrq +4, wkdw uhvxowv iurp d VGI prgho zlwk
d frqvwdqw uhodwlyh udwh ri ulvn dyhuvlrq +FUUD, xwlolw| ixqfwlrq uhdgv/ vhh e} Kdqvhq dqg Vlqjohwrq +4<;5,/ )Ewc t| ' B |n | 3q E ,n -|n , +67,
Wkh glvfrxqw idfwru lv jlyhq e| B dqg q lv wkh ulvn dyhuvlrq frh!flhqw vr w ' EB q Wkh ,
yhfwru -| frqwdlqv wkh uhwxuqv rq , glhuhqw dvvhwv dw wlph |/ | lv frqvxpswlrq dw wlph | dqg
, lv d , yhfwru ri rqhv1 Zh xvh d & lqvwuxphqw yhfwru f| wkdw frqvlvwv ri d frqvwdqw
dqg odjjhg ydoxhv ri frqvxpswlrq jurzwk dqg dvvhw uhwxuqv1 Ehfdxvh B lv d vfdodu/ +67, lv
htxlydohqw wr )Ewc t| ' |n | 3q E ,n -|n ,kc +68,
zlwk k ' B3 Wkh ghulydwlyh ri +68, zlwk uhvshfw wr k htxdov
, dqg lv vsdqqhg e| wkh
lqvwuxphqwv1 Dvvxpswlrq 5 lv wkxv dozd|v vdwlvhg iru k dqg Dvvxpswlrq 4 grhv wkhuhiruh qrw lpso| d fhqwudo olplw wkhruhp iru wkh ghulydwlyh ri )Ewc t| zlwk uhvshfw wr k G
I A [A |' s|Ewf 7^Ewf $ _ s q c +69,
zlwk s|Ew ' E)Ewc t| f|c ^Ew ' EYqY )Ewc t| f| ' E*L}E|n| E|n| 3qE ,n -|n f|c
7^Ew ' ^Ew.E^Ew dqg E
s Efc T Wkh fhqwudo olplw wkhruhp lq +69, uhvxowv iurp d
vwdwlvwlfdo prgho iru )Ewc t| dqg YqY )Ewc t| wkdw h{sodlqv wkhvh wzr vhulhv iurp f| Wklv vkrzv
wkdw Dvvxpswlrq 4 sxwv qr vwulqjhqw frqglwlrqv rq wkh dvvhw uhwxuq dqg frqvxpswlrq jurzwk vhulhv1
Vl}h dqg srzhu frpsdulvrq
Zh jhqhudwh wkh duwlfldo gdwd xvlqj wkh Prqwh0Fduor ghvljq ri Wdxfkhq +4<;9d/e,1 Zh fdoleudwh d 432 glphqvlrqdo Pdunry fkdlq wr dssur{lpdwh d Jdxvvldq Yhfwru DxwrUhjuhvvlrq
+YDU, ri rughu 4 wwhg wr frqvxpswlrq dqg glylghqg jurzwk1 Wklv YDU+4, lv dovr xvhg e| Nrfkhuodnrwd +4<<3,/ Kdqvhq e| @, +4<<9, dqg Vwrfn dqg Zuljkw +5333,=
S| _| ' ff2 fffe n fS ff. fee f. S|3 _|3 n 0Sc| 0_c| c +6:,
zlwk S| wkh orj0jurzwk udwh ri XV shu fdslwd uhdo dqqxdo frqvxpswlrq dqg _| wkh orj0jurzwk
udwh ri uhdo dqqxdo glylghqgv rq wkh Vwdqgdug ) Srru 8331 Wkh glvwxuedqfhv E0Sc| 0_c| duh
lqghshqghqwo| qrupdoo| glvwulexwhg zlwk phdq }hur dqg ydu+0Sc| ' ffec ydu+0Sc| ' fff2c
fru+0Sc|c 0_c| ' fe
Wkh YDU+4, +6:, lv xvhg wr jhqhudwh dvvhw uhwxuq dqg frqvxpswlrq jurzwk vhulhv wkdw vdwlvi| d VGI prgho1 Iljxuh 4 vkrzv wkh srzhu fxuyhv ri whvwlqj Kf G q ' xvlqj Zdog
vwdwlvwlfv edvhg rq wkh FXH dqg wzr vwhs hvwlpdwru/ wkh vwdqgdug JPP OP vwdwlvwlf dqg wkh V/ N/ M dqg JPP0P vwdwlvwlfv1 Zh xvh frqglwlrqdo fulwlfdo ydoxhv iru wkh JPP0P vwdwlvwlf1 Wkh lqvwuxphqwv frqvlvw ri d frqvwdqw dqg xs wr wkuhh shulrg odjjhg ydoxhv ri frqvxpswlrq jurzwk dqg dvvhw uhwxuqv1 Wkh qxpehu ri revhuydwlrqv htxdov 4331 Zh dvvxph wkdw wkhuh lv qr dxwrfruuhodwlrq ru khwhurvfhgdvwlflw| lq wkh prphqw htxdwlrqv vr zh fdq xvh d sdudphwulf fryduldqfh pdwul{ hvwlpdwru1 Hdfk sdqho lq Iljxuh 4 vkrzv wkh srzhu fxuyh iru d glhuhqw gdwd jhqhudwlqj surfhvv +GJS,1 Wkh GJS iru Sdqho 4 lv d VGI prgho fdoleudwhg wr dssur{lpdwh wkh YDU iurp +6:,1 Wkh vdph GJS lv xvhg lq Sdqho 5 exw zlwk fru+0Sc|c 0_c| ' fbD Wkh GJS
