14 Testing the Rational Expectations Hypothesis

47
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv
K|srwkhvlv
Lq wkh uhpdlqghu ri wklv errn/ zh frqvlghu wkh zd|v lq zklfk wkh UH
k|srwkhvlv kdv ehhq whvwhg hpslulfdoo|1 Zh ehjlq lq wklv fkdswhu zlwk
wkh h!flhqw pdunhw k|srwkhvlv/ zklfk lv d mrlqw k|srwkhvlv pdgh xs
ri udwlrqdo h{shfwdwlrqv dqg d k|srwkhvlv derxw krz h{shfwhg uhwxuqv
duh ghwhuplqhg1 Lq fkdswhu 48/ zh glvfxvv wkh hylghqfh dvvhpeohg e|
Urehuw Eduur dqg rwkhuv wr whvw zkhwkhu rxwsxw lv ghwhuplqhg e| wkh
Qhz Fodvvlfdo prgho ru qrw1 Lq fkdswhu 49/ zh uhylhz gluhfw whvwv ri
udwlrqdo h{shfwdwlrqv xvlqj vxuyh| gdwd rq dfwxdo h{shfwdwlrqv> zh wkhq
wxuq wr wkh hvwlpdwlrq ri frpsohwh udwlrqdo h{shfwdwlrqv prghov1 Vrph
vxfk prghov duh cghhs vwuxfwxuh* +zkhuh wkh sdudphwhuv duh wkrvh ri
wdvwhv dqg whfkqrorj|/ suhvxpdeo| lqyduldqw wr fkdqjhv lq wkh surfhvvhv
gulylqj wkh h{rjhqrxv yduldeohv lqfoxglqj srolf|, exw prvw duh cvkdoorz
vwuxfwxuh*/ wkh sdudphwhuv ehlqj wkrvh ri djjuhjdwh vxsso| dqg ghpdqg
fxuyhv1
Wkh sxusrvh ri wklv fkdswhu lv wr frqvlghu wkh uhodwlrqvkls ehwzhhq
wkh frqfhswv ri qdqfldo pdunhw h!flhqf| dqg udwlrqdo h{shfwdwlrqv1
D sduwlfxodu ihdwxuh ri qdqfldo pdunhwv lv wkdw wudglqj fdq rffxu/ lq
sulqflsoh/ doprvw frqwlqxrxvo|/ dqg wkh pdunhw sulfh lv iuhh wr pryh wr
hudglfdwh dq| lpedodqfh ehwzhhq ghpdqg dqg vxsso|1 Ixuwkhupruh vlqfh
wkh dvvhwv wudghg +vwrfnv/ erqgv/ frpprglwlhv, fdq eh uhvrog ru wudghg lq
ixwxuh shulrgv lw iroorzv wkdw qdqfldo pdunhwv duh/ pruh reylrxvo| wkdq
rwkhuv/ vshfxodwlyh lq wkh whfkqlfdo vhqvh wkdw h{shfwdwlrqv ri ixwxuh dvvhw
sulfhv dhfw fxuuhqw dvvhw sulfhv1
Zh ehjlq wklv fkdswhu e| ghqlqj wkh frqfhsw ri dq h!flhqw fdslwdo
pdunhw dqg vhwwlqj rxw prghov iru wkh ghwhuplqdwlrq ri qrupdo ru htxl0
oleulxp dvvhw uhwxuqv dqg dovr wkh lpsolfdwlrqv ri udwlrqdo h{shfwdwlrqv
iru wkh ghwhuplqdwlrq ri dvvhw sulfhv1 D eulhi uhylhz ri wkh hpslulfdo
hylghqfh iru h!flhqf| lq dvvhw pdunhwv wrjhwkhu zlwk wkh sureohpv idfhg
668
669
Frqiurqwlqj Prghov zlwk Idfwv
lq lqwhusuhwdwlrq ri wkh hpslulfdo hylghqfh lv dovr suhvhqwhg1
HIILFLHQF\/ SHUIHFWLRQ DQG DVVHW SULFLQJ
LQ FDSLWDO PDUNHWV1
D fdslwdo ru dvvhw pdunhw lv ghqhg wr eh h!flhqw zkhq sulfhv +h1j1 vwrfn
sulfhv/ erqg sulfhv ru h{fkdqjh udwhv, ixoo| dqg lqvwdqwdqhrxvo| uh hfw
doo dydlodeoh uhohydqw lqirupdwlrq1 Idpd +4<:3, kdv ghqhg wkuhh w|shv
ri pdunhw h!flhqf|/ dffruglqj wr wkh h{whqw ri wkh lqirupdwlrq uh hfwhg
lq wkh pdunhw=
41 Zhdn0irup h!flhqw= D pdunhw lv zhdn0irup h!flhqw li lw lv qrw
srvvleoh iru d wudghu wr pdnh deqrupdo uhwxuqv e| ghyhorslqj d
wudglqj uxoh edvhg rq wkh sdvw klvwru| ri sulfhv ru uhwxuqv1
51 Vhpl0vwurqj0irup h!flhqw= D pdunhw zkhuh d wudghu fdqqrw pdnh
deqrupdo uhwxuqv xvlqj d wudglqj uxoh edvhg rq sxeolfo| dydlodeoh
lqirupdwlrq1 H{dpsohv ri sxeolfo| dydlodeoh lqirupdwlrq duh sdvw
prqh| vxsso| gdwd/ frpsdq| qdqfldo dffrxqwv/ ru wlsvwhuv lq sh0
ulrglfdov1
61 Vwurqj0irup h!flhqw= Zkhuh d wudghu fdqqrw pdnh deqrupdo uh0
wxuqv xvlqj d wudglqj uxoh edvhg rq dq| lqirupdwlrq vrxufh/ zkhwkhu
sxeolf ru sulydwh1
Wkhvh wkuhh irupv ri h!flhqf| uhsuhvhqw d fuxgh sduwlwlrqlqj ri doo
srvvleoh lqirupdwlrq v|vwhpv lqwr wkuhh eurdg fdwhjrulhv/ wkh suhflvh
erxqgdulhv ri zklfk duh qrw hdvlo| ghqhg1 Krzhyhu/ wkh| duh xvhixo/
dv zh vkdoo vhh/ iru fodvvli|lqj hpslulfdo uhvhdufk rq pdunhw h!flhqf|1
Dv wkhlu qdphv vxjjhvw/ vwurqj0irup h!flhqf| lpsolhv vhpl0vwurqj h!0
flhqf| zklfk lq wxuq lpsolhv zhdn0irup h!flhqf|/ zkloh ri frxuvh wkh
uhyhuvh lpsolfdwlrqv gr qrw krog1
Lw lv xvhixo wr glvwlqjxlvk ehwzhhq wkh frqfhsw ri dq h!flhqw fdslwdo
pdunhw dqg wkdw ri d shuihfw fdslwdo pdunhw1 D shuihfw fdslwdo pdu0
nhw frxog eh ghqhg dv rqh lq zklfk wkh iroorzlqj frqglwlrqv krog +vhh
Frshodqg dqg Zhvwrq/ 4<;;,=
41 Pdunhwv duh lqirupdwlrqdoo| h!flhqw/ wkdw lv lqirupdwlrq lv frvwohvv
dqg lw lv uhfhlyhg vlpxowdqhrxvo| e| lqglylgxdov1
51 Pdunhwv duh iulfwlrqohvv/ wkdw lv wkhuh duh qr wudqvdfwlrqv frvwv ru
wd{hv/ dvvhwv duh shuihfwo| glylvleoh dqg pdunhwdeoh dqg wkhuh duh
qr frqvwudlqlqj uhjxodwlrqv1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
66:
61 Wkhuh lv shuihfw frpshwlwlrq lq surgxfw dqg vhfxulwlhv pdunhwv/
wkdw lv djhqwv duh sulfh0wdnhuv1
71 Doo lqglylgxdov duh udwlrqdo h{shfwhg0xwlolw| pd{lpl}huv1
Li frqglwlrqv 4 wr 7 zhuh phw +dqg dvvxplqj qr vljqlfdqw glvwruwlrqv
hovhzkhuh lq wkh hfrqrp|,/ wkh fdslwdo pdunhw zrxog eh doorfdwlrqdoo|
h!flhqw/ lq wkdw sulfhv zrxog eh vhw wr htxdwh wkh pdujlqdo udwhv ri uh0
wxuq iru doo surgxfhuv dqg vdyhuv/ dqg ri frxuvh frqvhtxhqwo| vdylqjv duh
rswlpdoo| doorfdwhg1 Wkh qrwlrq ri fdslwdo pdunhw h!flhqf| lv wkhuhiruh
pxfk ohvv uhvwulfwlyh1 Dq hohphqw ri lpshuihfw frpshwlwlrq lq surgxfw
pdunhwv zrxog lpso| fdslwdo pdunhw lpshuihfwlrq> qhyhuwkhohvv/ wkh vwrfn
pdunhw frxog ghwhuplqh d vhfxulw| sulfh zklfk ixoo| uh hfwhg wkh suhvhqw
ydoxh ri wkh vwuhdp ri h{shfwhg ixwxuh prqrsro| surwv1 Frqvhtxhqwo|
wkh vwrfn pdunhw frxog vwloo eh h!flhqw lq wkh suhvhqfh ri lpshuihfwlrq1
Dvvhw sulfhv/ lq rughu wr jlyh wkh fruuhfw vljqdov wr wudghuv/ pxvw
ixoo| dqg lqvwdqwdqhrxvo| uh hfw doo dydlodeoh lqirupdwlrq1 Krzhyhu/ dv
srlqwhg rxw e| Jurvvpdq dqg Vwljolw} +4<:9/ 4<;3,/ lw fdqqrw eh wkh fdvh
wkdw pdunhw sulfhv gr ixoo| dqg lqvwdqwdqhrxvo| uh hfw doo dydlodeoh lqiru0
pdwlrq1 Li wklv zhuh vr/ djhqwv zrxog kdyh qr lqfhqwlyh iru froohfwlqj dqg
surfhvvlqj lqirupdwlrq/ vlqfh lw zrxog douhdg| eh uh hfwhg lq wkh sulfh/
zklfk hdfk lqglylgxdo lv dvvxphg wr eh deoh wr revhuyh frvwohvvo|1 Lw lv
wkh srvvlelolw| ri rewdlqlqj deqrupdo surwv lq wkh frxuvh ri duelwudjlqj
zklfk surylghv wkh lqfhqwlyh wr froohfw dqg surfhvv qhz lqirupdwlrq1 Lq
wkh Jurvvpdq dqg Vwljolw} prgho/ lqglylgxdov fkrrvh wr ehfrph lqiruphg
ru uhpdlq xqlqiruphg/ dqg lq htxloleulxp hdfk lqglylgxdo lv lqglhuhqw
ehwzhhq uhpdlqlqj xqlqiruphg rq wkh rqh kdqg/ dqg froohfwlqj lqiru0
pdwlrq +ru ex|lqj wkh h{shuwlvh ri eurnhuv,/ vr ehfrplqj lqiruphg/ rq
wkh rwkhu1 Wklv lv ehfdxvh diwhu ghgxfwlqj lqirupdwlrq frvwv hdfk dfwlrq
rhuv wkh vdph h{shfwhg xwlolw|14
4 Khoozlj +4<;5, fkdoohqjhv wkh Jurvvpdq dqg Vwljolw} sursrvlwlrq wkdw wkh lqiru0
pdwlyhqhvv ri pdunhw sulfhv lq htxloleulxp lv erxqghg dzd| iurp ixoo lqirupdwlrqdo
h!flhqf|1 Khoozlj srlqwv rxw wkdw wklv sursrvlwlrq uhvwv rq wkh dvvxpswlrq wkdw
djhqwv ohduq iurp fxuuhqw sulfhv dw zklfk wudqvdfwlrqv kdyh dfwxdoo| ehhq frpsohwhg1
Wklv lv d prgho lq zklfk lqyhvwruv ohduq iurp sdvw htxloleulxp sulfhv exw qrw iurp
wkh dxfwlrqhhu*v fxuuhqw sulfh rhu1 Khoozlj lv deoh wr vkrz wkdw li wkh wlph vsdq
ehwzhhq vxffhvvlyh pdunhw wudqvdfwlrqv lv vkruw/ wkh pdunhw fdq dssur{lpdwh ixoo lq0
irupdwlrqdo h!flhqf| duelwudulo| forvho| dqg |hw wkh uhwxuq wr ehlqj lqiruphg uhpdlqv
erxqghg dzd| iurp }hur1 Wklv uhvxowv iurp wkh idfw wkdw lqiruphg djhqwv fdq xwl0
ol}h wklv lqirupdwlrq ehiruh xqlqiruphg djhqwv kdyh dq rssruwxqlw| wr lqihu lw iurp
fxuuhqw pdunhw sulfhv1
Khoozlj dovr sxuvxhv wkh lpsolfdwlrqv wkdw dulvh li rqh uhod{hv wkh dvvxpswlrq wkdw
djhqwv fdqqrw dvvxuh wkhpvhoyhv ri ehlqj lqiruphg lq d jlyhq shulrg/ exw udwkhu djhqwv
fkrrvh wkh iuhtxhqf| rq dyhudjh dw zklfk wkh| rewdlq lqirupdwlrq1 Lw dsshduv wkdw
66;
Frqiurqwlqj Prghov zlwk Idfwv
Qhyhuwkhohvv/ d uhdvrqdeoh lqwhusuhwdwlrq ri hpslulfdo whvwv ri wkh
h!flhqw prgho k|srwkhvlv lv/ jlyhq wkdw wkh gdwd duh froohfwhg dw glvfuhwh
lqwhuydov/ wkdw wkh surfhvv ri duelwudjh kdv rffxuuhg zlwklq wkh shulrg1
Frqvhtxhqwo| wkh lpsolfdwlrqv ri glhuhqw dydlodeoh lqirupdwlrq fdq wkhq
eh dqdo|vhg zlwkrxw prghoolqj wkh surfhvv ri duelwudjh lwvhoi +Ehjj/ 4<;5,
dqg wklv lv wkh xvxdo dvvxpswlrq lq hpslulfdo zrun1
UDWLRQDO H[SHFWDWLRQV DQG PDUNHW HIIL0
FLHQF\
Wkh vhpl0vwurqj0irup h!flhqw pdunhwv prgho/ l1h1 wkdw edvhg rq sxe0
olfo| dydlodeoh lqirupdwlrq/ lv dq dssolfdwlrq ri wkh frqfhsw ri udwlrqdo
h{shfwdwlrqv/ dowkrxjk wklv zdv qrw vwuhvvhg lq wkh hduo| olwhudwxuh rq
h!flhqf|/ zklfk jrhv edfn pxfk ixuwkhu wkdq wkh udwlrqdo h{shfwdwlrqv
olwhudwxuh1 Li h{shfwdwlrqv duh qrq0udwlrqdo/ wkhq sxeolfo| dydlodeoh lq0
irupdwlrq zloo qrw eh uh hfwhg lq dvvhw sulfhv dqg v|vwhpdwlf deqrupdo
surw rssruwxqlwlhv zloo eh dydlodeoh1 Wklv fdq eh vhhq vlpso| hqrxjk e|
qrwlqj wkdw pdunhw djhqwv kdyh wr nqrz wkh prgho jryhuqlqj sulfhv +ru
dfw dv li wkh| nqrz lw, lq rughu wr holplqdwh deqrupdo h{shfwhg uhwxuqv>
li wkh prgho jryhuqlqj h{shfwhg sulfhv lv glhuhqw iurp wkdw jryhuqlqj
dfwxdo sulfhv/ wkhuh zloo eh v|vwhpdwlf deqrupdo uhwxuqv dydlodeoh lq wkh
pdunhw1
Vwurqj0irup h!flhqf| dovr lpsolhv wkdw djhqwv kdyh udwlrqdo h{shf0
wdwlrqv/ vlqfh wkh| pxvw nqrz krz wr xvh doo vruwv ri sulydwh lqirupdwlrq
dv zhoo dv sxeolf> zkhuh vwurqj0irup h!flhqf| glhuv iurp vhpl0vwurqj lv
derxw wkh hhfwv ri sulydwh lqirupdwlrq rq wkh pdunhw +ixoo| glvfrxqwhg
lq vwurqj0irup/ qrw dw doo lq vhpl0vwurqj,1
Qrw txlwh wkh vdph lv wuxh ri zhdn0irup h!flhqf|1 Lq wklv fdvh/ vlqfh
wkh| pdnh h!flhqw xvh rqo| ri wkh sdvw klvwru| ri sulfhv/ wkh| pxvw nqrz
wkh wlph vhulhv prgho jryhuqlqj sulfhv> vwulfwo| vshdnlqj wklv grhv qrw
lpso| nqrzohgjh ri wkh xqghuo|lqj vwuxfwxudo prgho/ vlqfh wkhuh zloo eh
jhqhudoo| lqvx!flhqw lghqwli|lqj uhvwulfwlrqv1 Qhyhuwkhohvv/ lq sudfwlfh/
zlwk olplwhg vdpsohv dqg vwuxfwxudo fkdqjh/ uhfryhu| ri wkh wlph0vhulhv
sdudphwhuv e| pdunhw djhqwv iurp wkh gdwd fdq hhfwlyho| eh uxohg rxw1
Lw lv wkhuhiruh qdwxudo/ li qrw qhfhvvdu|/ wr dvvxph lq wklv fdvh wrr wkdw
djhqwv kdyh udwlrqdo h{shfwdwlrqv dqg vr nqrz wkh xqghuo|lqj prgho/
iurp zklfk wkh| duh wkhq deoh wr ghulyh wkh wlph0vhulhv sdudphwhuv1
Zkloh/ wkhuhiruh/ pdunhw h!flhqf| fdq eh uhjdughg dv lpso|lqj udwlr0