iru Sdqhov 6 dqg 7 lv d VGI fdoleudwhg wr wkh YDU iurp +6:, zlwk wkh YDU+4, sdudphwhu pdwul{ pxowlsolhg e| wzr1 Lq Sdqho 7/ fru+0Sc|c 0_c| ' fbD
Iljxuh 4 vkrzv wkdw wkh Zdog dqg JPP OP vwdwlvwlfv duh vl}h glvwruwhg hvshfldoo| zkhq wkh fruuhodwlrq sdudphwhu lv odujh1 Doo wkh rwkhu vwdwlvwlfv duh qrw vl}h glvwruwhg1 Wkh srzhu ri wkh M0vwdwlvwlf lv w|slfdoo| txlwh orz h{fhsw iru wkrvh ydoxhv zkhuh wkh N0vwdwlvwlf kdv lwv vsxulrxv ghfolqh ri srzhu1 Wkh frpelqhg M0N whvw surfhgxuh kdv wkhuhiruh jrrg srzhu surshuwlhv1 Wkh srzhu ri wkh JPP0P vwdwlvwlf lv w|slfdoo| frpsdudeoh wr wkh srzhu ri wkh M0N whvw zklfk uhvxowv ehfdxvh wkhvh vwdwlvwlfv xvh wkh M dqg N vwdwlvwlfv lq d vlplodu pdqqhu1 Lq Sdqho 4/ zklfk lv wkh GJS zkhuh wkh lghqwlfdwlrq ri q lv wkh zhdnhvw/ wkh V0vwdwlvwlf lv voljkwo| pruh
srzhuixo wkdq wkh rwkhu vwdwlvwlfv exw wkh V0vwdwlvwlf lv rq dyhudjh grplqdwhg e| wkh JPP0 P dqg M0N vwdwlvwlfv lq wkh rwkhu Sdqhov1 Wklv uhvxowv iurp wkh odujhu ghjuhhv ri iuhhgrp sdudphwhu ri wkh olplwlqj glvwulexwlrq ri wkh V0vwdwlvwlf frpsduhg wr wkh M dqg N vwdwlvwlfv1
Iljxuh 4= Srzhu fxuyhv ri vwdwlvwlfv wkdw whvw zlwk <8( +dv|pswrwlf, vljqlfdqfh Kf G q ' G V +vrolg olqh,/ N +grwwhg,/ M +vwduv,/ M0N zlwk kN ' ffec kM ' ff +gdvkhg,/
JPP0P +gdvkhg0grwwhg,/ Zdog0FXH +furvvhv,/ Zdog05vwhs +soxvvhv,/ OP +wuldqjohv,1
-8 -6 -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 β R e je c ti o n F re q u e n c y -8 -6 -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R e je c ti o n F re q u e n c y β
Sdqho 4= Zhdn vshflfdwlrq/ fru ' fe1 Sdqho 5= Zhdn vshflfdwlrq/ fru ' fbD1
-8 -6 -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 β R e je c ti o n F re q u e n c y -8 -6 -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R e je c ti o n F re q u e n c y β
Sdqho 6= Vwurqj vshflfdwlrq/ fru ' fe Sdqho 7= Vwurqj vshflfdwlrq/ fru ' fbD Vwrfkdvwlf glvfrxqw idfwru prgho iru frqvxpswlrq jurzwk dqg uhwxuq rq VS833
Zh frqvwuxfw dv|pswrwlf frqghqfh vhwv ri wkh ulvn dyhuvlrq sdudphwhu q lq d VGI prgho iru |hduo| revhuydwlrqv ri wkh jurzwk ri XV frqvxpswlrq dqg dq dvvhw uhwxuq vhulhv wkdw frqvlvwv ri wkh uhwxuqv rq wkh Frzohv Frpplvlrq lqgh{ iroorzhg e| wkh dqqxdo dyhudjh sulfh ri wkh Vwdqgdug dqg Srruv prqwko| frpsrvlwh lqgh{1 Wkh revhuydwlrqv fryhu wkh shulrg 4;:4 wr 4<<61 Wkh gdwd lv dq h{whqvlrq ri wkh gdwd iurp Fdpsehoo dqg Vkloohu +4<;:, dqg lv dovr xvhg e| Vwrfn dqg Zuljkw +5333,1 Dv lqvwuxphqwv zh xvh d frqvwdqw dqg xs wr wkuhh shulrg odjjhg revhuydwlrqv ri wkh wzr vhulhv1 Zh xvh wkh 5vwhs0Zdog/ Zdog0FXH/ OP/ V/ M/ JPP0P dqg N0vwdwlvwlfv wr frqvwuxfw dv|pswrwlf frqghqfh vhwv iru wkh ulvn dyhuvlrq sdudphwhu q1 Wkh dv|pswrwlf fulwlfdo ydoxhv uhvxow iurp d 2E glvwulexwlrq iru wkh 5vwhs0Zdog/ Zdog0FXH/ OP
dqg N vwdwlvwlfv zkloh wkh dv|pswrwlf fulwlfdo ydoxhv iru wkh M dqg V vwdwlvwlfv uhvxow iurp d 2EDc 2ES glvwulexwlrq uhvs11 Iru wkh JPP0P vwdwlvwlf zh xvh lwv frqglwlrqdo dv|pswrwlf
glvwulexwlrq1
Iljxuh 5 vkrzv rqh plqxv wkh +dv|pswrwlf, R0ydoxh iru wkh glhuhqw vwdwlvwlfv/ wkdw whvw wkh k|srwkhvlv Kf G q ' qfc iru d vhtxhqfh ri ydoxhv ri qf Iljxuh 5 frqwdlqv d olqh dw 31<8
zklfk hqdeohv xv wr frqvwuxfw wkh <8( dv|pswrwlf frqghqfh vhw lq d vwudljkwiruzdug pdqqhu e| xvlqj wkh lqwhuvhfwlrq ri wkh fxuyhv ri wkh glhuhqw vwdwlvwlfv zlwk wkh <8( olqh1
Iljxuh 5= Rqh plqxv wkh R0ydoxh iru vwdwlvwlfv wkdw whvw Kf G q ' qf iru glhuhqw ydoxhv ri
qf G V +vrolg olqh,/ N +grwwhg,/ M +vwduv,/ JPP0P +gdvkhg grwwhg,/ Zdog0FXH +furvvhv,
Zdog05vwhs +soxvvhv,/ OP +wuldqjohv,1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 β0 1 m in u s p -v a lu e
Wkh FXH dqg 5vwhs hvwlpdwru iru q kdsshq wr frlqflgh dw 0315:1 Wkh rqh plqxv R0ydoxh sorwv ri wkhvh vwdwlvwlfv duh lghqwlfdo1 Wkh rqh plqxv R0ydoxh sorwv vkrz wkdw wkh JPP0P dqg N vwdwlvwlfv duh htxdo wr }hur dw wkh FXH dqg wkdw wkh V dqg M vwdwlvwlfv dwwdlq wkhlu plqlpxp dw wkh FXH1 Wkh rqh plqxv R0ydoxh sorwv ri wkh N dqg OU vwdwlvwlfv duh yhu| vlplodu dqg dv|pphwulf1 Wkh rqh plqxv R0ydoxh sorw ri wkh M0vwdwlvwlf lv pruh lqirupdwlyh derxw wkh ydoxh ri q wkdq wkh N vwdwlvwlf1 Zh wkxv kdyh wr eh fdxwlrxv zlwk lqwhusuhwlqj wkh N dqg JPP0P vwdwlvwlf vlqfh wkh ydolglw| ri wkh prphqw htxdwlrq lv pruh dq lvvxh wkdq wkh ydoxh ri q Wkh frqghqfh vhwv wkdw uhvxow iurp wkh frpelqhg M0N whvw/ zlwk kN ' ffe dqg kM ' ffc
dqg wkh JPP0P vwdwlvwlf/ zlwk k ' ffDc wkhuhiruh glhu frqvlghudeo|1 Wkh V0vwdwlvwlf lv dovr pruh lqirupdwlyh wkdq wkh N dqg JPP0P vwdwlvwlfv1 Wkh ydolglw| ri wkh prphqw htxdwlrq ehfrphv yhu| sureohpdwlf zkhq zh lqfoxgh wkh ulvn iuhh lqwhuhvw udwh lq wkh VGI prgho dqg/ vlplodu wr Vwrfn dqg Zuljkw +5333,/ wkh M dqg V vwdwlvwlfv duh wkhq dozd|v vljqlfdqw dw wkh <<( ohyho1
; Frqfoxvlrqv