uhod{lqj vxfk dvvxpswlrqv ohdyhv klv pdlq uhvxow deryh xqlpsdluhg vhh dovr wkh
vxuyh| e| Mrugdq dqg Udgqhu +4<;5,1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
66<
qdo h{shfwdwlrqv/ udwlrqdo h{shfwdwlrqv grhv qrw lpso| pdunhw h!flhqf|1
Pdunhw h!flhqf| lv d mrlqw k|srwkhvlv derxw h{shfwdwlrqv dqg pdunhw
ehkdylrxu +vshflfdoo|/ wkh prgho ri htxloleulxp h{shfwhg uhwxuqv,1 Wkh
pdlq k|srwkhvlv derxw ehkdylrxu lv wkh fdslwdo dvvhw sulflqj prgho +vhh
ehorz,> dqg lq hpslulfdo whvwv pruh ghwdlohg dvvxpswlrqv pxvw dovr eh
pdgh derxw krz htxloleulxp uhwxuqv zloo pryh1 Ixuwkhu k|srwkhvhv frq0
fhuq wkh ehkdylrxu ri wkh djhqwv zlwk dffhvv wr glhuhqw vhwv ri cdydlodeoh*
lqirupdwlrq1 Xqghu zhdn0irup h!flhqf|/ dfwlyh pdunhw sduwlflsdqwv duh
dvvxphg wr pdnh hhfwlyh xvh rqo| ri wkh sdvw klvwru| ri sulfhv lq wkhlu
pdunhw> rqh wkhruhwlfdo edvlv iru wklv kdv ehhq lq wkh frvwv ri rewdlqlqj
dqg surfhvvlqj zlghu lqirupdwlrq +Ihljh dqg Slhufh/ 4<:9,1 Xqghu vhpl0
vwurqj0irup h!flhqf|/ wkh dvvxpswlrq pdgh lv wkh xvxdo rqh lq udwlrqdo
h{shfwdwlrqv pdfur prghov wkdw dfwlyh djhqwv xvh doo sxeolfo| dydlodeoh
lqirupdwlrq/ suhvxphg wr eh xvhdeoh dw }hur ru orz frvw1 Ilqdoo|/ lq wkh
vwurqj0irup fdvh lw lv dvvxphg wkdw wkrvh djhqwv zlwk dffhvv wr sulydwh
lqirupdwlrq ghsor| ru lqgluhfwo| lq xhqfh ixqgv wr holplqdwh h{shfwhg
uhwxuqv iurp wklv vrxufh ri lqirupdwlrq> wkhuh duh/ krzhyhu/ sureohpv
zlwk wklv/ vlqfh sulydwh lqirupdwlrq fdqqrw eh vrog dw d idlu sulfh +rqfh
glyxojhg lw lv ydoxhohvv/ exw ehiruh glyxojhqfh lw lv lpsrvvleoh iru wkh
ex|hu wr dvvhvv, dqg wkrvh zlwk dffhvv kdyh/ e| ghqlwlrq/ lq jhqhudo
rqo| olplwhg ixqgv1
Ehfdxvh doo ghqlwlrqv ri pdunhw h!flhqf| lqyrnh wkh frqfhsw ri de0
qrupdo uhwxuqv/ zh duh uhtxluhg iru hpslulfdo zrun wr kdyh d wkhru| ri
wkh htxloleulxp h{shfwhg udwh ri uhwxuq iru dvvhwv1 Whvwv ri pdunhw h!0
flhqf| duh frqgxfwhg diwhu doorzdqfh iru wkh htxloleulxp udwh ri uhwxuq1
Li wkh ulvnlqhvv ri dq dvvhw grhv qrw fkdqjh ryhu wlph +ru frqyhuvho| li lwv
ulvn fkdqjhv udqgrpo| ryhu wlph, wkhq/ iru h{dpsoh/ zhdn0irup h!flhqf|
lpsolhv wkdw wkhuh vkrxog qrw eh dq h{wudsrodwlyh sdwwhuq lq wkh wlph vh0
ulhv ri uhwxuqv1 Li wkhuh zhuh d uhfxuulqj sdwwhuq ri dq| w|sh/ wudghuv
zkr uhfrjql}hg wkh sdwwhuq zrxog xvh lw wr pdnh deqrupdo surwv1 Wkh
yhu| hruw wr xvh vxfk sdwwhuqv zrxog/ xqghu wkh h!flhqf| k|srwkhvlv/
ohdg wr wkhlu holplqdwlrq1
WKH FDSP PRGHO RI HTXLOLEULXP H[SHFWHG
UHWXUQV
Wkh htxloleulxp h{shfwhg uhwxuq rq dvvhwv lv d fhqwudo wrslf lq prghuq
sruwirolr wkhru| dqg wkh lqwhuhvwhg uhdghu lv gluhfwhg wr/ iru h{dpsoh/
Frshodqg dqg Zhvwrq +4<;;, iru d ixoo glvfxvvlrq +Doohq/ 4<;8/ surylghv
d khosixo lqwurgxfwlrq,1 Wkh iroorzlqj eulhi dffrxqw ri rqh vxfk dqg
zlgho| xvhg wkhru|/ wkh fdslwdo dvvhw sulflqj prgho +FDSP,/ pxvw vx!fh
673
Frqiurqwlqj Prghov zlwk Idfwv
khuh1
Rqo| li wudghuv duh ulvn0qhxwudo5 zloo wkh| eh lqglhuhqw wr wkh ydul0
delolw| ri wkh uhwxuqv +l1h1 wkh ulvn, rq wkhlu sruwirolr1 Ulvn0dyhuvh lqgl0
ylgxdov zloo eh frqfhuqhg derxw djjuhjdwh sruwirolr ulvn dqg zloo uhtxluh
d ulvn suhplxp rq hdfk dvvhw +ru fodvv ri dvvhwv,1 E| frpelqlqj dvvhwv
lq wkh sruwirolr lw lv srvvleoh wr glyhuvli| dzd| vrph ri wkh ulvn +wkh cxq0
v|vwhpdwlf* ulvn, dvvrfldwhg zlwk dq dvvhw1 Krzhyhu/ wr wkh h{whqw wkdw
wkh uhwxuqv rq dq dvvhw pryh zlwk wkh pdunhw/ wkhuh zloo eh d frpsr0
qhqw ri ulvn +v|vwhpdwlf ulvn, wkdw fdqqrw eh glyhuvlhg dzd|1 Dvvxplqj
rswlpdo sruwirolr glyhuvlfdwlrq/ wkh ulvn suhplxp uh hfwv wkh dvvhw*v
v|vwhpdwlf ulvn dqg khqfh lwv frqwulexwlrq wr wkh ryhudoo yduldelolw| ri uh0
wxuqv rq wkh sruwirolr1 Wklv suhplxp zloo eh lqfoxghg lq wkh htxloleulxp
h{shfwhg udwh ri uhwxuq rq wklv dvvhw/ lq dgglwlrq wr wkh jhqhudo udwh ri
uhwxuq rq wkh sruwirolr1
Wkh dojheud ri wkh FDSP wkhru| lv vwudljkwiruzdug1 Doo lqyhvwruv duh
dvvxphg/ dv lq vwdqgdug sruwirolr dqdo|vlv/ wr pd{lpl}h wkhlu h{shfwhg
xwlolw| vxemhfw wr wkhlu sruwirolr zhdowk frqvwudlqw= wklv |lhogv iru hdfk
lqyhvwru dq rswlpdoo| glyhuvlhg kroglqj ri hdfk dvvhw lq whupv ri wkh
h{shfwhg uhwxuqv rq doo dvvhwv1 Lw lv frqyhqlhqw wr djjuhjdwh wklv htxdwlrq
dfurvv lqyhvwruv dqg uhihu wr wkh uhvxow dv wkh dvvhw ghpdqg ri d cw|slfdo*
lqyhvwru +exw wklv grhv qrw lpso| wkdw doo lqyhvwruv duh wkh vdph> lw lv
phuho| dq h{suhvvlrq lqglfdwlqj dyhudjlqj,1 Wkh FDSP*v frqwulexwlrq
lv wkhq wr qrwh wkdw dw dq| wlph wkhuh zloo eh d fhuwdlq vwrfn ri hdfk
dvvhw rxwvwdqglqj lq wkh pdunhw zklfk pxvw eh khog lq htxloleulxp1 Wr
eh khog/ lwv h{shfwhg uhwxuq +dqg vr lwv sulfh, pxvw vdwlvi| wkh dvvhw
ghpdqg htxdwlrq ri wkh w|slfdo lqyhvwru= lq rwkhu zrugv/ wklv htxdwlrq
lv wxuqhg urxqg dqg vroyhg iru wkh qhfhvvdu| h{shfwhg uhwxuq lq whupv
ri wkh rxwvwdqglqj dvvhw txdqwlw|1 Dv wkh txdqwlw| ri hdfk dvvhw ulvhv/
krzhyhu ulvn| lw pd| eh/ lw frqwulexwhv rqo| d olwwoh wr wkh wrwdo ulvn rq
wkh zkroh pdunhw sruwirolr ehfdxvh lw irupv rqo| d vpdoo sduw ri wklv
glyhuvlhg zkroh> khqfh wkh ulvh lq qhfhvvdu| h{shfwhg uhwxuq +idoo lq
sulfh, lv qhjoljleoh dqg zh fdq wdon ri cwkh* uhtxluhg h{shfwhg uhwxuq rq
dq dvvhw/ lqghshqghqwo| ri lwv txdqwlw| rxwvwdqglqj1 Fohduo| wklv pxvw
qrw eh sxvkhg wrr idu= vrph dvvhwv +h1j1 groodu oldelolwlhv lq fxuuhqf|
5 Frqvlghu dq lqglylgxdo idfhg zlwk d srvvleoh jdpeoh> kh pd| fkrrvh hlwkhu wr
uhfhlyh 433 iru vxuh/ ru wr wrvv d frlq dqg uhfhlyh 83 li khdgv rffxu dqg 483 li wdlov
rffxu1 Wkh h{shfwhg rxwfrph ri wklv odwwhu fkrlfh lv 433 @ 318+83, . 318+483,1 Wkh
txhvwlrq lv= zloo wkh lqglylgxdo suhihu wkh dfwxduldo ydoxh ri wkh jdpeoh +wklv lw lwv
h{shfwhg rxwfrph, zlwk fhuwdlqw| ru zloo kh suhihu wkh jdpeoh lwvhoiB Li kh suhihuv wkh
jdpeoh kh lv d ulvn oryhu> li kh lv lqglhuhqw kh lv ulvn qhxwudo> dqg li kh suhihuv wkh
vxuh rxwfrph kh lv ulvn dyhuvh1 Lw lv dovr srvvleoh wr frpsxwh wkh pd{lpxp dprxqw
ri zhdowk dq lqglylgxdo zrxog eh zloolqj wr jlyh xs lq rughu wr dyrlg wkh jdpeoh1 Wklv
lv wkh qrwlrq ri d ulvn suhplxp +vhh Sudww/ 4<97> Duurz/ 4<:4> Pdunrzlw}/ 4<8<,1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
674
sruwirolrv, duh rxwvwdqglqj lq yhu| odujh txdqwlwlhv dqg wkhlu ulvn fdqqrw
eh ixoo| glyhuvlhg dzd|1
Dvvxph wkuhh dvvhwv/ rqh ri zklfk |lhogv d vdih uhwxuq 1 Wkh w|s0
lfdo lqyhvwru pd{lpl}hv wkh h{shfwhg xwlolw| ri hqg0ri0shulrg zhdowk/
Hw X+Zw.4 ,/ zlwk uhvshfw wr z4 dqg z5 vxemhfw wr wkh exgjhw frqvwudlqw
Zw.4 @ z4 U4w . z5 U5w . +4 z4 z5 ,Uw
+4,
zkhuh zl lv wkh vkduh lq wkh lqyhvwru*v sruwirolr ri dvvhw l/ Ulw lv wkh
dfwxdo uhwxuq gxulqj wkh shulrg/ Zw.4 lv klv hqg0shulrg zhdowk h{suhvvhg
dv dq lqgh{ +ehjlqqlqj shulrg zhdowk @ 4,/ X lv klv xwlolw| ixqfwlrq
+X ! A 3> X !! ? 3,/ dqg Hw lv klv h{shfwdwlrq iruphg dw wkh ehjlqqlqj ri
wkh shulrg zkhq kh wdnhv klv lqyhvwphqw ghflvlrq1
Wdnh d Wd|oru vhulhv h{sdqvlrq ri X+Zw.4 , durxqg Hw +Zw.4 ,=
X+Zw.4 , @ X+Hw Zw.4 , . X ! =+Zw.4 Hw Zw.4 , .
3=8X !! =Hw +Zw.4 Hw Zw.4 ,5 . ==== +5,
Ljqrulqj whupv ri kljkhu rughu wkdq wzr/ +vhh h1j1 Nudxv dqg Olw}hq0
ehujhu +4<:9,/ Kzdqj dqg Vdwfkhoo +4<<:, ru Kduyh| dqg Vlggltxh +5333,
iru vrph lpsolfdwlrqv ri kljkhu prphqwv, ru dowhuqdwlyho| dvvxplqj wkdw
wkh djhqw*v xwlolw| ixqfwlrq rqo| ghshqgv rq wkh uvw wzr prphqwv/ wkh
h{shfwdwlrq ri wklv lv=
Hw X+Zw.4 , @ X+Hw Zw.4 , . X ! =+Hw Zw.4 Hw Zw.4 , .
3=8X !! =Hw +Zw.4 Hw Zw.4 ,%
+6,
zklfk lv htxdo wr
X+z4 Hw U4w . z5 Hw U5w . i4 z4 z5 jUw , .
3=8X !! +z45 54 . z55 55 . 5z4 z5 45 , +7,
zkhuh lm @ Hw +^Ulw Hw Ulw `^Umw Hw Umw `,/ wkh fryduldqfh ehwzhhq wkh
uhwxuqv ri dvvhwv l dqg m/ dqg 5l @ wkh yduldqfh ri l*v uhwxuq1
Wkh uvw rughu frqglwlrqv zlwk uhvshfw wr z4 dqg z5 uhvshfwlyho| duh=
X ! ^Hw U4w Uw ` @ X !! ^z4 54 . 45 z5 `
+8,
X ! ^Hw U5w Uw ` @ X !! ^z5 55 . 45 z4 `
+9,
X !!
@s
X!
+:,
dqg
675
Frqiurqwlqj Prghov zlwk Idfwv
s lv ghqhg dv d phdvxuh ri devroxwh ulvn dyhuvlrq1
Zh fdq h{suhvv +8, dqg +9, lq wkh FDSP irup zlwk uhtxluhg h{shfwhg
dvvhw uhwxuqv dv wkh ghshqghqw yduldeoh +wkh vwdqgdug sruwirolr dqdo|vlv
kdv wkh dvvhw vkduhv dv wkh ghshqghqw yduldeohv,= z4 dqg z5 duh jlyhq
e| wkh rxwvwdqglqj vwrfnv ri dvvhwv 4/ 5 dqg wkh vdih dvvhw lq wkh ryhudoo
pdunhw1
Uhduudqjlqj +8, wkh h{shfwhg uhwxuq rq dvvhw 4 iru h{dpsoh lv =
Hw U4w @ Uw . s+z4 54 . z5 45 , @ Uw . sz+{4 ^54 45 ` . 45 ,
+;,
dqg rq dvvhw 5
Hw U5w @ Uw . s+z5 55 . z4 45 , @ Uw . sz+{5 ^55 45 ` . 45 ,
+<,
zkhuh {4 lv wkh vkduh ri dvvhw 4 lq wkh ulvn| sduw ri wkh pdunhw sruwirolr/
{4 @ zz4 > {5 @ zz5 > dqg z lv wkh vkduh ri wkh ulvn| sduw lq wkh wrwdo
pdunhw +z @ z4 . z5 ,= Vxssrvh zh uhjdug dvvhw 4 dv ehlqj d vlqjoh
dvvhw dqg dvvhw 5 dv ehlqj d sruwirolr ri doo dvvhwv lq wkh ulvn| pdunhw/
wkdw lv hvvhqwldoo| cwkh* ulvn| pdunhw sruwirolr1 Wkhq +;, uhyhdov wkdw
wkh h{shfwhg uhwxuq rq d vlqjoh dvvhw frqvlvwv ri wkh vdih uhwxuq soxv d
ulvn suhplxp uh hfwlqj ulvn dyhuvlrq +s,/ wkh ryhudoo vkduh ri ulvn| dvvhwv
lq wkh zkroh pdunhw +z, dqg wkh fryduldqfh ehwzhhq dvvhw 4 dqg wkh
ulvn| pdunhw +45 ,/ cv|vwhpdwlf ulvn*= wkhuh lv dovr d vpdoo frpsrqhqw
lq wkh ulvn suhplxp iru wkh h{whqw wr zklfk ulvn rq dvvhw 4 h{fhhgv wklv
fryduldqfh +cglyhuvldeoh ulvn*,/ exw wklv kdv d qhjoljleoh hhfw ehfdxvh
lw lv pxowlsolhg e| {4 / wkh vpdoo vkduh ri dvvhw 4 lq wkh ulvn| pdunhw1
Vr surylghg dvvhw 4 lv vpdoo lq uhodwlrq wr wkh pdunhw/ lwv ulvn fdq eh
yluwxdoo| wrwdoo| glyhuvlhg dzd| dqg eduho| dhfwv wkh h{shfwhg uhwxuq1
Vxssrvh zh qrz ohw dvvhw 5 hpeudfh wkh zkroh ulvn| pdunhw1 Wkhq
{5 @ 4 dqg zh kdyh iurp +<,=
Hw Upw @ Uw . sz5p
+43,
zkhuh p lv wkh zkroh ulvn| sruwirolr vr wkdw lq wklv fdvh 55 @ 5p 1 Iurp
wklv lw iroorzv wkdw=
sz @
+Hw Upw Uw ,
5p
+44,
Wklv phdqv zh fdq uhzulwh dvvhw 4*v uhtxluhg uhwxuq/ vxevwlwxwlqj +43,
lqwr +;,/ dv
Hw U4w @ Uw .