Xqghu d qryho dvvxpswlrq/ wkh olplw ehkdylru ri wkh N0vwdwlvwlf/ zklfk lv d JPP OP vwdwlvwlf wkdw xvhv d Mdfreldq hvwlpdwru wkdw lv lqghshqghqw ri wkh prphqw htxdwlrqv/ lv vkrzq wr eh urexvw wr zhdn dqg pdq| lqvwuxphqw dv|pswrwlfv1 Ehfdxvh lw lv txdgudwlf irup ri wkh ghuly0 dwlyh ri wkh remhfwlyh ixqfwlrq/ lw vxhuv iurp d vsxulrxv srzhu ghfolqh durxqg wkh pd{lpxp dqg lq h{lrq srlqwv ri wkh remhfwlyh ixqfwlrq vlqfh wkh prphqw frqglwlrqv duh wkhq qrw vdwlv0 hg1 Wklv vsxulrxv srzhu ghfolqh lv ryhufrph e| frpelqlqj wkh N0vwdwlvwlf zlwk d M0vwdwlvwlf wkdw whvwv wkh ydolglw| ri wkh prphqw htxdwlrqv1 D JPP h{whqvlrq ri Pruhlud*v frqglwlrqdo olnholkrrg udwlr vwdwlvwlf lv rewdlqhg zkhq zh xvh d vwdwlvwlf wkdw whvwv wkh udqn ri wkh Mdfreldq wr dwwdfk wkh zhljkwv wr wkh M dqg N vwdwlvwlfv lq wkh frpelqdwlrq1
Dsshqgl{
Ghulydwlrq ri Htxdwlrq +49,1 Zh xvh wkh h{suhvvlrq iru yhfETssEwf/
yhf+TssEwf ' *4A <". k A SA |' SA 's7|Ewf 7sEwf l c zkrvh ghulydwlyh zlwk uhvshfw wr w uhdgv
Yyhf+TssEw Yw mwf ' *4A <". k A SA |'SA' Y 7s|Ewf Yw 7sEwf ns7|Ewf Y 7sYwEwf l ' *4A <". k A SA |'SA' 7^cEwf 67^6cEwf 7sEwf n 7s|Ewf 7^cEwf 67^6cEwf
' E U&syhfETwscEwf E6 U&syhfETwsc6Ewf
n EU&s yhfETwscEwf EU&s 6yhfETwsc6Ewf
c
zlwk TwscEwf G & &sc ' c c 6 dqg TwsEwf ' ETwscEwf Twsc6Ewf Iru wkh ghulydwlyh
ri wkh remhfwlyh ixqfwlrq/ wklv lpsolhv
2Y'EwYw mwf ' sAEwfc t TssEwf3RAEwfc t 2 EsAEwfc t TssEwf3 sAEwfc t TssEwf3
E U&syhfETwscEwf E6 U&syhfETwsc6Ewf
n EU&s yhfETwscEwf EU&s 6yhfETwsc6Ewf
' sAEwfc t TssEwf3RAEwfc t yhfETwscEwfTssEwf3sAEwfc t
6yhfETwsc6EwfTssEwf3sAEwfc t
' sAEwfc t TssEwf3 RcAEwfc t TwscEwfTssEwf3sAEwfc t
R6cAEwfc t 6Twsc6EwfTssEwf3sAEwfc t c
zlwk RcAEwfc t G &s c ' c c 6c dqg RAEwfc t ' ERcAEwfc t R6cAEwfc t
Surri ri Ohppd 41 Zh suh0pxowlso| wkh fhqwudo olplw wkhruhp iurp Dvvxpswlrq 4 e| -Ewf ' U&s f TwsEwfTssEwf3 U&w wr rewdlq s A-Ewf AsAEwfc t A7^AEwfc t $ _ -Ewf s w / s A AsAEwfc t A7^AEwfc t ATwsEwfTssEwf3sAEwfc t $ _ s ws zkhuh ^AEwfc t 'SA|'7^|Ewfc dqg s ws Efc TssEwf f f TwwsEwf c zlwk TwwsEwf ' TwwEwf TwsEwfTssEwf3TswEwfc zklfk vkrzv wkdw ws lv lqghshqghqw ri
s Zh ixuwkhu suh0pxowlso| wkh olplwlqj h{suhvvlrq e| gldj+U&sc zklfk uhvxowv lq
s A AsAEwfc t yhf+ A(AEwfc t awEwf $ _ s ws c vlqfh E^cAEwfc t TwscEwfTssEwf3sAEwfc t ' RcAEwfc t TwscEwfTssEwf3sAEwfc t dqg *4A <"SA|'.E^|Ewf 'yhf+awEwf 48
Surri ri Ohppd 51 41 Zkhq awEwf kdv ixoo udqn/ A(AEwfc t ' awEwf n IA qs A A(AEwfc t awEwf r $ R awEwfc zklfk vkrzv wkdw wkh olplwlqj ehkdylru ri A(AEwfc t lv lqghshqghqw ri wkh olplwlqj ehkdylru ri I AsAEwfc t
51 awEwf kdv d zhdn ydoxh vxfk wkdw awEwf ' awcAc awcA ' IAc G &s6 dqg udqn+ ' 6
I
A(AEwc t '
s
A awcA nsAA(AEwfc t awEwf $_ n wsc 6wsc6 c
zkhuh wsc G & c ' c c 6 dqg ws ' Ewsc wsc6c zklfk vkrzv wkdw/ ehfdxvh ws
lv lqghshqghqw ri sc vhh Ohppd 4/ wkdw wkh olplwlqj ehkdylru ri IA(AEwc t lv lqghshqghqw
ri wkh olplwlqj ehkdylru ri I
AsAEwfc t
61 Uhvxowv gluhfwo| iurp Ohppd 41
Surri ri Wkhruhp 41 Lw uhvxowv iurp Ohppd 5 wkdw/
Av(AEwfc t $
_ (c
zkhuh lq fdvh 4 ri O hppd 5/ v ' dqg ( lv d {hg frqvwdqw htxdo wr awEwfc dqg lq fdvhv 5
dqg 6 ri Ohppd 5/ v '
2 dqg ( lv d udqgrp yduldeoh lqghshqghqw ri s Zh wkhq rewdlq I AsAEwfc t TssEwf3 Av(AEwfc t $ _ sTssEwf3( Wkh frqglwlrqdo glvwulexwlrq ri sTssEwf3( jlyhq ( uhdgv sTssEwf3(m( Efc (TssEwf3(
Vlqfh ( lv lqghshqghqw ri sc zh rewdlq d xqfrqglwlrqdo uhvxow e| qrupdol}lqj wkh h{suhvvlrq
e| E(T ssEwf3(32c I AsAEwfc t TssEwf3(AEwfc t d(AEwfc t TssEwf3(AEwfc t o 3 2 $ _ sTssEwf3(E(TssEwf3(32 ' '_wc zlwk '_w Efc U6
Surri ri Wkhruhp 61 Wr hvwdeolvk wkh olplw ehkdylru zkhq wkh qxpehu ri lqvwuxphqwv & dqg revhuydwlrqv A mrlqwo| frqyhujh wr lqqlw|/ zh xvh d olplw vhtxhqfh lq zklfk zh uvw ohw A frqyhujh wr lqqlw| dqg diwhuzdugv & Ehfdxvh wkh frqyhujhqfh ryhu A lv wrzdugv qrupdo glvwulexwhg udqgrp yduldeohv/ Ohppd 9 ri Skloolsv dqg Prrq +4<<<, dssolhv/ zklfk phdqv wkdw wkh vhtxhqwldo olplw zkhuh zh uvw ohw A frqyhujh wr lqqlw| dqg diwhuzdugv & lv lghqwlfdo wr wkh mrlqw olplw zkhuh erwk frqyhujh wr lqqlw| vlpxowdqhrxvo|1
Zh frqvlghu wkh olplwlqj ehkdylru ri sAEwfc t TssEwf3( AEwfc t c zlwk
(AEwfc t ' RcAEwfc t TwscEwf TssEwf3sAEwfc t
R6cAEwfc t 6 Twsc6Ewf TssEwf3sAEwfc t c
zkhuh TwscEwfc ' c c 6c dqg TssEwf duh hvwlpdwruv ri wkh fryduldqfh sdudphwhuv/ TwscEwfc
' c c 6c dqg TssEwf Zh dqdo|}h wkh olplwlqj ehkdylru ri wkh N0vwdwlvwlf xqghu d mrlqw
olplwlqj vhtxhqfh iru & E' &s
, dqg A
Zh vshfli| wkh hvwlpdwru TwscEwf TssEwf3 ri TwscEwfTssEwf3c ' c c 6 dv TwscEwf TssEwf3' U&s TwscEwfTssEwf3 TwscEwf TwscEwf TssEwf TssEwf TssEwf3 nTwscEwfTssEwf3c dqg TssEwf3 ' TssEwf3n TssEwf3 k TssEwf TssEwf l TssEwf3 Dv d frqvhtxhqfh/ (AEwfc t ' (AEwfc t n U&s TwscEwfTssEwf3 TwscEwf TwscEwf TssEwf TssEwf 6 U&s Twsc6EwfTssEwf3 Twsc6Ewf Twsc6Ewf TssEwf TssEwf TssEwf3sAEwfc t
Wkh frqyhujhqfh udwh ri wkh hvwlpdwruv TwscEwfc ' c c 6 dqg TssEwf htxdov A> vr
A> 3 E E E C TwscEwf TwscEwf 111 Twsc6Ewf Twsc6Ewf TssEwf TssEwf 4 F F F D$_ 3 E E E C [wsc 111 [wsc6 [ss 4 F F F Dc
zkhuh [wscc ' c c 6 dqg [ss duh &s &s glphqvlrqdo udqgrp pdwulfhv zlwk d qrq0
ghjhqhudwh +qrupdo, glvwulexwlrq1
Zh frqvlghu wkh fdvh wkdw awEwf lv d }hur pdwul{ dqg wkhuhiruh I A(AEwfc t $_ wsc 6wsc6