i{4 ^54 4p ` . 4p j
+Hw Upw Uw ,
5p
+45,
ru pruh jhqhudoo| iru dq| dvvhw
Hw Ulw @ Uw . l +Hw Upw Uw ,
+46,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
676
zkhuh zh dvvxph wkh zhljkw {l lv vpdoo hqrxjk wr eh ljqruhg dqg vr
p
l @ l5
+47,
p
Hw Upw Uw lv wkh h{fhvv uhwxuq uhtxluhg rq d xqlw ri wkh dyhudjh sruw0
irolr= lw lv wkhuhiruh cwkh frvw ri dyhudjh ulvn*1 Lqglylgxdo dvvhwv frp0
pdqg kljkhu ru orzhu h{fhvv uhwxuqv dffruglqj wr wkhlu cehwd*/ l / zklfk
phdvxuhv wkhlu v|vwhpdwlf ulvn/ wkh fryduldqfh ri wkh lwk dvvhw zlwk wkh
pdunhw sruwirolr/ glylghg e| wkh yduldqfh ri wkh pdunhw sruwirolr dv
looxvwudwhg lq jxuh 47141
(W 5LW
(W 5PW
WKHPDUNHW
SULFLQJOLQH
5W
βL
Iljxuh 4714= Wkh Sulflqj ri d Vhfxulw|
RWKHU PRGHOV RI HTXLOLEULXP H[SHFWHG
UHWXUQV
D uhodwhg lghd wr wkh FDSP dvvxphv wkdw djjuhjdwh pdunhw ulvn lv grp0
lqdwhg e| vhyhudo vrxufhv ri ulvn= sulqflsdo frpsrqhqw dqdo|vlv fdq eh
xvhg wr vhsdudwh wkhvh rxw1 Wkxv wkhuh lv qrw rqh exw vhyhudo pdunhw
culvnv* ru yduldeohv zklfk duh xqfruuhodwhg +iru h{dpsoh/ rqh yduldeoh
pljkw eh d phdvxuh ri wkh zruog exvlqhvv f|foh/ dqrwkhu pljkw eh wkh
Qruwk0Vrxwk whupv ri wudgh,= wkh h{shfwhg uhwxuq dvvrfldwhg zlwk wdn0
lqj hdfk ulvn lv sulfhg h{dfwo| dv deryh qrz wklqn ri cdvvhw 4* dqg
677
Frqiurqwlqj Prghov zlwk Idfwv
cdvvhw 5* dv ehlqj uhvshfwlyho| ch{srvxuh* wr culvn 4* dqg culvn 5*= wkdw lv/
|rx ex| Â4 zruwk ri dvvhwv frpelqhg lq vxfk d zd| wkdw wkhlu uhwxuq lv
fruuhodwhg shuihfwo| zlwk yduldeoh 4/ dqg vlploduo| iru yduldeoh 51 Wkhvh
ixqgdphqwdo ulvnv ehlqj sulfhg/ hdfk dvvhw lv wkhq sulfhg/ e| duelwudjh/
dffruglqj wr lwv dprxqw ri hdfk ulvn1 Wklv lv wkh duelwudjh sulflqj wkhru|
+DSW,/ gxh wr Urvv +4<:9,1
FRQVXPSWLRQ FDSP
Lq fkdswhuv 44 dqg 45 zh vhw rxw Oxfdv* wkhru| ri dvvhw sulflqj1 Wkh
uvw0rughu frqglwlrq iru wkh rswlpdo frqvxpswlrq dqg sruwirolr ghflvlrq
lv jlyhq e| htxdwlrq +<, lq fkdswhu 44 dv
x! +fw , @ Hw +4 . Ulw ,x! +fw.4 ,
+48,
zkhuh zh kdyh jhqhudolvhg wkh prgho wr doorz iru wkh uhdo jurvv uhwxuq/
4 . Ulw / rq wkh lwk dvvhw*6
Uhfdoolqj wkdw wkh fryduldqfh ri [ dqg \ ghqrwhg e| fry[\ lv e|
ghqlwlrq fry[\ H+[ H[,=+\ H\ , H[\ H[=H\ +zh
dovr qrwh lq sdvvlqj wkdw fry+4 . [> \ , @ fry[\ ,/ zh fdq zulwh wkh
xqfrqglwlrqdo irup ri +48, dv
H+4 . Ulw , @
+4 fry+4 . Ulw =Nw.4 ,,
HNw.4
+49,
6 Li zh zulwh wkh exgjhw frqvwudlqw dv
S| n R| r|n n ^| K|n ' ER| n _| r| n K|
zkhuh K| uhsuhvhqwv kroglqjv ri wkh ulvnohvv dvvhw/ r| uhsuhvhqwv kroglqjv ri wkh ulvn|
dvvhw/ R| lv wkh sulfh ri wkh ulvn| dvvhw/ ^| lv wkh qrupdolvhg sulfh ri wkh ulvnohvv dvvhw1
Wkh rswlpl}dwlrq |lhogv
ES| ! b| ' f
+4,
!b| ^| n q.| b|n ' f
+5,
!b| R| n q.| ER|n n _|n b|n ' f
+6,
zkhuh b| lv wkh Odjudqjh pxowlsolhu1
Wkhvh frqglwlrqv/ 406/ |lhog

'

'
ES|n ^| ES| ER|n n _|n ES|n q.|
R| ES| q.|
qrwlqj wkdw ^|3 ' E n - zkhuh - lv wkh ulvn0iuhh uhdo udwh1 Rc r fdq eh jhqhudol}hg
wr dq| dvvhw dv lq wkh wh{w1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
678
,
lv fdoohg wkh vwrfkdvwlf glvfrxqw idfwru ru sulflqj
zkhuh Nw.4 @ xx3+f+fww.4
,
nhuqho1
Iru d ulvn iuhh uhdo lqwhuhvw udwh Uwi wkh fryduldqfh whup lv }hur vr
wkdw
4
H+4 . Uwi , @
+4:,
HNw.4
3
Iru wkh sruwirolr ri dvvhwv zh kdyh wkdw
H+4 . Upw , @
+4 fry+Upw > Nw.4 ,,
HNw
+4;,
Zh fdq hpsor| +49,/ +4:, dqg +4;, wr rewdlq
H+Ulw , Uwi @ +4 . Uwi ,fry+Ulw > Nw.4 ,
+4<,
dqg
H+Ulw , Uwi @
^fryUlw > Nw.4 `+HUpw Uwi ,
fry+Upw > Nw.4 ,
+53,
Htxdwlrq +53, looxvwudwhv wkdw dq dvvhw*v h{shfwhg uhdo uhwxuq lq h{fhvv ri
wkh ulvn iuhh udwh lv kljkhu wkh pruh qhjdwlyh wkh fryduldqfh ehwzhhq wkh
dvvhw dqg wkh udwlr ri pdujlqdo xwlolwlhv1 Wklv lv ehfdxvh zkhq frqvxps0
wlrq lv vpdoohu/ pdujlqdo xwlolw| lv kljkhu1 Dq dvvhw zlwk d pruh qhjdwlyh
fryduldqfh wkxv lpsolhv wkdw wkh uhwxuq lv orzhu zkhq frqvxpswlrq lv
orz1 Wklv lv wkh suhflvho| wkh vwdwh krzhyhu zkhq zhdowk lv pruh ydoxhg>
frqvhtxhqwo| wkh djhqw uhtxluhv d kljkhu ulvn suhplxp wr krog wkh dv0
vhw1 Wklv prgho ri wkh uhodwlrqvkls ehwzhhq h{shfwhg uhwxuqv lv nqrzq
dv wkh frqvxpswlrq FDSP1 Htxdwlrq +53, vkrzv wkh uhodwlrqvkls wr wkh
ruglqdu| FDSP zkhuh fryduldqfhv zlwk wkh sulflqj nhuqho uhsodfh wkrvh
zlwk wkh pdunhw sruwirolr1
Wr pdnh wkh prgho hpslulfdoo| rshudwlrqdo zh kdyh wr vshfli| wkh
huuru vwuxfwxuh ri wkh prgho dqg wkh irup ri wkh xwlolw| ixqfwlrq1
Frqvlghu wkh lpsolfdwlrqv ri wkh prgho li
x+fw , @
f4
4
w
4
+54,
zkhuh lv d frqvwdqw dqg lv fdoohg wkh frh!flhqw ri uhodwlyh ulvn dyhuvlrq17
Wklv ixqfwlrq kdv wkh frqyhqlhqw surshuw| wkdw lw qhvwv wkh olqhdu fdvh/
@ 3/ dqg wkh orjdulwkplf irup/ @ 41 +Wkh odwwhu fdvh lv irxqg e|
hpsor|lqj O*Krslwdo*v uxoh zkhq @ 4=,
Iru wklv ixqfwlrq
x! +fw.4 ,
fw.4 @+
,
+55,
x! +fw ,
fw
679
Frqiurqwlqj Prghov zlwk Idfwv
Lq dgglwlrq zh dvvxph wkdw frqvxpswlrq dqg dvvhw uhwxuqv duh mrlqwo|
frqglwlrqdoo| orjqrupdoo| glvwulexwhg1 Wklv doorzv xv wr pdnh xvh ri wkh
idfw wkdw iru d frqglwlrqdoo| orjqrupdoo| glvwulexwhg yduldeoh/ \>
oq Hw +\ , @ Hw ^oq \ ` . 3=8yduw ^oq \ `
+56,
Wdnlqj orjdulwkpv ri +48, dqg hpsor|lqj +56, zh rewdlq
Hw oq+4 . Ulw , @ oq . Hw +oq fw.4 oq fw ,
3=8iydu oq+4 . Ulw , . 5 ydu+oq
fw.4
,,
fw
5fry+oq+4 . Ulw , oq+
fw.4
,,j +57,
fw
+fw.4 ,
+Qrwh ohw \w @ +4.Uxlw3,x
vr iurp +48,/ oq+4, @ oq . oq+Hw \w ,= Wkhq
+fw ,
xvh +55, dqg +56, wr rewdlq
3
fw.4
,, . 3=8iydu+4 . Ulw , .
fw
fw.4
fw.4
, 5fry+oq+4 . Ulw , oq+
,,j
5 ydu+
fw
fw
3 @ oq . Hw +oq+4 . Ulw , oq+
Uhduudqjlqj wklv h{suhvvlrq jlyhv +58,,
Li wkh uhdo udwh ri lqwhuhvw rq dq dvvhw/ Uwi / lv ulvn iuhh +57, vlpsolhv
wr
fw.4
oq+4 . Uwi , @ oq . Hw +oq fw.4 oq fw , 3=8+ 5 ydu+oq
, +58,
fw
Zkhq wkhuh lv d qrplqdo dvvhw |lhoglqj d fhuwdlq qrplqdo udwh/ Qw /
zh fdq hpsor| wkh ghqlwlrq ri wkh uhdo udwh ri lqwhuhvw iru wklv dvvhw/
vd|/ 4 . UQw / dv
4 . UQ w +4 . Qw ,Sw
Sw.4
+59,
zkhuh S lv wkh sulfh ohyho1
Vxevwlwxwlrq ri +59, lq +48, dqg hpsor|lqj wkh vdph phwkrg dv deryh
jlyhv xv wkh uhtxluhg qrplqdo uhwxuq rq wkh vdih qrplqdo dvvhw/ dvvxplqj
frqglwlrqdo qrupdolw|/ dv=
oq+4 . Qw , @ oq . Hw +oq Sw.4 oq Sw , . Hw +oq fw.4 oq fw ,
fw.4
fw.4
, . ydu+, . 5fry+oq
> ,, +5:,
3=8+ 5 ydu+oq
fw
fw
zkhuh @ oq+Sw.4 , oq+Sw ,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
67:
Zh revhuyh iurp +58, wkdw wkh ulvnohvv uhdo udwh kdv d olqhdu uhodwlrq0
vkls zlwk wkh h{shfwhg fkdqjh lq uhdo frqvxpswlrq1 Li wkh ulvnohvv uhdo
udwh lv dvvxphg frqvwdqw zh fdq uhduudqjh +58, zlwk h{shfwhg fkdqjhv lq
frqvxpswlrq dv wkh ohiw kdqg vlgh yduldeoh1 Wklv lv wkh uhodwlrqvkls hvwl0
pdwhg e| Kdoo +4<:;,= xqghu udwlrqdo h{shfwdwlrqv fkdqjhv lq frqvxps0
wlrq vkrxog eh ruwkrjrqdo wr lqirupdwlrq gdwhg sulru wr wkh lqirupdwlrq
vhw frqglwlrqlqj h{shfwdwlrqv1
Iurp +5:, zh revhuyh wkdw wkh vdih qrplqdo dvvhw uhvsrqgv zlwk d
xqlw frh!flhqw wr h{shfwhg lq dwlrq dqg lv dovr ghwhuplqhg e| whupv
zklfk uh hfw lq dwlrq ulvn suhpld1
Vxewudfwlqj +58, iurp +57, zh rewdlq
Hw oq+4 . Ulw , oq+4 . Uwi , . 3=8ydu+4 . Ulw ,
fw.4
@ fry oq+4 . Ulw ,> oq+
,
fw
ru wkdw
fw.4
oq Hw
@ fry oq+4 . Ulw ,> oq+
,
fw
+4 . Uwi ,
+4 . Ulw ,
+5;,
+5<,
Wkh lpsolfdwlrq ri wklv prgho lv wkdw ulvn suhpld duh ghwhuplqhg e|
wkh fryduldwlrq ehwzhhq wkh dvvhw dqg fkdqjhv lq frqvxpswlrq wlphv wkh
frh!flhqw ri uhodwlyh ulvn dyhuvlrq1
Ehfdxvh wkh fryduldqfh whup lv uhodwlyho| vpdoo lq dfwxdo gdwd/ hpslu0
lfdo whvwv ri wklv prgho kdyh qrw ehhq deoh wr h{sodlq wkh h{fhvv uhwxuq
rq vwrfnv ryhu erqgv/ rq dyhudjh vrph 9( lq wkh XV ryhu 433 |hduv dqg
odujh lq rwkhu frxqwulhv dv zhoo +vhh Fdpsehoo/ 4<<9,/ zlwkrxw dvvxplqj
d srvvleo| lpsodxvleo| odujh frh!flhqw ri uhodwlyh ulvn dyhuvlrq +zhoo lq
h{fhvv ri 43,1 Wklv kdv ehfrph nqrzq dv wkh htxlw| suhplxp sx}}oh diwhu
wkh slrqhhulqj frqwulexwlrq ri Phkud dqg Suhvfrww +4<;8,1 Uhfhqw frq0
wulexwlrqv kdyh vxjjhvwhg d ydulhw| ri phfkdqlvpv wkdw pljkw uhvroyh wkh
sx}}oh1 Wkhvh lqfoxgh pruh jhqhudo vshflfdwlrqv ri wkh xwlolw| ixqfwlrq
+h1j1 Hsvwhlq dqg ]lq/ 4<;</ 4<<4,/ wkh lqwurgxfwlrq ri kdelw irupd0
wlrq +h1j1 Frqvwdqwlqlghv/ 4<<3> Fdpsehoo dqg Frfkudqh/ 4<<</ 5333,/
vxuylyru eldv +h1j1 Eurzq/ Jrhw}pdqq dqg Urvv/ 4<<8,/ shvr sureohpv
+vhh ehorz> Ulhw}/ 4<;;,/ khwhurjhqhrxv djhqwv +h1j1 Frqvwdqwlqlghv dqg
Gx!h/ 4<<9,1 Pdq| ri wkhvh lghdv duh vxshueo| vhw rxw lq Fdpsehoo/ Or
dqg PdfNlqod| +4<<:,1
67;
Frqiurqwlqj Prghov zlwk Idfwv
GHWHUPLQDWLRQ RI DVVHW SULFH EHKDYLRXU
XQGHU UDWLRQDO H[SHFWDWLRQV
Zh qh{w frqvlghu wkh lpsolfdwlrqv ri wkh dvvxpswlrq ri udwlrqdo h{shf0
wdwlrqv iru wkh ehkdylrxu ri dvvhw sulfhv1 Frqvlghu wkh uhwxuq wr kroglqj
dq dvvhw ryhu wkh shulrg w wr w . 4/ vd| d vkduh1 Zh kdyh
Uw.4 @
Sw.4 . Gw.4
4
Sw
+63,
zkhuh Gw.4 lv dq| glylghqg ru sd|phqw lq wkh shulrg dqg Sw > Sw.4 lv
wkh sulfh ri wkh dvvhw dw wlph w dqg w . 4=
Zh dvvxph lqlwldoo| iru vlpsolflw| wkdw lqyhvwruv duh ulvn qhxwudo vr
wkdw yld duelwudjh h{shfwhg uhwxuqv duh htxdo wr wkrvh rq d ulvnohvv dvvhw
zlwk d uhdo udwh ri lqwhuhvw/ U/ dvvxphg frqvwdqw/ vr wkdw Hw Uw.4 @ U1
Vlqfh Sw lv sduw ri wkh fxuuhqw lqirupdwlrq vhw/ wkh h{shfwdwlrq ri
+63, zkhq uhduudqjhg lv jlyhq e|
Sw @
4
4
Hw Sw.4 .
Hw Gw.4
+4 . U,
+4 . U,
+64,
Dvvxplqj udwlrqdo h{shfwdwlrqv dqg vroylqj wklv prgho iruzdugv Q
shulrgv zh rewdlq
Sw @ Hw
Q
[
l@4
Hw Sw.Q
Gw.l
.
l
+4 . U,
+4 . U,Q
+65,
Lq wkh devhqfh ri vshfxodwlyh exeeohv +vhh ehorz, zh dvvxph wkdw wkh
odvw whup jrhv wr }hur dv zh ohw Q jr wr lqqlw|1 Lq wklv fdvh wkh fxuuhqw
dvvhw sulfh lv htxdo wr wkh h{shfwhg ydoxh ri wkh vwuhdp ri glylghqgv lqwr
wkh lqghqlwh ixwxuh1 Wklv whup lv wkh ixqgdphqwdo ri wkh surfhvv zklfk
zh fdq fdoo Iw 1
Xqghu wkhvh dvvxpswlrqv zh kdyh wkdw=
%"
&
[ Gw.l
Sw @ Hw
+66,
+4 . U,l
l@4
Zh fdq zulwh +66, lq wkh htxlydohqw irup
%"
&
[ Gw.l
Gw
Hw Gw.4
Hw Gw.4
Gw
.
.
Sw @ Hw
.
l
5
+4
.
U,
4
.
U
4
.
U
+4
.
U,
+4
. U,5
l@4
Hw Gw.5
Hw Gw.5
. ===
6
+4 . U,
+4 . U,6
+67,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
Uhduudqjlqj +67, zh kdyh wkdw
%"
&
[ Gw.l
Hw Gw.4
Gw
Hw Gw.5
.
.
Sw @ Hw
.
. ===
l
5
+4 . U,
4 . U +4 . U,
+4 . U,6
l@4
vr wkdw xvlqj +66, zh fdq uhzulwh +68, dv
%"
&
[ Gw.l
Sw
Gw
.
.
Sw @ Hw
l
+4
.
U,
4
.
U
4
.
U
l@4
67<
+68,
+69,
Pxowlso|lqj wkurxjk e| 4 . U dqg uhduudqjlqj zh rewdlq wkh dowhu0
qdwlyh irup
%"
&
[ Gw.l
4
Gw
. Hw
Sw @
+6:,
U
U
+4 . U,l4
l@4
Htxdwlrq +6:, vkrzv wkdw wkh fxuuhqw sulfh ri wkh vwrfn lv htxdo wr
wkh glylghqg glylghg e| U soxv d whup lq wkh glvfrxqwhg vwuhdp ri h{0
shfwhg ixwxuh fkdqjhv lq glylghqgv1 Lq wklv irup wkh prgho lv dphqdeoh
wr hpslulfdo whvwlqj lq wkh irup ri frlqwhjudwlrq dqdo|vlv +vhh Wlph0Vhulhv
Dqqh{,1 Li glylghqgv duh d qrq0vwdwlrqdu| surfhvv exw fkdqjhv lq glyl0
ghqgv duh vwdwlrqdu| wkhq wkh vwrfn sulfh lv frlqwhjudwhg zlwk glylghqgv
zlwk frh!flhqw U4 1
Dq lqvljkwixo vshfldo fdvh ri wkh deryh prgho dulvhv zkhq glylghqgv
duh h{shfwhg wr jurz dw d frqvwdqw udwh j=
Iru wklv fdvh
Hw Gw.l @ +4 . j,Hw Gw.l4 @ +4 . j,l Gw
+6;,
Vxevwlwxwlqj +6;, lq +66, zh rewdlq
Sw @
+4 . j,Gw
+4 . j,5 Gw
+4 . j,6 Gw +4 . j,7 Gw
.
.
.
. ===
+4 . U,
+4 . U,5
+4 . U,6
+4 . U,7
Zh fdq uhzulwh +6<, dv
+4 . j,Gw
+4 . j,
+4 . j,5
+4 . j,6
4.
.
.
.===
Sw @
+4 . U,
+4 . U, +4 . U,5 +4 . U,6
+6<,
+73,
4
@ 4 . { . {5 . {6 . ===== iru m { m? 4 dqg dvvxplqj
Uhfdoolqj wkdw 4{
wkdw j ? U +dv lw pxvw eh vlqfh wkh vwrfn sulfh lv qrw lqqlwh, zh fdq
uhzulwh +73, dv
;
<
@ +4 . j,G
+4 . j,Gw ?
4
w
Sw @
+74,
= 4 +4.j, > @ U j
+4 . U,
+4.U,
683
Frqiurqwlqj Prghov zlwk Idfwv
Wklv irup lv wkh Jrugrq jurzwk prgho1 Lw ghprqvwudwhv wkh lpsru0
lv forvh wr j wkhq vpdoo shupdqhqw fkdqjhv lq U fdq
wdqw srlqw wkdw li U
kdyh d pdmru lpsdfw rq wkh vwrfn sulfh1
Wkh dvvxpswlrq pdgh lq wkh ghulydwlrq ri wkh dvvhw sulfh +66, lv wkdw
h{shfwhg vwrfn uhwxuqv duh htxdo wr d frqvwdqw ulvn0iuhh uhdo udwh ri lq0
whuhvw +lq wkh frqwh{w ri FDSP vd| d frqvwdqw uhdo udwh soxv d frqvwdqw
ulvn suhplxp,1 Uhod{lqj wklv dvvxpswlrq lq wkh deryh iudphzrun uhvxowv
lq wkh orvv ri dqdo|wlf wudfwdelolw| vlqfh h{shfwdwlrqv zrxog eh ri d qrq0
olqhdu irup1 Lw lv lqwhuhvwlqj wr qrwh dq dssur{lpdwlrq iru wkh fdvh ri d
yduldeoh uhwxuq wkdw suhvhuyhv wudfwdelolw| zklfk zdv lqlwldoo| hpsor|hg
e| Fdpsehoo dqg Vkloohu +4<;; d/ e,1
Wdnlqj orjdulwkpv ri +63, zh kdyh wkdw
oq+4 . Uw.4 , @ oq+Sw.4 . Gw.4 , oq+Sw , @ oq+Sw.4 , oq+Sw ,
Gw.4
. oq 4 .
Sw.4
Qrwlqj wkdw
Gw.4
Sw.4
+75,
@ h{s^oq Gw.4 oq Sw.4 ` zh fdq uhzulwh +75, dv
uw.4 @ sw.4 sw . oq+4 . h{s+gw.4 sw.4 ,,
+76,
zkhuh orzhu fdvh ohwwhuv qrz uhsuhvhqw orjdulwkpv vr wkdw uw @ oq+4 .Uw ,
Ohwwlqj }w @ gw.4 sw.4 dqg wdnlqj d uvw0rughu Wd|oru h{sdqvlrq ri
wkh odvw whup lq +76, durxqg wkh phdq ydoxh ri } zh rewdlq
uw.4 @ sw.4 sw . oq+4 . h{s+},, .
h{s+},
^gw.4 sw.4 }` +77,
4 . h{s+},
Uhduudqjlqj +77, zh rewdlq
uw.4 @
sw.4
h{s+},gw.4
} h{s+},
.
sw . oq+4 . h{s+},, 4 . h{s+},
4 . h{s+},
4 . h{s+},
+78,
Ohwwlqj @
h{s+},
+4.h{s+},,
4
4.h{s+},
dqg qrwlqj wkdw
4
@ h{s+}, dqg 4 @
zh fdq uhzulwh +78, dv
uw.4 @ sw.4 . +4 ,gw.4 sw oq+, +4 , oq
4
+79,
ru
uw.4 @ sw.4 . +4 ,gw.4 sw . zkhuh @ oq+, +4 , oq 4
+7:,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
684
Iru d frqvwdqw glylghqg0sulfh udwlr
@
4
4. G
S
+7;,
Wkh dyhudjh ydoxh ri wkh glylghqg sulfh udwlr revhuyhg lq hfrqrplhv
vkrzv wkdw wkh ydoxh ri zloo eh forvh wr xqlw|/ durxqg 31<9 lq wkh XV
iru h{dpsoh/ vr wkdw lq +7:, wkh zhljkw rq wkh orj sulfh lv forvh wr rqh
dqg wkdw rq orj glylghqgv forvhu wr }hur1 Fohduo| wkh dssur{lpdwlrq
zloo eh pruh dffxudwh wkh vpdoohu wkh yduldwlrq lq wkh orj glylghqg sulfh
udwlr1 Dqdo|vlv e| Fdpsehoo dqg Vkloohu +4<;;d, lv vxjjhvwlyh wkdw wkh
dssur{lpdwlrq grhv qrw surgxfh jurvv ylrodwlrqv ri uhdolw|/ sduwlfxoduo|
dw wkh prqwko| ohyho ri dqdo|vlv1 Hvvhqwldoo| wkh dssur{lpdwlrq surylghv
dqdo|wlfdo wudfwdelolw| dw wkh frvw ri vrph huuru lq wkh vwdwhphqw ri dyhu0
djh uhwxuqv1 Iurp dq hpslulfdo shuvshfwlyh glylghqgv dsshdu wr eh pruh
sduvlprqlrxvo| h{sodlqhg dv d orjolqhdu udwkhu wkdq d olqhdu wlph0vhulhv
surfhvv dqg wklv lv dq dgydqwdjh ri wkh dssur{lpdwlrq phwkrg1
Zh fdq wdnh h{shfwdwlrqv ri +7:, dqg vroyh iruzdug wr rewdlq
%"
&
[ l
. Hw
sw @
^+4 , gw.l.4 uw.l.4 `
+7<,
4
l@3
zkhuh zh dvvxph rqfh djdlq lq wkh devhqfh ri vshfxodwlyh exeeohv wkdw
wkh whup Hw " sw." lv }hur1
Zh revhuyh iurp +7<, wkdw wkh fxuuhqw orj vwrfn sulfh lv kljkhu wkh
kljkhu h{shfwhg ixwxuh glylghqgv dqg wkh orzhu h{shfwhg ixwxuh uhwxuqv
+l1h1 wkh vwrfn glvfrxqw idfwruv,1
Zh fdq wudqvsrvh +7<, iroorzlqj wkh w|sh ri surfhgxuh rxwolqhg deryh
+67, wr rewdlq wkh htxlydohqw irup
%"
&
[ l
sw @
+gw.l.4 uw.l.4 ,
. gw . Hw
+83,
4
l@3
+Klqw= lq +7<, dgg lqvlgh vtxduh eudfnhwv wkh whup gw gw . gw.4 gw.4 . 5 gw.5 5 gw.5 . . q gw.q q gw.q / uhduudqjh lq whupv ri
gw.4.l 1 Wkh whupv lq gw . gw.4 . 5 gw.5 . === . q gw.q wkhq fdq eh
vxevwlwxwhg rxw1,
Htxdwlrq +7:, lv xvhixo iru looxvwudwlqj dqrwkhu srlqw1Wdnlqj h{shfwd0
wlrqv dw wlph w ri +7:, dqg vxewudfwlqj wkh uhvxowdqw iurp +7:, zh rewdlq
uw.4 Hw uw.4 @ +sw.4 Hw sw.4 , . +4 ,^gw.4 Hw gw.4 `
+84,
Li zh ohdg +83, rqh shulrg/ wdnh h{shfwdwlrqv ri lw dw wlph w dqg
vxewudfw wkh uhvxowdqw iurp lw/ zh rewdlq dq h{suhvvlrq iru sw.4 Hw sw.4
zklfk zh vxevwlwxwh lq +84, wr rewdlq
685
Frqiurqwlqj Prghov zlwk Idfwv
uw.4 Hw uw.4 @
;
"
S l
A
A
g
g
.
H
+g
,
H
u
? w.4
w w.4
w.4
w.5.l
w.5.l
l@3
"
S l
A
A
+gw.5.l uw.5.l ,
Hw
=
l@3
<
A
A
@
A
A
>
. +4 , +gw.4 Hw gw.4 ,
+85,
Uhfrjqlvlqj wkdw gw.4 Hw gw.4 @ Hw.4 gw.4 Hw.4 gw +Hw gw.4 Hw gw ,
xqghu wkh lqirupdwlrq dvvxpswlrqv ri wkh prgho/ l1h1 gw lv revhuydeoh dw
wlph w/ gw.4 dw w . 4> vr Hw.4 gw.4 @ gw.4 / Hw gw @ gw > zh fdq uhzulwh +85,
lq wkh irup suhvhqwhg e| Fdpsehoo +4<<4, qdpho|
uw.4 Hw uw.4 @ Hw.4
"
[
l
gw.4.l Hw
l@3
"
[
^Hw.4
"
[
l gw.4.l
l@3
l uw.4.l Hw
l@4
@ +Hw.4 Hw ,
"
[
"
[
l uw.4.l `
l@4
l +gw.4.l uw.5.l , +86,
l@3
Htxdwlrq +86, lv lqirupdwlyh1 Zh revhuyh wkdw xqh{shfwhg vwrfn uh0
wxuqv duh d ixqfwlrq ri uhylvlrqv ri h{shfwhg ixwxuh fkdqjhv lq glylghqgv
dqg uhylvlrqv ri vwrfn glvfrxqw idfwruv1 Ixuwkhu/ fhwhulv sdulexv dq xs0
zdug uhylvlrq lq h{shfwdwlrqv ri ixwxuh uhwxuqv ohdgv wr d idoo lq wkh sulfh
wrgd| vlqfh iru d jlyhq glylghqg vwuhdp wklv fdq rqo| eh jhqhudwhg iurp
d orzhu sulfh wrgd|1 Wkh htxdwlrq dovr ghprqvwudwhv krz uhylvlrqv ri h{0
shfwdwlrqv lqwr wkh lqghqlwh ixwxuh lpsdfw rq wkh fxuuhqw lqqrydwlrq lq
uhwxuqv1 Xqghu udwlrqdo h{shfwdwlrqv wkhvh uhylvlrqv duh qhzv1 Krzhyhu
wklv fuhdwhv d sureohp lq hpslulfdo zrun zklfk hqghdyrxuv wr hvwlpdwh
wkh lpsdfw ri hpslulfdo phdvxuhv ri qhzv rq xqdqwlflsdwhg dvvhw uhwxuqv1
Fohduo| wkh lpsdfw ri fxuuhqw qhzv fdq kdyh dq dpeljxrxv lpsdfw rq
xqdqwlflsdwhg uhwxuqv ghshqglqj rq wkh lpsolfdwlrqv lw kdv iru ixwxuh
uhylvlrqv wr h{shfwdwlrqv1 Iru h{dpsoh wkh qhzv wkdw wkh fxuuhqw prqh|
vwrfn lv kljkhu wkdq dqwlflsdwhg frxog eh lqwhusuhwhg dv d vljqdo iru ixwxuh
wljkwhqlqj ri prqh| vwrfn ru d pryh wr d pruh uhod{hg uhjlph1 Vlploduo|
rxwsxw jxuhv kljkhu wkdq dqwlflsdwhg frxog vljqdo wkh hqg ri d voxps
ru wkh ehjlqqlqj ri dq lq dwlrqdu| hslvrgh1 Wklv lpsolhv wkdw zlwkrxw
nqrzohgjh ri fxuuhqw dqg ixwxuh srolf| uhjlphv hpslulfdo hvwlpdwhv ri
wkh lpsdfw ri fxuuhqw qhzv zloo eh cqrq0vwuxfwxudo* vr wkdw wkh hvwlpdwhg
frh!flhqwv pd| eh gl!fxow wr lqwhusuhw1
Zh fdq looxvwudwh wzr rwkhu lpsruwdqw lpsolfdwlrqv iru wkh srwhqwldo
ehkdylrxu ri dvvhw sulfhv dqg uhwxuqv xvlqj d prglhg h{dpsoh eruurzhg
iurp Fdpsehoo/ Or dqg Pdfnlqod| +4<<:,1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
686
Dvvxph wkdw wkh orj ri glylghqgv lv jhqhudwhg e| wkh surfhvv
gw.4 @ gw . xw.4
+87,
zkhuh xw.4 lv d udqgrp yduldeoh dqgm m? 4= Vr zh dvvxph wkdw glylghqgv
duh d vwdwlrqdu| surfhvv lq wklv h{dpsoh1
Ixuwkhu dvvxph wkdw wkh vwrfn glvfrxqw idfwru lv jhqhudwhg e|
Hw uw.4 @ u . |w
+88,
zkhuh u lv d frqvwdqw dqg |w lv jhqhudwhg e| wkh surfhvv
|w @ |w4 . yw
+89,
zkhuh m m? 4 dqg yw lv d udqgrp yduldeoh1
|w lv dq revhuydeoh yduldeoh dqg pljkw uhsuhvhqw fkdqjlqj ulvn1 Jlyhq
wkhvh dvvxpswlrqv zh fdq vxevwlwxwh lq +83, wr rewdlq wkh vroxwlrq iru
wkh sulfh ri wkh dvvhw1 Wkh orj sulfh ri wkh dvvhw +83, lv jlyhq e|
+"
,
[ l
. Hw
sw @
^+4 , gw.4.l uw.4.l `
+8:,
4
l@3
e|
Wkh uvw whup lq wkh eudfhv/ vxevwlwxwlqj iru ixwxuh glylghqgv/ lv jlyhq
+4 ,^Hw gw.4 . Hw gw.5 . 5 Hw gw.6 . = = = ` @
+4 ,^gw . 5 gw . 5 6 gw . = = = ` +8;,
dqg wkh vhfrqg whup e|
iHw uw.4 . Hw uw.5 . 5 Hw uw.6 . = = = j @
iu . |w . +u . |w , . 5 +u . 5 |w , . = = = j +8<,
vr wkdw vxevwlwxwlqj lq +8:, zh rewdlq
sw @
|w
+4 ,
u
. gw
4
4 4 4 +93,
4
@ 4 . { . {5 . {6 . = = = iru m { m? 4,
+uhfdoolqj djdlq wkdw 4{
Zh fdq vxevwlwxwh +93, lqwr +7:, wr rewdlq wkh vroxwlrq iru wkh rqh
shulrg vwrfn uhwxuq dv
|w.4
+4 ,
u
. gw.4
. +4 ,gw.4
uw.4 @ 4
4 4 4 +4 ,
u
|w
. gw
. +94,
4
4 4 4 687
Frqiurqwlqj Prghov zlwk Idfwv
Xvlqj +87, dqg +89, dqg vlpsoli|lqj wklv jlyhv
uw.4 @ u . xw.4 . |w yw.4
+4 ,
+95,
Wkh vroxwlrq iru orj sulfhv looxvwudwhv wkdw zkhq wkh yduldelolw| ri |w
lv vpdoo/ vr wkdw wkh yduldelolw| ri h{shfwhg uhwxuqv lv vpdoo/ wklv grhv qrw
lpso| wkdw wkh yduldelolw| ri sulfhv qhhg eh vpdoo1 Wklv lv ehfdxvh wkh
ghqrplqdwru ri wkh whup lq |w fdq eh vpdoo1 Pruh irupdoo|/ dvvxplqj
wkh fryduldqfh ehwzhhq |w dqg gw lv }hur/ zh kdyh iurp +93, wkdw wkh
yduldqfh ri sulfhv lv jlyhq e|
5s @
5|
+4 ,5 5 5
5y
.
g zkhuh 5| @
5
5
+4 ,
+4 ,
4 5
+96,
5
x lv wkh yduldqfh ri glylghqgv1
zkhuh 5g @ 4
5
5
Fohduo| | fdq eh vpdoo exw 5s odujh1
Zh fdq dovr vroyh iru wkh uhgxfhg irup DUPD wlph0vhulhv uhsuh0
vhqwdwlrq +vhh Wlph0Vhulhv Dqqh{, ri orj uhwxuqv dqg orj sulfhv lq rxu
h{dpsoh1
Xvlqj wkh odj rshudwru/ dqg vxevwlwxwlqj iru |w zh rewdlq iru uhwxuqv
uw.4 @ u . xw.4 .
yw
yw.4
+4 O, +4 ,
+97,
Pxowlso|lqj rxw wkh odj rshudwru zh rewdlq
uw.4 @ uw . u+4 , . xw.4 xw .
yw
yw.4
+4 , +4 ,
+98,
Wkh vxppdwlrq ri wkh wzr huuru whupv rq wkh uljkw kdqg vlgh ri +98,
fdq eh uhzulwwhq dv d prylqj dyhudjh huuru surfhvv ri rughu rqh vr wkdw
revhuyhg uhwxuqv iroorz dq DUPD +4/ 4, surfhvv1 Ghshqglqj xsrq wkh
fryduldqfh ehwzhhq qhzv lq glylghqgv dqg qhzv lq uhwxuqv wkh surfhvv
fdq h{klelw srvlwlyh ru qhjdwlyh dxwrfruuhodwlrq1
Hpsor|lqj d vlplodu surfhvv iru fkdqjhv lq dvvhw sulfhv zh rewdlq wkh
DUPD uhsuhvhqwdwlrq iru wkh ohyho ri wkh orj dvvhw sulfh dv +zkhuh O lv
wkh odj rshudwru,
sw +4 O,+4 O,
@
+ u,+4 ,+4 , +4 ,xw +4 O,
yw +4 O,
xw .