Zh uvw ohw A frqyhujh wr lqqlw| dqg diwhuzdugv wkh qxpehu ri lqvwuxphqwv & E' &s
,
Vlqfh , lv {hg/ zkhq & frqyhujhv wr lqqlw| dovr &s frqyhujhv wr lqqlw|1 Xvlqj wkh vh0
txhqwldo olplwlqj vhtxhqfh/ zh frqvwuxfw wkh olplwlqj ehkdylru ri irxu glhuhqw hohphqwv ri sAEwfc t TssEwf3( AEwfc t G sAEwfc t TssEwf3( AEwfc t ' E sAEwfc t TssEwf3(AEwfc t n E2 sAEwfc t TssEwf3 k TssEwf TssEwf l TssEwf3(AEwfc t n E sAEwfc t TssEwf3 U&s TwscEwfTssEwf3 TwscEwf TwscEwf TssEwf TssEwf 6 U&s Twsc6EwfTssEwf3 Twsc6Ewf Twsc6Ewf TssEwf TssEwf TssEwf3sAEwfc t n Ee sAEwfc t TssEwf3 k TssEwf TssEwf l TssEwf3 U&s TwscEwfTssEwf3 Twsc T Ewf TwscEwf ssEwf TssEwf 6 U&s Twsc6EwfTssEwf3 Twsc6 T Ewf Twsc6Ewf ssEwf TssEwf 4:
41 Zh rewdlq iurp wkh olplwlqj ehkdylru zlwk uhvshfw wr A rqo| dqg &s {hg/ vhh Ohppd 4/ wkdw AsAEwfc t TssEwf3(AEwfc t $_ sTssEwf3 wsc 6wsc6 Zh wkhq ohw &s frqyhujh wr lqqlw|/
s &s sTssEwf3 wsc 6wsc6 $_ 6c zlwk 6 G 6 c 6 Efc \wT w dqg \wT w ' *4&s<" &s wsc 6wsc6 TssEwf3 wsc 6wsc6 51 Zkhq A frqyhujhv wr lqqlw| zlwk &s {hg/ A> A sAEwfc t TssEwf3 k TssEwf TssEwf l TssEwf3(AEwfc t $ _ sTssEwf3[ssTssEwf3 wsc 6wsc6
Wkh frqyhujhqfh udwh vkrzv wkdw wkh uvw hohphqw dozd|v grplqdwhv wklv hohphqw dv|pswrwl0 fdoo|1 61 Zkhq A frqyhujhv wr lqqlw| zlwk &s {hg/ A> A sAEwfc t TssEwf3 U&s TwscEwfTssEwf3 TwscEwf TwscEwf TssEwf TssEwf TssEwf3sAEwfc t $ _ sTssEwf3E[wsc TwscEwfTssEwf3[ss TssEwf3s Zkhq &s frqyhujhv wr lqqlw|/ &s sTssEwf3E[wsc TwscEwfTssEwf3[ss TssEwf3s $_ bc
zkhuh b G c ' c c 6c lv d udqgrp yduldeoh zlwk d qrq0ghjhqhudwh glvwulexwlrq1
71 Zkhq A frqyhujhv wr lqqlw| zlwk &s {hg/ A2> A sAEwfc t TssEwf3 k TssEwf TssEwf l TssEwf3 U&s TwscEwfTssEwf3 TwscEwf TwscEwf TssEwf TssEwf TssEwf3sAEwfc t $ _ sTssEwf3[ssTssEwf3E[wsc TwscEwfTssEwf3[ss TssEwf3s
Wkh frqyhujhqfh udwh vkrzv wkdw wkh wklug hohphqw dozd|v grplqdwhv wklv hohphqw dv|pswrw0 lfdoo|1
Wkh olplwlqj ehkdylru ri wkh uvw hohphqw dozd|v grplqdwhv wkh olplwlqj ehkdylru ri wkh vhfrqg hohphqw zkloh wkh olplwlqj ehkdylru ri wkh wklug hohphqw dozd|v grplqdwhv wkh olplwlqj ehkdylru ri wkh irxuwk hohphqw1 Hohphqw 4 kdv wr eh vfdohg e|
As&s dqg hohphqw 6 kdv wr eh vfdohg
e| A>
&sA wr kdyh frqyhujhqfh ri hdfk ri wkhp wr d qrq0}hur qlwh udqgrp yduldeoh zkhq &s
dqg A mrlqwo| frqyhujh wr lqqlw|1 Zkhq &s lv sursruwlrqdo wr A2>c wkhvh vfdolqj idfwruv duh
sursruwlrqdo wr rqh dqrwkhu1 Zkhq wkh mrlqw olplwlqj vhtxhqfh ri &s dqg A lv vxfk wkdw &s
A2> $ fc zh vfdoh hohphqw 4 e| d vpdoohu idfwru1 Khqfh/ wkh olplwlqj ehkdylru ri hohphqw 4