+4 ,
4 +4 ,
+99,
vr wkdw wkh ohyho ri wkh orj ri wkh dvvhw sulfh lv ghvfulehg e| dq DUPD+5/
4, lq wklv h{dpsoh1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
688
Dowkrxjk erwk uhwxuqv dqg dvvhw sulfhv duh iruhfdvwdeoh lq wklv h{0
dpsoh wkhuh fdq eh qr suhvxpswlrq wkdw wkh pdunhw lv lqh!flhqw1 Wkh
fuxfldo hohphqw lq wkh ghqlwlrq lv wkdw deqrupdo uhwxuqv vkrxog qrw
eh iruhfdvwdeoh lq dq h!flhqw pdunhw1 Zh zloo revhuyh dqrwkhu h{dp0
soh ri wklv qh{w zkhuh wkh h{fkdqjh udwh kdv d suhglfwdeoh sdwk lq wkh
Gruqexvfk ryhuvkrrwlqj prgho hyhq wkrxjk wkh pdunhw lv h!flhqw e|
frqvwuxfwlrq1
RSHQ HFRQRP\ PRGHOV ZLWK HIILFLHQW IL0
QDQFLDO PDUNHWV
Lq Fkdswhu 43 zh vhw rxw wkh jhqhudo ehkdylrxu ri pdfur prghov ri wkh
rshq hfrqrp| zlwk h!flhqw pdunhwv dqg Qhz Fodvvlfdo sulfh ehkdylrxu/
xqghu {hg dqg rdwlqj h{fkdqjh udwhv1 Khuh zh irfxv lq pruh ghwdlo
rq wkh ehkdylrxu ri qrplqdo h{fkdqjh udwhv xqghu rdwlqj/ xqghu ydu|0
lqj dvvxpswlrqv derxw sulfh ehkdylrxu1 Wklv zloo looxvwudwh wkh uroh ri
qdqfldo h!flhqf| shu vh lq rshq pdfur prghov1
Rxu prgho lv edvhg rq wkdw rxwolqhg e| Gruqexvfk +4<:9, lq klv
vhplqdo sdshu1 Iru vlpsolflw| lw lv dvvxphg wkdw wkhuh lv shuihfw fdslwdo
prelolw| ehwzhhq frxqwulhv +l1h1 wudqvdfwlrqv frvwv duh qhjoljleoh dqg
lqwhuqdwlrqdo dvvhwv duh shuihfw vxevwlwxwhv,1 Frqvlghu lqlwldoo| d ulvn0
qhxwudo djhqw zkr lv idfhg zlwk wkh fkrlfh ehwzhhq kroglqj d grphvwlf
ru iruhljq erqg iru wkh gxudwlrq ri rqh shulrg +vd| <3 gd|v,1 Wkh qrplqdo
udwhv ri lqwhuhvw lq wkh iruhljq frxqwu| dqg lq wkh grphvwlf frxqwu| duh
jlyhq e| Uw dqg UwI uhvshfwlyho|1 Vlqfh wkh erqgv duh shuihfw vxevwlwxwhv/
dvvhw pdunhw htxloleulxp uhtxluhv wkdw wkh h{shfwhg udwhv ri uhwxuq rq wkh
wzr erqgv eh htxdo1 Wklv h{shfwhg udwh ri uhwxuq kdv wzr frpsrqhqwv1
Wkh uvw frpsrqhqw lv wkh lqwhuhvw udwh rq wkh erqg/ zklfk zh fdq
dvvxph wr eh nqrzq dw w> wkh vhfrqg frpsrqhqw lv wkh h{shfwhg fdslwdo
jdlq ru orvv iurp h{fkdqjh udwh fkdqjhv gxulqj wkh <30gd| shulrg1
Lw iroorzv wkdw wkh vshfxodwlyh frqglwlrq iru htxloleulxp/ nqrzq dv
xqfryhuhg lqwhuhvw duelwudjh lv=
Uw @ UwI . +Hw Vw.4 Vw ,
+9:,
zkhuh Uw > UwI duh wkh grphvwlf dqg iruhljq qrplqdo lqwhuhvw udwh/ Vw
lv wkh orjdulwkp ri wkh fxuuhqw h{fkdqjh udwh +khuh grphvwlf xqlwv shu
iruhljq xqlw, dqg Hw Vw.4 lv wkh h{shfwdwlrq ri wkh udwh lq shulrg rqh +<3
gd|v lq rxu h{dpsoh,1 D ulvh lq Vw lq rxu qrwdwlrq khuh uhsuhvhqwv d
ghsuhfldwlrq ri wkh krph fxuuhqf|1 Htxdwlrq +9:, wkhuhiruh lpsolhv wkdw
wkh lqwhuhvw udwh glhuhqwldo lq idyrxu ri grphvwlf erqgv pxvw eh htxdo
wr wkh h{shfwhg ghsuhfldwlrq ri wkh h{fkdqjh udwh1
689
Frqiurqwlqj Prghov zlwk Idfwv
Iru h{dpsoh/ li wkh grphvwlf fxuuhqf| sd|v 45 shu fhqw lqwhuhvw +shu
<3 gd|v, dqg wkh iruhljq fxuuhqf| sd|v 7 shu fhqw lqwhuhvw/ d grphv0
wlf lqyhvwru ex|lqj iruhljq fxuuhqf| dw wkh ehjlqqlqj ri wkh shulrg dqg
frqyhuwlqj edfn dw wkh hqg ri wkh shulrg zloo/ dvvxplqj wkh grphvwlf
fxuuhqf| ghsuhfldwhv e| ; shu fhqw/ h{shfw wr qlvk xs zlwk vx!flhqw
grphvwlf fxuuhqf| wr pdnh klp lqglhuhqw ehwzhhq kroglqj grphvwlf ru
iruhljq erqgv1
Wkhuh lv dovr d iruzdug pdunhw iru iruhljq h{fkdqjh lq pdq| h{fkdqjh
udwhv +l1h1 wudghuv fdq dw wlph w frqwudfw wr wudgh iruhljq fxuuhqf| dw wlph
w . 4,1 Lq wkh iruzdug pdunhw/ odujh wudqvdfwruv duh uhtxluhg wr sxw xs
rqo| yhu| vpdoo dprxqwv ri prqh| dv cpdujlq uhtxluhphqwv*/ vr wkhuh lv
qr qhhg wr glvfrxqw1 Frqvhtxhqwo|/ xvlqj d vlplodu dujxphqw wr wkh rqh
deryh/ duelwudjh lpsolhv wkh fryhuhg lqwhuhvw duelwudjh frqglwlrq wkdw
Uw @ UwI . +Iw Vw ,
+9;,
zkhuh Iw lv wkh orjdulwkp ri wkh iruzdug h{fkdqjh udwh dw wlph w iru
shulrg w . 41 Zh qrwh wkdw wkh fryhuhg frqglwlrq lv ulvnohvv dqg krogv
yld duelwudjh uhjdugohvv ri wkh pdqqhu lq zklfk h{shfwdwlrqv duh iruphg1
Xqfryhuhg duelwudjh lv d vshfxodwlyh frqglwlrq/ khqfh wkh h{solflw dv0
vxpswlrq ri ulvn0qhxwudo lqyhvwruv1 Htxdwlqj wkh fryhuhg dqg xqfryhuhg
frqglwlrq zh rewdlq wkdw Iw @ Hw Vw.4 vr wkdw wkh iruzdug udwh lv d gluhfw
phdvxuh ri wkh pdunhw*v h{shfwdwlrq ri wkh ixwxuh h{fkdqjh udwh1 Wkh
surshuwlhv ri wkh iruzdug udwh dv d suhglfwru ri ixwxuh vsrw udwhv kdv ehhq
d irfxv ri pxfk hpslulfdo uhvhdufk dv zh zloo glvfxvv ehorz1
Lq wkh Gruqexvfk prgho lw lv dvvxphg wkdw sulfhv lq jrrgv ru oderxu
pdunhwv duh lq wkh vkruw whup cvwlfn|* zlwk uhvshfw wr fkdqjhv lq pdunhw
frqglwlrqv1 Wklv frxog/ iru lqvwdqfh/ eh ehfdxvh ri wkh h{lvwhqfh ri pxowl0
shulrg zdjh ru sulfh frqwudfwv dv lq wkh Qhz Nh|qhvldq prgho1 Lw iroorzv
iurp wklv dvvxpswlrq wkdw sxufkdvlqj srzhu sdulw| grhv qrw krog lq wkh
vkruw uxq1 Sxufkdvlqj srzhu sdulw| +SSS, ru wkh codz ri rqh sulfh*/
vwdwhv wkdw lq wkh devhqfh ri wudqvsruw frvwv dqg rwkhu wudqvdfwlrqv frvwv
lqwhuqdwlrqdo duelwudjh lq jrrgv vkrxog holplqdwh glhuhqwldov ehwzhhq
wkh sulfhv ri jrrgv lq glhuhqw frxqwulhv1 Zh glvfxvvhg lq fkdswhu 43
rq wkh rshq hfrqrp| krz wklv zrxog qrw krog lq wkh vkruw uxq exw
vkrxog krog lq wkh orqj uxq/ zkhwkhu wudghg jrrgv duh krprjhqhrxv
dfurvv frxqwulhv ru glhuhqwldwhg dqg lpshuihfwo| frpshwlwlyh +SSS lq
wkh orqj uxq exw qrw wkh vkruw lv frquphg e| qxphurxv hpslulfdo whvwv/
h1j1 Wd|oru hw do1/ 5334,1
Xqghu SSS zh zrxog kdyh=
Vw @ sw sI
w
+9<,
zkhuh s/ sI duh wkh orjdulwkpv ri wkh grphvwlf dqg iruhljq sulfh ohyho1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
68:
Zlwk shuihfw fdslwdo prelolw| dqg SSS/ iurp +9:, dqg +9<,/ wkh htxdo0
l}dwlrq ri h{shfwhg uhdo udwhv ri lqwhuhvw lpphgldwho| iroorzv=
I
Uw +Hw sw.4 sw , @ UwI +Hw sI
w.4 sw ,
+:3,
Devhqfh ri SSS lq wkh vkruw uxq phdqv wkdw zh pd| h{dplqh wkh
ehkdylrxu ri wkh uhdo h{fkdqjh udwh/ {/ ghqhg dv=
{w @ Vw sw . sI
w
+:4,
Iru vlpsolflw| zh dvvxph wkdw rxwsxw +lq orjv,/ uhdo lqwhuhvw udwhv/
iruhljq lqwhuhvw udwhv dqg wkh iruhljq sulfh ohyho +lq orjv, duh {hg dqg
qrupdolvhg wr }hur> exw wklv grhv qrw dhfw wkh ihdwxuhv ri wkh prgho rq
zklfk zh irfxv khuh1 Wkh prgho lv jlyhq e|
Uw @ Hw Vw.4 Vw
+:5,
pvw @ pw . p @ sw Uw
+:6,
sw sw4 @ n+Vw sw4 , . xw
+:7,
zkhuh pw lv d udqgrp prqhwdu| vkrfn durxqg p/ wkh frqvwdqw dyhudjh
prqh| vxsso|/ dqg xw lv d vxsso| vkrfn1 Htxdwlrq +:6, lv d frqyhqwlrqdo
ghpdqg iru prqh| ixqfwlrq htxdwhg wr prqh| vxsso|1 Htxdwlrq +:7, lv
dq dg krf sulfh dgmxvwphqw phfkdqlvp zklfk fdswxuhv wkh k|srwkhvlv
wkdw wkh sulfh ohyho uhvsrqgv voxjjlvko| wr fkdqjhv lq wkh h{fkdqjh udwh/
yld djjuhjdwh ghpdqg dqg lwv hhfwv rq wkh yroxph ri qhw wudgh1 Wkh
vshflfdwlrq kdv wkh frqyhqlhqw surshuw| wkdw zkhq n @ 4> SSS krogv
lqvwdqwdqhrxvo| dsduw iurp wkh fxuuhqw vxsso| vkrfn1
Zkhq SSS krogv lqvwdqwdqhrxvo| wkh prgho fdq eh uhduudqjhg wr
jlyh
Vw @
Hw Vw.4 pvw xw
.
+4 . ,
+4 . ,
+:8,
Htxdwlrq +:8, zloo eh uhfrjqlvhg wr kdyh wkh vdph irup dv +64, deryh1
Zh vroyh wklv prgho xqghu wkh dvvxpswlrq wkdw djhqwv kdyh ixoo fxuuhqw
lqirupdwlrq1 Wkh uhgxfhg irup iru wkh prgho lv jlyhq e|
np . pw +4 n,pw4 @ nVw . xw +Hw Vw.4 Vw , .
+4 n,+Hw4 Vw Vw4 , +:9,
Iroorzlqj wkh surfhgxuhv rxwolqhg lq fkdswhu 5/ wkh vroxwlrq iru wkh
68;
Frqiurqwlqj Prghov zlwk Idfwv
h{fkdqjh udwh lq wkh ixoo0lqirupdwlrq fdvh lv jlyhq e|7
Vw @ +4 },p . }Vw4 . dr pw . d4 pw4 . d5 xw
+::,
zkhuh } lv wkh vwdeoh urrw ri wkh htxdwlrq=
} 5 }++4 n, . n . , . +4 n, @ 3
+:;,
4
4
> d4 @ +4 n,d3 > d5 @ n.+4},
dqg d3 @ n.+4},.+4n,
Zh qrwlfh iurp +::, wkdw/ lq wkh orqj uxq/ zkhq Vw @ Vw4 > wkh hodv0
wlflw| ri wkh h{fkdqjh udwh zlwk uhvshfw wr dq lqfuhdvh lq wkh shupdqhqw
ohyho ri prqh| vxsso| lv xqlw|1 Lq rwkhu zrugv/ lq wkh orqj uxq wkh h{0
fkdqjh udwh ghsuhfldwhv e| wkh fkdqjh lq p1 Jlyhq wkdw p lv d frqvwdqw
+:5, lpsolhv wkdw U lv }hur lq wkh orqj uxq/ dqg khqfh +:6, wkdw sulfhv
kdyh wkh vdph uhvsrqvh wr p dv wkh h{fkdqjh udwh1 Htxdwlrq +:7, lpsolhv
wkdw lq wkh orqj uxq SSS pxvw krog1 Ohdglqj +::, rqh shulrg dqg wdnlqj
h{shfwdwlrqv zh rewdlq=
Hw Vw.4 Vw @ +4 },+p Vw , . d4 pw
+:<,
Htxdwlrq +::, ghqhv dq h{shfwdwlrqv phfkdqlvp nqrzq dv uhjuhvvlyh
h{shfwdwlrqv1 Lw lqirupv xv wkdw zkhq wkh htxloleulxp h{fkdqjh udwh lv
deryh wkh fxuuhqw h{fkdqjh udwh/ wkhq h{shfwdwlrqv duh uhylvhg xszdugv
dqg ylfh yhuvd1 Wkh idfw wkdw uhjuhvvlyh h{shfwdwlrqv fdq e| fkrlfh ri
wkh uhjuhvvlyh sdudphwhu +4 }, eh udwlrqdo lv rqh lpsolfdwlrq ri wklv
Gruqexvfk prgho1 Krzhyhu/ zh vkrxog qrwh wkdw wklv surshuw| fdq krog
lq udwlrqdo h{shfwdwlrqv prghov rqo| zkhuh wkhuh lv rqh vwdeoh urrw> lw
grhv qrw krog jhqhudoo|18
Zh fdq vxevwlwxwh +::, lqwr +:4, dqg +:6, wr rewdlq
Vw @
p^4 . +4 },` . pw xw +4 n,sw4
n . +4 },
+;3,
7 Iru wkh uhdo lqwhuhvw udwh glhuhqwldo wr eh frqvwdqw/ dv lqyhvwljdwhg e| Plvknlq
+4<;4,/ dq lqqlwho| odujh lqwhuwhpsrudo vxevwlwxwlrq uhvsrqvh lv uhtxluhg iru hlwkhu
k ru q1 Wkh hylghqfh grhv qrw vxssruw vxfk uhvsrqvhv> lw lv wkhuhiruh qrw vxusulvlqj
wkdw Plvknlq frqfoxghv iurp klv uhgxfhg0irup zrun wkdw wkh glhuhqwldo ydulhv ryhu
wlph1 Vlplodu dujxphqwv dsso| wr yduldwlrq ri wkh uhdo lqwhuhvw udwh zlwklq d forvhg
hfrqrp|/ dv lqyhvwljdwhg e| Idpd +4<:8, iru wkh XVD> wklv hylghqfh kdv qrz ehhq
irxqg +e| Qhovrq dqg Vfkzhuw/ 4<::, wr vxssruw qrq0frqvwdqf|/ zklfk djdlq lv qrw
vxusulvlqj1 Zh pd| dovr qrwh wkdw wkh vl}h ri yduldwlrq lq erwk wkh glhuhqwldo dqg
wkh forvhg +h1j1 zruog, hfrqrp| ohyho ri uhdo lqwhuhvw udwhv fdqqrw eh vxjjhvwhg d
sulrul> qru fdq wkh ohqjwk ri wlph wr frqyhujhqfh +ghwhuplqhg e| 5 lq wklv prgho dv
lq xhqfhg e| doo wkh sdudphwhuv,1
8 Uhodwlyh
ulvn dyhuvlrq +UUD, lv ghqhg dv -- '
fdwlrq ri wkh xwlolw| ixqfwlrq1
3Sw 33 ESw 3 ESw ' B iru wkh vshfl0
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
68<
Htxdwlrq +;3, looxvwudwhv dqrwkhu nh| lqvljkw ri Gruqexvfk1
Wkh lpsdfw ri d fkdqjh lq wkh shupdqhqw ohyho ri prqh| vxsso| +p,
lq wkh vkruw uxq/ fhwhulv sdulexv/ lv juhdwhu wkdq xqlw| h{fhsw zkhq n @ 4
dqg frqvhtxhqwo| juhdwhu wkdq wkh orqj0uxq lpsdfw1 Wklv skhqrphqrq lv
nqrzq dv cryhuvkrrwlqj*1 Wkh udwlrqdoh iru wklv hhfw lv wkdw ehfdxvh/ lq
wkh vkruw uxq/ sulfhv duh dw d srlqw lq wlph vwlfn| ru dgmxvwlqj vorzo|/ wkh
rqo| zd| wkh prqh| pdunhw fdq uhpdlq lq htxloleulxp dv wkh shupdqhqw
ohyho ri wkh prqh| vxsso| lv lqfuhdvhg lv iru wkh lqwhuhvw udwh wr idoo1
Krzhyhu/ d idoolqj lqwhuhvw udwh kdv wr eh dvvrfldwhg zlwk dq h{shfwhg
dssuhfldwlrq ri wkh fxuuhqf|1 Frqvhtxhqwo| wkh fxuuhqw h{fkdqjh udwh
kdv wr ghsuhfldwh ixuwkhu wkdq lwv orqj0uxq ydoxh lq rughu wr jlyh ulvh
wr dqwlflsdwlrqv ri dq dssuhfldwlrq dv lw pryhv wr lwv xowlpdwh orqj0uxq
ydoxh1
Wklv phfkdqlvp lv looxvwudwhg gldjudppdwlfdoo| lq jxuh 47151 Vxs0
srvh wkdw dw wlph w @ 3 wkh srxqg2groodu udwh lv '4 @ Â41 Dw wlph w @ Q
wkh dxwkrulwlhv lqfuhdvh wkh ohyho ri wkh prqh| vxsso| e| 433 shu fhqw1
Lq wkh orqj uxq wklv fdxvhv wkh srxqg wr ghsuhfldwh djdlqvw wkh groodu
wr '4 @Â5131 Krzhyhu/ lq wkh vkruw uxq wkh srxqg ghsuhfldwhv ixuwkhu/
wr vd| '4 @ Â5183 dqg wkhq iroorzv wkh duurzhg sdwk edfn wr orqj0uxq
htxloleulxp1 Zh qrwlfh wkdw dorqj wkh duurzhg sdwk wkh srxqg lv ds0
suhfldwlqj/ exw kdv dozd|v ghsuhfldwhg uhodwlyh wr w @ 31 Wkh srvvlelolw|
wkdw h!flhqw dvvhwv pdunhwv/ lq frqmxqfwlrq zlwk vwlfn| zdjhv ru sulfhv/
frxog jlyh ulvh wr yrodwloh ehkdylrxu ri dvvhw sulfhv zdv d sulqflsdo lqvljkw
ri Gruqexvfk dqg kdv ehhq lq xhqwldo +vhh h1j1 Exlwhu dqg Ploohu/ 4<;4,1
…SHU
B
P
…
…
1W
Iljxuh 4715= H{fkdqjh Udwh Ryhuvkrrwlqj
693
Frqiurqwlqj Prghov zlwk Idfwv
Zh vkrxog dovr qrwh iurp wkh vroxwlrq iru wkh h{fkdqjh udwh wkdw/
zkloh d srvlwlyh prqhwdu| vkrfn fdxvhv d ghsuhfldwlrq ri wkh fxuuhqf|/ lwv
lpsdfw fdq eh juhdwhu ru ohvv wkdq xqlw|/ exw vlqfh p kdv qrw fkdqjhg/
wkhuh zloo dozd|v eh dq cryhuvkrrw* ri wkh orqj0uxq htxloleulxp1 Wkh
Gruqexvfk uhvxow dssolhv wr shupdqhqw +xqdqwlflsdwhg, fkdqjhv lq wkh
prqh| vwrfn +p,1
Dqrwkhu lpsruwdqw ihdwxuh ri wkh prgho lv wkdw wkh h{fkdqjh udwh kdv
d suhglfwdeoh sdwk +::,1 Krzhyhu xqfryhuhg duelwudjh zdv dvvxphg lq wkh
prgho frqvhtxhqwo| wkh dvvhw pdunhw lv h!flhqw vlqfh h{shfwhg deqrupdo
uhwxuqv +lq wklv h{dpsoh ghyldwlrqv iurp xqfryhuhg lqwhuhvw duelwudjh,
duh qrw suhglfwdeoh1 Wklv lv dqrwkhu h{dpsoh zkhuh suhglfwdelolw| ri dq
dvvhw sulfh grhv qrw ylrodwh pdunhw h!flhqf|1 Lq sduwlfxodu wkh h{fkdqjh
udwh grhv qrw iroorz d udqgrp zdon/ wkrxjk wklv lv vrphwlphv dvvxphg
lq hpslulfdo zrun1
D qxpehu ri dxwkruv kdyh dwwhpswhg wr whvw wkh Gruqexvfk prgho
+vhh h1j1 Gulvnloo/ 4<;4> Iudqnho/ 4<:<> Kddfkh dqg Wrzqvhqg/ 4<;4>
Ghphu|/ 4<;7,/ e| h{dplqlqj wkh surshuwlhv ri uhgxfhg0irup h{fkdqjh
udwh htxdwlrqv ghulyhg iurp vwuxfwxudo prghov ri wkh Gruqexvfk w|sh1
Wkh hpslulfdo uhvxowv wkh dxwkruv uhsruw duh xqidyrxudeoh wr wkh prgho1
Krzhyhu wkhuh duh d qxpehu ri sureohpv zlwk wkhvh whvwv/ wkh pdlq
rqh ehlqj wkdw wkh| doo dvvxph wkh uhjuhvvlyh irup iru h{shfwdwlrqv/
zklfk lv lq jhqhudo lqfruuhfw +Plqirug dqg Shho/ 4<;6,1 Iru h{dpsoh/
Kddfkh dqg Wrzqvhqg +4<;4, dqg Iudqnho +4<:</ 4<;5, vshfli| prghov
lq zklfk odjjhg dgmxvwphqw ru zhdowk hhfwv duh lqwurgxfhg lqwr wkh
ghpdqg iru prqh| ixqfwlrq ru odjjhg dgmxvwphqw lv lqwurgxfhg lqwr wkh
lqwhuhvw duelwudjh frqglwlrq> lq wkhvh vshflfdwlrqv/ h{shfwdwlrqv zloo qrw
eh uhjuhvvlyh1 Er{ 4714 vkrzv wkdw ryhuvkrrwlqj dovr fdq/ exw grhv qrw
qhfhvvdulo|/ rffxu lq htxloleulxp prghov1
Er{ 4714
H[FKDQJH UDWH RYHUVKRRWLQJ LQ HTXL0
OLEULXP PRGHOV
Zh qrz h{dplqh zkhwkhu h{fkdqjh udwh ryhuvkrrwlqj/ zklfk rffxuv
lq wkh Gruqexvfk prgho dv d uhvxow ri glvhtxloleulxp lq jrrgv ru
oderxu pdunhwv/ pxvw dozd|v eh uhjdughg lq d uhdo hfrqrp| dv rf0
fxuulqj dv d frqvhtxhqfh ri vxfk ihdwxuhv1 Zh zloo ghprqvwudwh wkdw
wklv lv qrw wkh fdvh1 Ryhuvkrrwlqj fdq lqghhg rffxu lq htxloleulxp
rshq hfrqrp| prghov1 Lq rughu wr h{sodlq wklv/ zh dgrsw wkh prgho
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
694
ri wkh suhylrxv vhfwlrq1
Lq nhhslqj zlwk wkh htxloleulxp iudphzrun/ zh dvvxph wkdw doo
djhqwv irup h{shfwdwlrqv rq wkh edvlv ri wkh vdph vhw ri pdfur lq0
irupdwlrq1 Wklv zh gdwh dw w 4/ dqg iru vlpsolflw| ljqruh fxuuhqw
joredo lqirupdwlrq +exw wklv grhv qrw dowhu rxu srlqw,1
Khqfh wkh h{fkdqjh pdunhw htxloleulxp frqglwlrq ehfrphv=
Hw4 Vw.4 Vw @ Uw
+4,
dqg wkh uhdo lqwhuhvw udwh glhuhqwldo/ u/ lv ghqhg dv=
uw @ Uw Hw4 sw.4 . sw
+5,
+Zh kdyh iru vlpsolflw| vhw wkh iruhljq uhdo lqwhuhvw udwh wr }hur1,
Ghpdqg iru prqh| lv=
pw @ sw . |w Uw
+6,
pw @ w . pw . yw
+7,
pw @ xw
+8,
Prqh| vxsso| lv=
zkhuh
Wklv prqh| vxsso| ixqfwlrq qrz doorzv qrw rqo| iru +xqdqwlflsdwhg,
whpsrudu| fkdqjhv lq wkh ohyho ri prqh| +y, dqg rqfh0iru0doo fkdqjhv
lq wkh ohyho +, exw dovr iru rqfh0iru0doo fkdqjhv lq wkh vwhdg| vwdwh
udwh ri lqfuhdvh +x,1 Lw zloo wkxv shuplw xv wr h{dplqh wkh glhuhqw
"
S
wl . pw . yw zkhuh
uhdfwlrqv wr wkhvh vkrfnv1 +qrwh pw @
pw @
"
S
l@3
l@3
xwl ,
Iurp wkh ghqlwlrq ri uhdo lqwhuhvw udwhv dqg wkh uhdo h{fkdqjh udwh/
{w @ Vw sw > dqg iurp +4, zh kdyh=
+Uw Hw4 sw.4 . sw , @ Hw4 Vw.4 Hw4 sw.4 +Vw sw , @
Hw4 {w.4 {w
+9,
l1h1 wkh uhdo lqwhuhvw glhuhqwldo pxvw htxdo wkh h{shfwhg uhdo ghsuh0
fldwlrq1
Qrz frpsohwh wkh prgho zlwk dq LV dqg Skloolsv fxuyh=
|w @ uw . {w
+:,
695
Frqiurqwlqj Prghov zlwk Idfwv
|w @ uw . +sw Hw4 sw , . |w4
+;,
Htxdwlrq +;, kdv wkh ixoo fodvvlfdo irup glvfxvvhg lq fkdswhu 61 Qrwh
wkdw qhlwkhu qru lv lqqlwh/ vr wkdw u zloo ydu|1 Wkh prgho kdv
ehhq vhw xs vr wkdw | @ u @ { @ 3 lq htxloleulxp1 Qrwlfh wkdw lw
ehorqjv wr wkh vdph Pxqghoo0Iohplqj idplo| dv wkdw ri fkdswhu 431
Ghqh wkh vxshuvfulsw cxh * dv cxqdqwlflsdwhg dw w04*> khqfh iru h{dp0
soh sxh
w @ sw Hw4 sw 1 Htxdwlrqv +9,+;, fdq eh vroyhg dv d uhfxuvlyh
eorfn lq whupv ri sxh
w wr jlyh=
xh
{w @ {w4 . 3 sxh
w . + 4 3 ,sw4
+<,
.
> 4 @ 3 +.,+4,.
zkhuh 3 @ ..
dqg lv wkh +dvvxphg
xqltxh, vwdeoh urrw ri wkh fkdudfwhulvwlf htxdwlrq
5 +4 .
. + . ,
@3
, .
.
.
+43,
u dqg | kdyh vlplodu vroxwlrqv= d uvw0rughu prylqj dyhudjh lq sxh
dqg uvw0rughu dxwruhjuhvvlyh frh!flhqw 1 Iurp +6, xvlqj wkhvh/ zh
rewdlq=
xh
sxh
w @ tpw
+44,
xh
{xh
w @ 3 tpw
+45,
zkhuh t @ ^4..34.+.,3 ` lv juhdwhu wkdq 3 dqg ohvv wkdq 41 Wkh
qrplqdo h{fkdqjh udwh ghsuhfldwlrq lv=
xh
xh
Vwxh @ {xh
w . sw @ t+4 . 3 ,pw
+46,
zkhuh pxh
w @ w . xw . yw
Zh fdq xvhixoo| uhzulwh
t+4 . 3 , @
. ..
+ . . ,+4 . , . + . . ,
+47,
zklfk pdnhv lw fohdu wkdw wkh ydoxh lv srvlwlyh dqg juhdwhu ru ohvv wkdq
xqlw| ghshqglqj rq doo wkh lpsdfw sdudphwhuv +krzhyhu li . . lv juhdwhu wkdq 4 wkhq lw pxvw eh ohvv,1 Qrwlfh wkdw ryhuvkrrwlqj
surshuwlhv rffxu lq wkh eurdg vhqvh wkdw erwk wkh qrplqdo dqg wkh
uhdo h{fkdqjh udwh ghsuhfldwh lq d cyrodwloh* pdqqhu lq uhvsrqvh wr
srvlwlyh prqh| vkrfnv1
Iru wkh fdvh zkhuh wkh prqh| vxsso| lv jurzlqj ryhu wlph zh ghqh
ryhuvkrrwlqj dv d uhdfwlrq ri wkh qrplqdo h{fkdqjh udwh e| d juhdwhu
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
696
sursruwlrq wkdq wkh fkdqjh lq wkh +fxuuhqw, htxloleulxp qrplqdo h{0
fkdqjh udwh/ l1h1 wkdw zklfk zrxog suhydlo zhuh wkh suhvhqw prqh|
vxsso| glhuhqfh wr eh pdlqwdlqhg lq shushwxlw|/ dsduw iurp hoh0
phqwv h{shfwhg wr eh uhyhuvhg +yw ,1
Rq wklv edvlv/ zh fdq ghwhuplqh iurp +46, wkdw=
41 Wkh h{fkdqjh udwh pd| uhvsrqg pruh wkdq sursruwlrqdwho| wr
d rqfh0iru0doo fkdqjh lq p +w ,/ dqg vr wkh htxloleulxp h{fkdqjh
udwh1 Wklv lv wkh ryhuvkrrwlqj frqvlghuhg e| Gruqexvfk +o<:9,/
zklfk ghdov zlwk vxusulvh vkliwv lq wkh shupdqhqw ohyho ri p1
51 Lw dovr pd| uhvsrqg pruh wkdq sursruwlrqdwho| wr d ulvh lq
p zklfk lv gxh wr d shupdqhqw ulvh lq lwv jurzwk udwh +x,/
Krzhyhu/ vlqfh wkhuh lv qr zd| wkdw vshfxodwruv fdq glvwlqjxlvk
ehwzhhq dqg x vkrfnv zkhq wkh| rffxu/ wkh uhdfwlrqv wr erwk
duh wkh vdph1
61 Lw dovr uhvsrqgv wr d whpsrudu| fkdqjh lq p +y,/ zklfk rq
rxu ghqlwlrq grhv qrw fkdqjh wkh htxloleulxp h{fkdqjh udwh1
Wklv lv dovr d irup ri ryhuvkrrwlqj/ wkrxjk qrw wkdw ghdow zlwk
e| Gruqexvfk1
Doo wkhvh w|shv ri ryhuvkrrwlqj lq uhvsrqvh wr prqhwdu| vkrfnv duh
txdolwdwlyho| wkh vdph dv wkrvh lq wkh cvwlfn| sulfh* prghov ri Gruq0
exvfk dqg Iudqnho/ |hw wkh| hphujh iurp dq htxloleulxp prgho1 E|
dowhulqj rxu dvvxpswlrqv derxw wkh dydlodelolw| ri fxuuhqw joredo lq0
irupdwlrq/ wkhvh uhvxowv frxog eh hdvlo| chqulfkhg* wr jlyh d ydulhw|
ri srwhqwldo ryhuvkrrwlqj uhvsrqvhv> vxevwdqwldo ryhuvkrrwlqj lv h{0
klelwhg lq hpslulfdo dssolfdwlrq e| dq htxloleulxp prgho ri wkh XN
hfrqrp| +Plqirug/ o<;3/ wkh Olyhusrro prgho,1 Wr vxp xs/ yrodwlolw|
ri wkh qrplqdo h{fkdqjh udwh +ryhuvkrrwlqj,/ dv zhoo dv ri wkh uhdo
h{fkdqjh udwh dqg uhdo lqwhuhvw glhuhqwldov/ lv qrw sulpd idflh hyl0
ghqfh ri csulfh vwlfnlqhvv*/ cglvhtxloleulxp* ru clqh!flhqf|* lq jrrgv
ru oderxu pdunhwv1
Glvwlqjxlvklqj Htxloleulxp iurp Glvhtxloleulxp Prghov ri wkh H{0
fkdqjh UdwhB
Iru jrrg phdvxuh zh fdq vkrz wkdw/ rq wkh edvlv ri d uhgxfhg irup
h{fkdqjh udwh htxdwlrq rq lwv rzq/ lw lv qrw srvvleoh wr ghwhuplqh
zkhwkhu lw frphv iurp dq htxloleulxp ru glvhtxloleulxp prgho1 Iru
wklv sxusrvh/ zh vhw xs wzr prghov lghqwlfdo lq doo uhvshfwv h{fhsw
lq wkhlu cvxsso|* ehkdylrxu1
Wkh uvw prgho lv wkh rqh mxvw ghdow zlwk> lw frqvlvwv ri wkh ghpdqg
iru prqh| ixqfwlrq +6,/ prqh| vxsso| ixqfwlrq +7,/ h!flhqw pdunhw
697
Frqiurqwlqj Prghov zlwk Idfwv
frqglwlrq +4,/ dqg LV fxuyh +:, dqg lwv vxsso| fxuyh lv dq htxloleulxp
rqh +;,1 Wkh vhfrqg prgho frqvlvwv ri wkh vdph htxdwlrqv dsduw iurp
+;, zkhuh lw kdv d vwlfn| sulfh Skloolsv fxuyh olnh Iudqnho*v/ qdpho|=
sw sw4 @ e{w4 . pw
+48,
+Zh dovr dvvxph lq wkh vslulw ri glvhtxloleulxp prghov wkdw vshfx0
odwruv kdyh ixoo fxuuhqw lqirupdwlrq/ dqg frqglwlrq wkh h{shfwdwlrqv
rshudwru wkurxjkrxw rq wkh edvlv ri fxuuhqw lqirupdwlrq1, Lw wxuqv
rxw wkdw wkh vroxwlrq iru {w lq wklv prgho lv=
{w @ 5 {w4 .
^w . xw . +4 4
4 ,+yw yw4 ,`
5 + . ,
+49,
zkhuh 5 lv wkh vwdeoh/ 4 wkh xqvwdeoh urrw ri wkh fkdudfwhulvwlf
htxdwlrq=
5 i5 ^4 . ` . @3
j . 4 .
.