wkhq grplqdwhv wkh olplwlqj ehkdylru ri doo wkh rwkhu hohphqwv1 Wkh 2E6 olplwlqj glvwulexwlrq
ri wkh N0vwdwlvwlf wkhuhiruh uhpdlqv ydolg zkhq awEwf lv d }hur pdwul{ dqg A&2>s $ f
Iru ixoo udqn ydoxhv ri awEwfc zh kdyh wr vfdoh (AEwfc t e| Ac vhh Ohppd 5/ vxfk wkdw
wkh 2E6 olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf dovr uhpdlqv ydolg zkhq &s
A2> $ f lq wkdw
fdvh1 Zhdn ydoxhv ri awEwf frqvwlwxwh dq lqwhuphgldwh fdvh ehwzhhq ixoo udqn dqg }hur ydoxhv
ri awEwf vxfk wkdw dovr lq wkh zhdn fdvh/ wkh 2E6 olplwlqj glvwulexwlrq ri wkh N0vwdwlvwlf
uhpdlqv ydolg1
Uhihuhqfhv
^4` Dqghuvrq/ W1Z1 Hvwlpdwlqj olqhdu uhvwulfwlrqv rq uhjuhvvlrq frh!flhqwv iru pxowlyduldwh qrupdo glvwulexwlrqv1 Dqqdov ri Pdwkhpdwlfdo Vwdwlvwlfv/ 55=65:684/ 4<841
^5` Dqguhzv/ G1Z1N1 Khwhurvnhgdvwlflw| dqg Dxwrfruuhodwlrq Frqvlvwhqw Fryduldqfh Pdwul{ Hvwlpdwlrq1 Hfrqrphwulfd/ 8<=;4:;8;/ 4<<41
^6` Ehnnhu/ S1D1 Dowhuqdwlyh Dssur{lpdwlrqv wr wkh Glvwulexwlrqv ri Lqvwuxphqwdo Yduldeoh Hvwlpdwruv1 Hfrqrphwulfd/ 95=98:9;4/ 4<<71
^7` Ehnnhu/ S1D1 dqg I1 Nohlehujhq1 Ilqlwh0Vdpsoh Lqvwuxphqwdo Yduldeohv Lqihuhqfh xvlqj dq Dv|pswrwlfdoo| Slyrwdo Vwdwlvwlf1 Hfrqrphwulf Wkhru|/ 53351 Iruwkfrplqj1
^8` Eurzq/ E1Z1 dqg Z1N1 Qhzh|1 H!flhqw Vhplsdudphwulf Hvwlpdwlrq ri H{shfwdwlrqv1
Hfrqrphwulfd/ 99=786797/ 4<<;1
^9` Fdpsehoo/ M1\1 dqg U1M1 Vkloohu1 Frlqwhjudwlrq dqg Whvwv ri Suhvhqw Ydoxh Prghov1 Mrxuqdo
ri Srolwlfdo Hfrqrp|/ <8=439543;;/ 4<;:1
^:` Fudjj/ M1F1 dqg V1J1 Grqdog1 Rq wkh Dv|pswrwlf Surshuwlhv ri OGX0Edvhg Whvwv ri wkh Udqn ri d Pdwul{1 Mrxuqdo ri wkh Dphulfdq Vwdwlvwlfdo Dvvrfldwlrq/ <4=4634463</ 4<<91 ^;` Fudjj/ M1F1 dqg V1J1 Grqdog1 Lqihuulqj wkh udqn ri d pdwul{1 Mrxuqdo ri Hfrqrphwulfv/
:9=556583/ 4<<:1
^<` Grqdog/ V1J1 dqg Z1N1 Qhzh|1 D Mdfnnqlih Lqwhusuhwdwlrq ri wkh Frqwlqxrxv Xsgdwlqj Hvwlpdwru1 Hfrqrplfv Ohwwhuv/ 9:=56<576/ 53331
^43` Grqdog/ V1J1 dqg Z1N1 Qhzh|1 Fkrrvlqj wkh Qxpehu ri Lqvwuxphqwv1 Hfrqrphwulfd/ 9<=449444<5/ 53341
^44` Gxirxu/ M10P1 Vrph Lpsrvvlelolw| Wkhruhpv lq Hfrqrphwulfv zlwk Dssolfdwlrqv wr Vwuxf0 wxudo dqg G|qdplf Prghov1 Hfrqrphwulfd/ 98=46986;;/ 4<<:1
^45` Kdqvhq/ O1S1 Odujh Vdpsoh Surshuwlhv ri Jhqhudol}hg Phwkrg Prphqwv Hvwlpdwruv1
Hfrqrphwulfd/ 83=435<4387/ 4<;51
^46` Kdqvhq/ O1S1 dqg N1 Vlqjohwrq1 Jhqhudol}hg Lqvwuxphqwdo Yduldeoh Hvwlpdwlrq ri Qrqolq0 hdu Udwlrqdo H{shfwdwlrqv Prghov1 Hfrqrphwulfd/ 83=459<45;9/ 4<;51
^47` Kdqvhq/ O1S1/ M1 Khdwrq dqg D1 \durq1 Ilqlwh Vdpsoh Surshuwlhv ri Vrph Dowhuqdwlyh JPP Hvwlpdwruv1 Mrxuqdo ri Exvlqhvv dqg Hfrqrplf Vwdwlvwlfv/ 47=5955;3/ 4<<91