+4:,
Frpsduh wklv wr wkh vroxwlrq iru wkh htxloleulxp prgho=
xh
{w @ {w4 . 3 tpxh
w . t+ 4 3 ,pw4
+4;,
Wklv vkrzv wkdw lw lv qrw srvvleoh wr glvwlqjxlvk ehwzhhq wkh htxl0
oleulxp dqg glvhtxloleulxp prghov rq wkh edvlv ri wkh uhgxfhg0irup
+uhdo, h{fkdqjh udwh htxdwlrqv dorqh> erwk duh DUPD+4/3/4, wlph0
vhulhv prghov +vhh Wlph Vhulhv Dqqh{,1 Lw iroorzv wkdw wkh prghov
fdq rqo| eh glvwlqjxlvkhg/ li dw doo/ rq wkh edvlv ri ixoo vwuxfwxudo
hvwlpdwlrq1 Wklv lv dqrwkhu h{dpsoh ri crevhuydwlrqdo htxlydohqfh*
+vhh fkdswhu 48,1
HPSLULFDO HYLGHQFH RQ PDUNHW HIILFLHQF\
Rxu lqwhuhvw khuh lv lq wkh hpslulfdo hylghqfh wkdw whvwlqj iru pdunhw
h!flhqf| vkhgv rq wkh udwlrqdo h{shfwdwlrqv k|srwkhvlv1 Lq wkhvh whvwv/
hlwkhu sduw ri wkh mrlqw k|srwkhvlv pd| idlo= wkh prgho ri htxloleulxp
h{shfwhg uhwxuqv ru wkh UH k|srwkhvlv1 Exw wklv lv xqdyrlgdeoh lq whvwlqj
d k|srwkhvlv derxw h{shfwdwlrqv zklfk duh qrw gluhfwo| revhuydeoh1
Prghoolqj ri wkh htxloleulxp h{shfwhg uhwxuq lv fohduo| fuxfldo lq hp0
slulfdo whvwv ri pdunhw h!flhqf|1 Dv wkh h{dpsohv deryh kdyh looxvwudwhg
dvvhw sulfhv ru uhwxuqv fdq kdyh suhglfwdeoh sdwwhuqv zlwkrxw qhfhvvdulo|
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
698
ylrodwlqj wkh h!flhqw pdunhwv k|srwkhvlv1 Wkh fuxfldo hohphqw lv wkdw
deqrupdo uhwxuqv vkrxog qrw eh v|vwhpdwlfdoo| suhglfwdeoh1
Vlqfh d surshuw| ri udwlrqdo h{shfwdwlrqv lv wkdw dq| glhuhqfh ri
rxwfrph iurp wkh h{shfwhg rxwfrph lv xqiruhfdvwdeoh iurp dydlodeoh
lqirupdwlrq/ zh kdyh wkdw =
Ulw @ Hw Ulw . ylw
+;4,
vr wkdw ylw lv lqghshqghqw ri Hw Ulw / wkh udwlrqdo h{shfwdwlrq1 Vxevwlwxw0
lqj wkh ghwhuplqdqwv ri Hw Ulw iurp FDSP/ iru h{dpsoh/ jlyhv=
Ulw @ Uw . tlw . ylw
+;5,
zkhuh tlw lv wkh ulvn suhplxp1 Li tlw zhuh frqvwdqw +;5, fdq eh hvwl0
pdwhg e| ruglqdu| ohdvw vtxduhv +zlwk wkh frh!flhqw rq Uw frqvwudlqhg
wr xqlw|,/ dqg wkh hvwlpdwhg huuru whup/ yf
lw / vkrxog eh lqghshqghqw ri doo
lqirupdwlrq dydlodeoh dw wkh ehjlqqlqj ri shulrg w= qrw phuho| sdvw yf
lw
+zhdn0irup,/ exw dovr doo uhohydqw gdwd +vxfk dv prqh| vxsso|/ lq dwlrq
dqg jurzwk,1 Wklv lv nqrzq dv dq ruwkrjrqdolw| whvw1 Lw lv dovr srvvleoh
wr hvwlpdwh +;4, iuhho|/ lq zklfk fdvh wkh frh!flhqw rq Uw vkrxog qrw eh
vljqlfdqwo| glhuhqw iurp xqlw|/ d ixuwkhu fkhfn rq wkh mrlqw k|srwkh0
vlv= wkh rwkhu whvwv dsso| dv ehiruh1 D ixuwkhu lpsolfdwlrq ri +;4, lv wkdw
dq| wudglqj uxoh/ W U +d v|vwhpdwlf uxoh iru wudglqj dvvhwv,/ zklfk xvhv
lqirupdwlrq dw wkh vwduw ri w/ lqfoxglqj sdvw yf
lw / wr ex| dqg vhoo dvvhw l/
lqwhqglqj wr pdnh surwv ehfdxvh
H^Ulw Hw Ulw ` m W U A 3
+;6,
pxvw idlo xqghu wkh h!flhqf| k|srwkhvlv vlqfh e| +;4,
H^Ulw Hw Ulw ` @ 3
+;7,
Vlqfh xqghu wkh h!flhqf| dvvxpswlrq dq| wudglqj vwudwhj| kdv dq
h{shfwhg deqrupdo uhwxuq ri }hur lw pxvw gr zruvh/ jlyhq wkh lqfuhdvhg
wudqvdfwlrqv frvwv dvvrfldwhg zlwk dq dfwlyh uxoh frpsduhg zlwk d wudglqj
vwudwhj| ri ex| dqg krog +gr qrwklqj,1 Wkh uhwxuqv wr hdfk uxoh glhu
e| h{shfwhg wudqvdfwlrqv frvwv1
Dq lpsruwdqw fodvv ri prghov lq zklfk h!flhqf| suhydlov +deqrupdo
uhwxuqv duh xqiruhfdvwdeoh, lv wkdw ri pduwlqjdohv1 Irupdoo| li d yduldeoh
]w lv ghvfulehg e| wkh vwrfkdvwlf surfhvv
H^]w.4 m ]w> ]w4> = = = @ ]w
+;8,
H^]w.4 ]w m ]w> ]w4> = = = ` @ 3
+;9,
ru
699
Frqiurqwlqj Prghov zlwk Idfwv
lw lv nqrzq dv wkh pduwlqjdoh surshuw|1
Wkh surfhvv
]w.4 @ ]w . xw.4
+;:,
lv wkhuhiruh d pduwlqjdoh1 Wkh huuru whup kdv wkh surshuw| wkdw Hw xw.4 @
3= Wkh huuru fdq h{klelw vwuxfwxuh vxfk dv wlph ydu|lqj khwhurvnhgdvwlflw|
+h1j1 DUFK hhfwv Wlph0Vhulhv Dqqh{,1 Li d frqvwdqw whup lv dgghg
wr +;:, lw lv nqrzq dv d vxepduwlqjdoh1 Wkh whuplqrorj| ri udqgrp
zdon dqg pduwlqjdoh duh riwhq lqwhufkdqjhg1 Lq idfw d udqgrp zdon dv
frqyhqwlrqdoo| ghqhg lv d vwurqjhu frqfhsw lq wkdw wkh huuru whup lv
dvvxphg wr eh lqghshqghqwo| dqg lghqwlfdoo| glvwulexwhg +llg,1
Vr wkh surfhvv
]w.4 @ ]w . . xw.4
+;;,
zkhuh lv d frqvwdqw dqg Hw xw.4 @ 3/ lv d vxepduwlqjdoh dqg d udqgrp
zdon li xw.4 lv llg1
WKH UDWLRQDOH IRU PDUNHW LQHIILFLHQF\
Eurdgo| wkh lpsolflw phfkdqlvp lqyroyhg iru wkh pdunhw wr eh h!flhqw
lq wkh vhqvh ri Idpd lv wkdw li dvvhw pdunhwv zhuh qrw h!flhqwo| djjuh0
jdwlqj dqg surfhvvlqj lqirupdwlrq wkh glvsdulw| ehwzhhq ixqgdphqwdo
ydoxhv dqg pdunhw sulfhv zrxog suhvhqw wudghuv zlwk surw rssruwxql0
wlhv1 Udwlrqdo vshfxodwruv lw lv dujxhg zloo/ hvvhqwldoo| lqvwdqwdqhrxvo|/
gulyh dvvhw sulfhv edfn wr wkhlu ixqgdphqwdo ydoxhv1
Wkh fdvh iru pdunhw lqh!flhqf| uhvwv rq hlwkhu ru erwk ri wkh suh0
plvvhv wkdw sulfhv ri dvvhwv pryh zkhq wkhuh lv qr qhz lqirupdwlrq frq0
fhuqlqj ixqgdphqwdov dqg wkdw wkh dfwlrq ri vshfxodwruv pd| qrw pryh
dvvhw sulfhv wrzdugv wkhlu ixqgdphqwdo ydoxhv1
Vrph hpslulfdo revhuydwlrqv wkdw duh rhuhg dv frqvlvwhqw zlwk wkhvh
suhplvvhv duh pdunhw fudvkhv vxfk dv Rfwrehu 4<wk 4<;: zkhq zruog
pdunhwv ihoo durxqg 53 shu fhqw zlwkrxw dssduhqwo| dq| lpsruwdqw qhzv
ehlqj hylghqw zklfk frxog dffrxqw iru wkh idoo/ ru vxvwdlqhg ulvhv lq sulfhv/
nqrzq dv exeeohv/ wkdw duh vhhplqjo| lqh{solfdeoh lq whupv ri pdunhw
ixqgdphqwdov1
Dovr lw lv revhuyhg wkdw pdq| wudglqj ghflvlrqv duh edvhg rq sdvw
sulfhv1 Fkduwlvp/ wkh h{wudsrodwlrq ri sdvw sulfhv/ lv zlgho| hpsor|hg dv
wkh edvlv iru wudglqj uxohv1 Lq dgglwlrq/ wkhuh lv h{whqvlyh xvh ri vwrs0orvv
rughuv/ zkhuhe| dq dvvhw lv vrog li lwv sulfh idoov e| d suh0vshflhg dprxqw
dv zhoo dv wkh jurzwk ri g|qdplf khgjlqj vwudwhjlhv vxfk dv sruwirolr
lqvxudqfh zkhuh lqyhvwruv ex| +vhoo, lqwr ulvlqj +idoolqj, pdunhwv1 Zklovw
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
69:
qrqh ri wkhvh dfwlylwlhv qhfhvvdulo| lpso| pdunhw lqh!flhqf| wkh| udlvh
wkh txhvwlrq dv wr zkhwkhu lqiruphg udwlrqdo vshfxodwruv duh duelwudjlqj
rxw wkh wudglqj uxohv ri dq| xqlqiruphg ru luudwlrqdo wudghuv> li wkh| duh
qrw/ wkhq lqh!flhqf| pd| eh suhvhqw1
Wkh uvw wkhruhwlfdo prgho ri lqh!flhqf| zh h{dplqh lv wkdw ri ud0
wlrqdo exeeohv1
VSHFXODWLYH EXEEOHV
Klvwru| lv uhsohwh zlwk h{dpsohv zkhuh dvvhw sulfhv kdyh h{klelwhg gud0
pdwlf lqfuhdvhv wkhq idoov zklfk lw lv dujxhg duh qrw uhdglo| h{sodlqhg
e| pryhphqwv lq wkh cixqgdphqwdov* ri wkh dvvhw vxfk dv wkh h{shfwhg
ixwxuh glylghqg vwuhdp1 Rqh h{dpsoh lv wkh Vrxwk Vhd exeeoh lq wkh XN=
lq wkh 4;wk fhqwxu| wkh vwrfn ri d frpsdq| zklfk wudghg lq wkh Vrxwk
Vhd h{shulhqfhg h{srqhqwldo sulfh lqfuhdvhv ehiruh soxpphwlqj lq ydoxh1
Dqrwkhu lv wkh wxols exoe hslvrgh lq Kroodqg lq wkh odwh vl{whhqwk fhqwxu|
zkhuh +vlqjoh, wxols exoev zhuh h{fkdqjhg iru odqg dqg jrog ehiruh wxolsv
xowlpdwho| ehfdph qhdu zruwkohvv +wkrxjk vhh Jduehu/ 4<;<,1
Lw lv dujxhg wkdw wkhvh hslvrghv uhsuhvhqw vshfxodwlyh exeeohv lq zklfk
wkh dqwlflsdwlrq ri ixwxuh fdslwdo jdlqv ohdgv wr vsludoolqj xszdug sulfh
pryhphqwv ehiruh wkh exeeoh hyhqwxdoo| froodsvhv ru srsv dqg wkh sulfh
h{klelwv gudpdwlf idoov1 Vlqfh wkh 4<;3v wkhuh kdv ehhq d frqvlghudeoh
dprxqw ri wkhruhwlfdo dqg hpslulfdo zrun rq vshfxodwlyh exeeohv1 Rxu
sxusrvh lq wklv vhfwlrq lv wr surylgh dq lqwurgxfwlrq wr wklv olwhudwxuh>
rxu glvfxvvlrq lv uhodwhg wr rxu hduolhu rqh lq fkdswhu 5 zkhuh zh vkrzhg
wkdw udwlrqdo h{shfwdwlrqv zlwk h{shfwdwlrqv ri ixwxuh yduldeohv fdq kdyh
exeeohv lq wkhlu vroxwlrq1
Zh dvvxph iru vlpsolflw| wkdw lqyhvwruv duh ulvn qhxwudo vr wkdw yld
duelwudjh h{shfwhg uhwxuqv duh htxdo wr wkrvh rq d ulvnohvv dvvhw zlwk
udwh ri uhwxuq U/ dvvxphg frqvwdqw/ vr wkdw Hw Uw.4 @ U1 Jlyhq wkhvh
dvvxpswlrqv zh rewdlq +64, deryh/ zklfk zh uhsurgxfh iru frqyhqlhqfh=
Sw @
4
4
Hw Sw.4 .
Hw Gw.4
+4 . U,
+4 . U,
+;<,
Dvvxplqj udwlrqdo h{shfwdwlrqv dqg vroylqj wklv prgho iruzdugv Q
shulrgv dv lq 66 zh rewdlq
%Q
&
[ Gw.l
Hw Sw.Q
Sw @ Hw
+<3,
l .
+4
. U,l
l@4 4 . U
Wkh vhfrqg whup lq +<3, lv wkh glvfrxqwhg ydoxh ri wkh vwrfn sulfh
Q shulrgv lq wkh ixwxuh1 Lq wkh devhqfh ri exeeohv dv zh ohw Q jr wr
69;
Frqiurqwlqj Prghov zlwk Idfwv
lqqlw| zh dvvxph wkdw wklv whup jrhv wr }hur1 Li vr/ wkh fxuuhqw dvvhw
sulfh lv htxdo wr wkh h{shfwhg ydoxh ri wkh vwuhdp ri glylghqgv lqwr wkh
lqghqlwh ixwxuh= wklv h{suhvvlrq lv wkh ixqgdphqwdo ri wkh surfhvv/ Iw 1
Wkh lghd ri vshfxodwlyh exeeohv lv wkdw htxdwlrq +<3, lv dovr frqvlvwhqw
zlwk udwlrqdo h{shfwdwlrqv vroxwlrqv rwkhu wkdq wkh ixqgdphqwdo vroxwlrq1
Li zh wu| wkh vroxwlrq
Sw @ Iw . Ew
+<4,
lq +;<, zh qg wkdw dv orqj dv Ew iroorzv wkh surfhvv
+4 . U,Ew @ Hw Ew.4
+<5,
lw lv d ydolg pdwkhpdwlfdo vroxwlrq wr +;<,1
E| vxevwlwxwlrq ri +<4, lq +;<, zh rewdlq
Sw @ Iw . Ew @
4
4
4
Hw Ew.4 .
Hw Iw.4 .
Hw Gw.4
+4 . U,
+4 . U,
+4 . U,
+<5, lv frqvlvwhqw zlwk wklv dqg wkh vroxwlrq irup iru
Iw @
4
4
Hw Iw.4 .
Hw Gw.4
+4 . U,
+4 . U,
w Iw.Q
lv wkh vdph dv +<3, zkhuh H+4.U,
l whqgv wr }hur dv Q whqgv wr 41 Lq
rwkhu zrugv li djhqwv eholhyh wkdw wkh surfhvv Ew lv gulylqj dvvhw sulfhv
wkhq lw zloo eh d udwlrqdo vroxwlrq/ d vhoi0ixooolqj surskhf|> qrwlfh
wkdw lw lpsolhv Hw Sw.Q @ Hw Iw.Q . Hw Ew.Q zklfk h{sorghv hqgohvvo|/
vr wkdw wkh exeeoh fdqqrw eh h{shfwhg wr exuvw hyhu1
Dv d frqvhtxhqfh ri wkh udwlrqdo h{shfwdwlrqv dvvxpswlrq
Ew.4 @ Hw Ew.4 . w.4
+<6,
zkhuh w.4 lv d udqgrp huuru1
Vxevwlwxwlrq ri +<5, lqwr +<6, jlyhv
Ew.4 @ +4 . U,Ew . w.4
+<7,
+<7, pdnhv fohdu wkdw wkh ghwhuplqlvwlf vroxwlrq iru E lv dq dv|ps0
wrwlfdoo| h{sorvlyh surfhvv1 Lw lv wkhuhiruh lpsruwdqw wkdw wkhuh eh qr
wudqvyhuvdlolw| frqglwlrq sxwwlqj d olplw rq wklv surfhvv> vxfk d frqgl0
wlrq zdv krz zh uxohg rxw exeeohv lq fkdswhu 51
Zh fdq dovr zulwh wkh vroxwlrq iru wkh exeeoh lq wkh irup +vhh Vdojh/
4<<:,
Ew @
Pw
w
+<8,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
zkhuh @
4
+4.U,
69<
dqg
Hw Pw.4 @ Pw
lpso|lqj wkdw Pw lv d pduwlqjdoh surfhvv1
Ohdglqj +<8, rqh shulrg dqg wdnlqj h{shfwdwlrqv/ qrwlqj wkdw
ghwhuplqlvwlf surfhvv/ zh rewdlq
Hw Ew.4 @
Pw
Hw Pw.4
w Ew
Ew
@ iurp +<9, w.4 @ w.4 @
w.4
+<9,
4
w
lv d
+<:,
Frqvhtxhqwo| d exeeoh frqwdlqv d pduwlqjdoh frpsrqhqw dqg dq|
surfhvv iru Pw vdwlvi|lqj wklv frqglwlrq lv d ydolg udwlrqdo exeeoh1
D jhqhudo irup ri d vwrfkdvwlf pduwlqjdoh surfhvv lv jlyhq e|
Pw @ w Pw4 . xw yw
+<;,
zkhuh wkh udqgrp yduldeoh w / kdv frqglwlrqdo h{shfwdwlrq wkdw H^w.4 m
Lw ` @ 4= Rwkhu dvvxpswlrqv duh wkdw Hiyw.l j @ 3> l @ 4 q> > H^w Pw m
Lw ` @ 3> H^w yw m Lw ` @ 3> H^w xw m Lw ` @ 3> H^xw Pw m Lw ` @ 3> H^xw yw m
Lw ` @ 3> H^yw Pw m Lw ` @ 3=
Wkhvh dvvxpswlrqv hvwdeolvk wkh lqghshqghqfh ri doo vwrfkdvwlf ydul0
deohv lq doo ohdgv dqg odjv1 Qr uhvwulfwlqj dvvxpswlrqv duh uhtxluhg frq0
fhuqlqj wkh qdwxuh ri wkh udqgrp yduldeoh xw =
Vxssrvh xw @ 4> ghqh d udqgrp yduldeoh kw wkdw lv qrupdoo| glv0
wulexwhg zlwk phdq dqg yduldqfh 5k 1 Zh fdq wkhq h{sorlw wkh surshuw|
ri frqglwlrqdo orj qrupdolw| dqg ghqh w iru dq| duelwudu| frqvwdqw dv
w @ h
ehfdxvh
5 5
kw . 5 k
5 5k
Hhkw @ Hhkw.4 @ h. 5
+<<,
+433,
Wdnlqj h{shfwdwlrqv ri +<<, wkh frqglwlrq Hw.4 @ 4 lv vdwlvhg1
Frqvhtxhqwo| zh fdq zulwh wkh pduwlqjdoh surfhvv ri +<;, dv=
5 5
kw . 5 k
Pw @ h
Pw4 . yw
+434,
Vxevwlwxwlrq ri +434, lqwr +<8, jlyhv wkh lpsolhg exeeoh dv=
5 5
kw . 5 k .oq Ew4 . w
Ew @ h
zkhuh w @ yww dqg h orj @
4
1
+435,
6:3
Frqiurqwlqj Prghov zlwk Idfwv
+Hw w.l @ 3 iru l 41 Qrwh vxevwlwxwlrq ri Pw iurp +<8, lqwr +<;,
+zlwk xw @ 4, jlyhv Ew @ 4 w Ew4 . yww > hpsor|lqj wkh ghqwlrq ri w
iurp +<<, zh kdyh +435,,1
Vhyhudo glhuhqw w|shv ri exeeohv fdq eh rewdlqhg h{sorlwlqj wkh lq0
vljkw wkdw udwlrqdo exeeohv lqfoxgh d pduwlqjdoh surfhvv1 Lq idfw dq|
surfhvv wkdw iroorzv d pduwlqjdoh surfhvv fdq eh lqfoxghg lq wkh exeeoh1
Iru h{dpsoh vxssrvh wkh ixqgdphqwdo fdq eh ghvfulehg e| d pduwlq0
jdoh surfhvv
Iw @ Iw4 . w
+436,
zkhuh w lv d udqgrp yduldeoh1
Lq wklv fdvh zh fdq ohw Pw eh=
Pw @ Iw vr wkdw wkh exeeoh lv Ew @
Iw
w
+437,
Iru pruh jhqhudo vshflfdwlrqv ri ixqgdphqwdov wkh phwkrg lv wr
vxevwlwxwh dssursuldwh udqgrp yduldeohv iru kw = Iru h{dpsoh ohw xv dgg
d frqvwdqw frpsrqhqw wr ixqgdphqwdov vr wkdw wkh| iroorz wkh surfhvv
Iw @ Iw4 . . w
+438,
zkhuh w lv Q+3> 5k ,1 Lq rughu wr qg wkh exeeoh wkdw fruuhvsrqgv wr
wklv surfhvv zh lqvshfw wkh jhqhudo exeeoh irupxodwlrq +435,1 Zh fdq
vxevwlwxwh iru wkh yduldeoh kw lq wklv irupxodwlrq vr orqj dv wkh surfhvv
zh vxevwlwxwh lq kdv d phdq ri dqg d yduldqfh 5k / ehfdxvh iru wkhvh
sdudphwhu ydoxhv zh fdq rewdlq wkh pduwlqjdoh surfhvv +434,1 Iw Iw4
lq +438, kdv d frqvwdqw phdq +, dqg yduldqfh 5k vr zh fdq vxevwlwxwh
Iw Iw4 iru kw lqwr wkh jhqhudo exeeoh irupxodwlrq +435, +zlwk w @ 3
dvvxphg }hur khuh iru vlpsolflw|, wr rewdlq
Ew
@h
Ew4
5 5
+Iw Iw4 , . 5 k .oq 5 5
Iw . 5 k .oq w
@
h
5 5
Iw4 . 5 k .oq +w4,
h
+439,
+439, lpsolhv wkdw wkh exeeoh surfhvv lq ixqgdphqwdov fruuhvsrqglqj
wr wkh ixqgdphqwdo surfhvv +438, lv jlyhq e|
Ew @ h
5 5
Iw . 5 k .oq w
@h
5 5
Iw . 5 k woq w
+43:,
Li ixqgdphqwdov iroorz wkh jhrphwulf surfhvv
oq Iw oq Iw4 @ . w
+43;,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6:4
zkhuh w @ Q+3> 5k ,/ zh rewdlq iroorzlqj wkh deryh surfhgxuh
Ew
@h
Ew4
5 5
+oq Iw oq Iw4 , . 5 k .oq @
5 5
oq Iw . 5 k .oq w
h
5 5
oq Iw4 . 5 k .oq +w4,
+43<,
h
zklfk lpsolhv
5 5
oq Iw . 5 k .oq w
Ew @ h
@
5 5
. 5 k woq w
Iw h
+qrwh h oq Iw @ Iw ,1 Zh revhuyh wkdw li .