^48` Nohlehujhq/ I1 Slyrwdo Vwdwlvwlfv iru whvwlqj Vxevhwv ri Vwuxfwxudo Sdudphwhuv lq wkh LY Uhjuhvvlrq Prgho1 Wlqehujhq Lqvwlwxwh Glvfxvvlrq Sdshu WL 533303;;27/ 53331
^49` Nohlehujhq/ I1 Slyrwdo Vwdwlvwlfv iru whvwlqj Vwuxfwxudo Sdudphwhuv lq Lqvwuxphqwdo Ydul0 deohv Uhjuhvvlrq1 Hfrqrphwulfd/ :3=4:;44;36/ 5335d1
^4:` Nohlehujhq/ I1 Wzr Lqghshqghqw Slyrwdo Vwdwlvwlfv wkdw Whvw Orfdwlrq dqg Plvvshflfd0 wlrq dqg dgg0xs wr wkh Dqghuvrq0Uxelq Vwdwlvwlf1 Wlqehujhq Lqvwlwxwh Glvfxvvlrq Sdshu/ WL 5335039727/ 5335e1
^4;` Nohlehujhq/ I1 dqg U1 Sdds1 Jhqhudol}hg Uhgxfhg Udqn Whvwv xvlqj wkh Vlqjxodu Ydoxh Ghfrpsrvlwlrq1 Xqsxeolvkhg Pdqxvfulsw/ Xqlyhuvlw| ri Dpvwhugdp/ 53351
^4<` Nrfkhuodnrwd/ Q1U1 Rq whvwv ri uhsuhvhqwdwlyh frqvxphu dvvhw sulflqj prghov1 Mrxuqdo ri
Prqhwdu| Hfrqrplfv/ 59=5;8637/ 4<<31
^53` Pruhlud/ P1M1 D Frqglwlrqdo Olnholkrrg Udwlr Whvw iru Vwuxfwxudo Prghov1 Xqsxeolvkhg Pdqxvfulsw/ Xqlyhuvlw| ri Fdoliruqld dw Ehunhoh|/ 53341
^54` Qhzh|/ Z1N1 dqg G1 PfIdgghq1 Odujh Vdpsoh Hvwlpdwlrq dqg K|srwkhvlv Whvwlqj1 Lq U1 Hqjoh dqg G1 PfIdgghq/ hglwru/ Kdqgerrn ri Hfrqrphwulfv/ Yroxph 7/ fkdswhu 69/ sdjhv 5446547;1 Hovhylhu Vflhqfh E1Y1/ 4<<71
^55` Qhzh|/ Z1N1 dqg U1M1 Vplwk1 Kljkhu Rughu Surshuwlhv ri JPP dqg Jhqhudol}hg Hpslu0 lfdo Olnholkrrg Hvwlpdwruv1 Zrunlqj sdshu/ 53341
^56` Qhzh|/ Z1N1 dqg N1G1 Zhvw1 D Vlpsoh Srvlwlyh Vhpl0Ghqlwh Khwhurvnhgdvwlflw| dqg Dxwrfruuhodwlrq Frqvlvwhqw Fryduldqfh Pdwul{1 Hfrqrphwulfd/ 88=:36:3;/ 4<;:1
^57` Qhzh|/ Z1N1 dqg N1G1 Zhvw1 K|srwkhvlv Whvwlqj zlwk H!flhqw Phwkrg ri Prphqwv Hvwlpdwlrq1 Lqwhuqdwlrqdo Hfrqrplf Uhylhz/ 5;=::::;:/ 4<;:1
^58` Skloolsv/ S1F1E1 dqg K1 Prrq1 Olqhdu Uhjuhvvlrq Olplw Wkhru| iru Qrqvwdwlrqdu| Sdqho Gdwd1 Hfrqrphwulfd/ 9:=438:4444/ 4<<<1
^59` Vwdljhu/ G1 dqg M1K1 Vwrfn1 Lqvwuxphqwdo Yduldeohv Uhjuhvvlrq zlwk Zhdn Lqvwuxphqwv1
Hfrqrphwulfd/ 98=88:8;9/ 4<<:1
^5:` Vwrfn/ M1K1 dqg M1K1 Zuljkw1 JPP zlwk Zhdn Lghqwlfdwlrq1 Hfrqrphwulfd/ 9;=4388 43<9/ 53331
^5;` Wdxfkhq/ J1 Vwdwlvwlfdo Surshuwlhv ri Jhqhudol}hg Phwkrg0ri0Prphqwv Hvwlpdwruv ri Vwuxfwxudo Sdudphwhuv rewdlqhg iurp Ilqdqfldo Pdunhw Gdwd1 Mrxuqdo ri Exvlqhvv dqg
Hfrqrplf Vwdwlvwlfv/ 7=6<:758/ 4<;9d1
^5<` Wdxfkhq/ J1 Ilqlwh Vwdwh Pdunry0Fkdlq Dssur{lpdwlrqv wr Xqlyduldwh dqg Yhfwru Dx0 wruhjuhvvlrqv1 Hfrqrplf Ohwwhuv/ 53=4::4;4/ 4<;9e1
^63` Zklwh/ K1 D Khwhurvnhgdvwlflw|0Frqvlvwhqw Fryduldqfh Pdwul{ Hvwlpdwru dqg d Gluhfw Whvw iru Khwhurvfhgdvwlflw|1 Hfrqrphwulfd/ 7;=;4:;6;/ 4<;31
^64` Zklwh/ K1 Dv|pswrwlf Wkhru| iru Hfrqrphwulfldqv1 Dfdghplf Suhvv/ 4<;71 53