exeeoh surfhvv lv jlyhq e|
Ew @ Iw
5 5k
5
+443,
. oq @ 3> wkhq wkh
+444,
Dowhuqdwlyho| li @ 3
4
+445,
w
Zkhq d exeeoh ghshqgv rq lwv rzq ydoxh lq wkh suhylrxv shulrg lw
lv fdoohg d Pdunryldq exeeoh1 Zkhq wkh exeeoh ghshqgv rq ixqgdphq0
wdov lw lv fdoohg dq lqwulqvlf exeeoh +Iurrw dqg Revwihog/ 4<<4,1 Zkhq
wkh exeeoh ghshqgv rq duelwudu| surfhvvhv lw lv fdoohg dq h{wulqvlf ru
h{wudqhrxv exeeoh1Wkh vroxwlrq iru h{wudqhrxv exeeohv iroorzv wkh vdph
surfhgxuh dv iru lqwulqvlf exeeohv1 Iru lqvwdqfh li wkh h{wudqhrxv sur0
fhvv/ Vw / iroorzv d pduwlqjdoh surfhvv wkhq wkh udwlrqdo exeeoh lv jlyhq
Vw
e| Ew @ w = Vdojh +4<<:, ghprqvwudwhv krz wr vroyh iru exeeohv iru wkh
jhqhudo DUPD vshflfdwlrq ri ixqgdphqwdov ru h{wudqhrxv yduldeohv1
Ghshqglqj xsrq wkh vshflfdwlrq ri d ydulhw| ri glhuhqw exeeoh sur0
fhvvhv duh ihdvleoh1
Li exeeohv zhuh qrq0vwrfkdvwlf/ w.4 @ 3/ wkhq wkh vroxwlrq wr +<5, lv
vlpso|
Ew @
Ew @ E3 +4 . U,w
+446,
vr wkdw wkh vroxwlrq iru wkh sulfh zrxog vlpso| hperg| d ghwhuplqlv0
wlf h{sorvlyh frpsrqhqw dqg zrxog lq sulqflsoh eh uhdglo| dphqdeoh wr
vwdwlvwlfdo whvwv1
Zklovw wkh srvvlelolw| ri vxfk ghwhuplqlvwlf exeeohv kdyh ehhq lqyhv0
wljdwhg hpslulfdoo| iru shulrgv ri k|shulq dwlrq ehiruh prqhwdu| uhirup
+Iorrg dqg Jduehu/ 4<;3,/ lw zrxog dsshdu wkdw exeeohv ri d ghwhuplq0
lvwlf irup duh qrw ihdwxuhv ri dvvhw ru rwkhu sulfhv1 Dv d frqvhtxhqfh
6:5
Frqiurqwlqj Prghov zlwk Idfwv
exeeohv zrxog qrw dshdu wr eh sodxvleoh hpslulfdoo| xqohvv wkhuh lv d
vljqlfdqw suredelolw| wkdw wkh| zloo froodsvh diwhu uhdfklqj kljk ohyhov1
Eodqfkdug +4<:<, dqg Eodqfkdug dqg Zdwvrq +4<;5, sursrvhg d sure0
delolvwlf exeeoh wkdw fdq hperg| wklv ihdwxuh1 Hvvhqwldoo| wkhuh duh wzr
uhjlphv zklfk rffxu zlwk frqvwdqw suredelolwlhv t dqg 4 t= Lq wkh uvw
vwdwh +D, wkh exeeoh vxuylyhv zlwk suredelolw| t dqg frqwlqxhv wr lq0
w
1 Lq wkh vhfrqg vwdwh
fuhdvh dw dq h{shfwhg udwh ri Hw Ew.4 @D @ +4.U,E
t
+F,/ zlwk suredelolw| 4 t/ wkh exeeoh froodsvhv vr wkdw Hw Ew.4 @F @ 3=
Dgglqj vwrfkdvwlf whupv zh kdyh
Ew.4 @
+4 . U,Ew
. w.4 zlwk suredelolw| t lq vwdwh D
t
+447,
dqg=
Ew.4 @ w.4 zlwk suredelolw| +4 t, lq vwdwh F
+448,
zkhuh wkh huuru kdv wkh surshuw| wkdw Hw w.4 @ 3 Dv d frqvhtxhqfh zh
kdyh
+4 . U,Ew
. +4 t,3
Hw Ew.4 @ t
+449,
t
Zh qrwh iurp +449, wkdw wkh exeeoh kdv wkh irup ri htxdwlrq +<5, dqg
lv wkhuhiruh d ydolg udwlrqdo h{shfwdwlrqv vroxwlrq1 Lq dgglwlrq zh revhuyh
wkdw lq vwdwh D wkh exeeoh jurzv dw d idvwhu udwh rq dyhudjh + 4t , wkdq wkh
qrq0suredelolvwlf exeeoh lq rughu wr frpshqvdwh iru wkh suredelolw| ri
froodsvh1 Froodsvlqj exeeohv duh frqvlvwhqw zlwk wkh dqdo|vlv hpsor|lqj
pduwlqjdohv> zh vlpso| ghqh wkh surfhvv
Pw @ w Pw4 . xw yw zlwk suredelolw| +44:,
dqg
Pw @ xw yw zlwk suredelolw| 4 +44;,
Zh fdq dovr ghulyh wkh h{shfwhg h{fhvv uhwxuq/ zklfk lv ghqhg dv
wkh uhwxuq rq wkh dvvhw zklfk lqfrusrudwhv wkh exeeoh plqxv wkh udwh ri
uhwxuq rq wkh ulvnohvv dvvhw lq uhjlph D dqg F= Ghqlqj wkh h{shfwhg
h{fhvv uhwxuq dw wkh hqg ri shulrg rqh dv [w.4 @ Hw Uw.4 U/ wkhq iurp
+;<,
[w.4 @
Hw Sw.4 . Hw Gw.4
+4 . U,
Sw
+44<,
Wdnlqj h{shfwdwlrqv dw wlph w ri +<4, dw wlph w . 4 zh rewdlq
Hw Sw.4 @ Hw Iw.4 . Hw Ew.4
+453,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
zkhuh
%
Hw Iw.4 @ Hw
+"
,&
+"
,
[ Gw.l.4
[ Gw.l.4
Hw.4
@ Hw
+4 . U,l
+4 . U,l
l@4
l@4
6:6
+454,
qrz
Hw Iw.4 . Hw Gw.4
"
[
Gw.l.4
@ Hw
. Hw Gw.4 @
+4 . U,l
l@4
+4 . U,Iw @ +4 . U,+Sw Ew , +455,
vr vxevwlwxwlqj +455, dqg +453, lqwr +44<, zh rewdlq
[w.4 @
+4 . U,+Sw Ew , . Hw Ew.4
+4 . U,
Sw
+456,
Lq uhjlph D zh vxevwlwxh iru Hw Ew.4 iurp +449, lqwr +456, wr rewdlq
h{shfwhg h{fhvv uhwxuqv dv
[w.4 @
+4 . U,Ew +4 t,
tSw
+457,
Lq uhjlph F zh vxevwlwxwh iru Hw Ew.4 iurp +449, lqwr +456, wr rewdlq
h{shfwhg uhwxuqv dv
[w.4 @ +4 . U,Ew
Sw
+458,
Zh revhuyh iurp +457, dqg +458, wkdw h{shfwhg h{fhvv uhwxuqv glhu
vxevwdqwldoo| lq wkh wzr shulrgv1 Krzhyhu h{shfwhg h{fhvv uhwxuqv duh
}hur dfurvv wkh wzr uhjlphv1 Udwlrqdo exeeohv gr qrw fuhdwh d suhglfwdeoh
sdwwhuq lq h{fhvv uhwxuqv> udwkhu wkh| fuhdwh yrodwlolw| lq sulfhv1
Zklovw wkh wkh deryh suredelolvwlf exeeoh kdv wkh surshuw| ri froodsv0
lqj/ Gled dqg Jurvvpdq +4<;;, vkrz wkdw wkh lpsrvvlelolw| ri qhjdwlyh
udwlrqdo exeeohv lq vwrfn pdunhwv +ehfdxvh ri wkh lpsolfdwlrq ri d qhj0
dwlyh h{shfwhg dvvhw sulfh lq wkh ixwxuh zklfk ylrodwhv olplwhg oldelolw|,
lpsolhv wkdw d exeeoh rqfh froodsvhg fdq qhyhu uhvwduw1 Hvvhqwldoo| li
wkh exeeoh wdnhv d }hur ydoxh/ wkhq iurp +<5, lwv h{shfwhg ydoxh lq wkh
ixwxuh lv }hur1 Ehfdxvh qhjdwlyh ydoxhv duh uxohg rxw iru vwrfn sulfhv wklv
lpsolhv wkdw wkh dyhudjh ri wkh srvlwlyh ydoxhv pxvw eh }hur/ exw wklv lv
d frqwudglfwlrq/ vr lw fdq rqo| eh }hur li lw wdnhv wkh ydoxh ri }hur lq doo
ixwxuh shulrgv1 Wklv srlqw dovr fduulhv wkh lpsolfdwlrq wkdw li d exeeoh
h{lvwv wrgd| lw pxvw kdyh dozd|v h{lvwhg/ l1h1 vlqfh wkh prphqw wudglqj
ehjdq1
Hydqv +4<<4, dqg Ydq Qrughq +4<<9, kdyh irupxodwhg surfhvvhv iru
exeeohv wkdw phhw wklv wkhruhwlfdo srlqw1 Ydq Qrughq doorzv iru wkh
6:7
Frqiurqwlqj Prghov zlwk Idfwv
srvvlelolw| wkdw wkh exeeoh lv h{shfwhg wr froodsvh rqo| sduwldoo| lq vwdwh
F dqg wkdw wkh suredelolw| ri d exeeoh*v frqwlqxhg jurzwk idoov dv wkh
exeeoh jurzv
Ydq Qrughq vshflhv wkdw wkh suredelolw| ri wkh exeeoh*v frqwlqxhg
jurzwk lv jlyhq e|
tw @ t+Ew , zlwk
gt+Ew ,
?3
g m Ew m
+459,
+wkh devroxwh ydoxh ri wkh exeeoh wr doorz iru qhjdwlyh exeeohv lq pdunhwv
vxfk dv h{fkdqjh udwhv1,1
Lq uhjlph F wkh exeeoh lv h{shfwhg wr froodsvh rqo| sduwldoo| dqg lv
jlyhq e|
Hw Ew.4 @ x+Ew , lq vwdwh F zlwk suredelolw| tw
+45:,
zkhuh x+=, lv d frqwlqxrxv dqg hyhu|zkhuh glhuhqwldeoh ixqfwlrq vxfk
w,
wkdw x+3, @ 3 dqg 4 gx+E
gEw 3=
Lq vwdwh D wkh exeeoh lv h{shfwhg wr jurz dw udwh
Hw Ew.4 @
+4 . U,Ew
^4 t+Ew ,`x+Ew ,
zlwk suredelolw| 4 tw +45;,
t+Ew ,
t+Ew ,
Lw lv hdv| wr yhuli| wkdw +45;, dqg +45:, lpso| htxdwlrq +<5,1 Zh fdq
ghgxfh iurp +45;, wkdw wkh h{shfwhg ydoxh ri wkh exeeoh lq wkh vxuylylqj
vwdwh lv d ghfuhdvlqj ixqfwlrq ri wkh suredelolw| ri vxuylydo t+Ew ,= Dv d
frqvhtxhqfh/ dv zlwk wkh Eodqfkdug exeeoh/ wkh juhdwhu wkh suredelolw|
ri froodsvh wkh odujhu lv wkh h{shfwhg jdlq rq d vxuylylqj exeeoh lq rughu
wr frpshqvdwh wkh lqyhvwru iru wkh srvvlelolw| ri froodsvh1
Ydq Qrughq*v vshflfdwlrq ri wkh exeeoh surfhvv lv lqwhuhvwlqj lq
ghprqvwudwlqj krz wkh suredelolw| ri froodsvh fdq eh uhdglo| hqgrjhqlvhg/
dqg dovr krz wkh surfhvv fdq eh prglhg vr wkdw wkh exeeoh lv qrw qhf0
hvvdulo| }hur lq wkh froodsvhg vwdwh1 Erwk wkhvh ihdwxuhv frxog eh uhohydqw
lq hpslulfdo zrun rq ghwhfwlqj exeeohv1
Hydqv irupxodwhv d exeeoh wkdw lv dozd|v srvlwlyh exw qhyhuwkhohvv
shulrglfdoo| froodsvhv1 Wkh Hydqv exeeoh wdnhv wkh irup
Ew.4 @ +4 . U,Ew xw.4 li Ew n
dqg
w.4 +4 . U,^Ew +4 . U,4 `
xw.4 li Ew A n
Ew.4 @ .
t
+45<,
+463,
zkhuh n dqg duh srvlwlyh sdudphwhuv zlwk 3 ? ? +4 . U,n dqg xw.4
lv dq h{rjhqrxv lqghshqghqwo| dqg lghqwlfdoo| glvwulexwhg +llg, srvlwlyh
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6:8
udqgrp yduldeoh zlwk Hw xw.4 @ 4= w.4 lv dq h{rjhqrxv l1lg Ehuqrxool
surfhvv+ lqghshqghqw ri x, zklfk wdnhv wkh ydoxh xqlw| zlwk suredelolw|
t dqg }hur zlwk suredelolw| 4 t=
Wdnlqj h{shfwdwlrqv dw w ri +45<, zh revhuyh wkdw wkh exeeoh lv htxd0
wlrq +<5, zkhq lw lv lq wkh uhjlph zkhuh Ew n= Zkhq Ew A n wkh exeeoh
kdv wkh h{shfwhg ydoxh ri
+4 . U,^Ew +4 . U,4 `
+4 . U,Ew
+4 t,
@
Hw Ew.4 @ .
t
t
t
zlwk suredelolw| t +464,
dqg
Hw Ew.4 @ zlwk suredelolw| 4 t
vr wkdw zkhq Ew A n
+4 . U,Ew
+4 t,
Hw Ew.4 @ t
. +4 t, @ +4 . U,Ew
t
t
+465,
+466,
Frqvhtxhqwo| wkh exeeoh vdwlvhv frqglwlrq +<5,1
Wkh Hydqv exeeoh kdv wkh surshuw| wkdw zkhq Ew n lw jurzv dw
phdq udwh 4 . U= Zkhq Ew A n wkh exeeoh chuxswv* dqg jurzv dw d idvwhu
phdq udwh dv orqj dv wkh surfhvv frqwlqxhv1 Zkhq wkh exeeoh froodsvhv
lw idoov wr d phdq ydoxh ri dqg wkh exeeoh surfhvv ehjlqv djdlq1 Hydqv
qrwhv wkdw e| ydu|lqj wkh sdudphwhuv > n dqg t rqh fdq fuhdwh exeeohv
zkhuh wkh iuhtxhqf| zlwk zklfk wkh exeeoh huxswv/ wkh dyhudjh ohqjwk
ri wlph ehiruh froodsvh dqg wkh vfdoh ri wkh exeeoh ydu|1 Dq h{dpsoh ri
dq Hydqv exeeoh lv ghslfwhg lq Iljxuh 47161
Dq lpsruwdqw ihdwxuh ri wkh Hydqv exeeoh lv wkdw vwdqgdug hpslulfdo
whvwv iru exeeohv edvhg rq xqlw urrw dqg frlqwhjudwlrq phwkrgv +Wlph0
Vhulhv Dqqh{, pd| qrw ghwhfw exeeohv zkhq wkh| duh suhvhqw1 Hydqv
vlpxodwhg exeeohv dqg dssolhg xqlw urrw whvwv wr wkh vlpxodwhg exeeohv1
Hyhq wkrxjk exeeohv duh dv|pswrwlfdoo| h{sorvlyh iru t 3=:8 iru pruh
wkdq <3( ri wkh vlpxodwhg exeeohv ri ohqjwk 433 revhuydwlrqv/ vwdwlvwlfdo
whvwv iru xqlw urrwv uhmhfwhg wkh k|srwkhvlv ri d exeeoh lq idyrxu ri d
vwdeoh dowhuqdwlyh1
Pruh uhfhqw hpslulfdo zrun kdv hqghdyrxuhg wr dsso| dowhuqdwlyh
hfrqrphwulf dssurdfkv vxfk dv vzlwfklqj uhjlph uhjuhvvlrq prghov wr dv0
fhuwdlq wkh suhvhqfh ri exeeohv +Kdplowrq/ 4<<7> Ydq Qrughq 4<<9> Ydq
Qrughq dqg Yljixvvrq/ 4<<;, jlyhq wkdw wkh Hydqv0w|sh prgho ghvfulehv
ehkdylrxu iru glhuhqw uhjlphv1 Wkhvh vwdwlvwlfdo phwkrgv dsshdu wr rhu
juhdwhu surplvh ri ghwhfwlqj exeeohv1 Dv |hw krzhyhu lw lv suredeo| idlu
wr vd| wkdw wkh glhuhqw hpslulfdo phwkrgv zklfk kdyh ehhq dssolhg wr d
6:9
Frqiurqwlqj Prghov zlwk Idfwv
Iljxuh 4716= H{dpsoh ri dq Hydqv Exeeoh
ydulhw| ri glhuhqw dvvhw pdunhwv lqfoxglqj vwrfn dqg h{fkdqjh pdunhwv
kdyh qrw jhqhudwhg dq| frqvlvwhqw hylghqfh ri vshfxodwlyh exeeohv1
Lw vkrxog eh qrwhg wkdw wkhruhwlfdo zrun e| Wluroh +4<;5/ 4<;8, kdv
ghprqvwudwhg wkdw exeeohv fdqqrw dulvh lq fhuwdlq prghov1 Iru h{dpsoh
lq dq ryhuodsslqj jhqhudwlrqv prgho zlwk dq lqqlwh qxpehu ri qlwh0
olyhg uhsuhvhqwdwlyh djhqwv/ Wluroh vkrzv wkdw zkhq wkh lqwhuhvw udwh h{0
fhhgv wkh jurzwk udwh lq wkh hfrqrp| +vr wkdw lw lv g|qdplfdoo| h!flhqw,
d exeeoh zrxog xowlpdwho| ylrodwh vrph djhqwv* exgjhw frqvwudlqwv1 Lw
zrxog dsshdu wkdw exeeohv frxog rqo| h{lvw lq hfrqrplhv zklfk duh g|0
qdplfdoo| lqh!flhqw li wkh prgho lv ri wkh uhsuhvhqwdwlyh0djhqw w|sh dqg
djhqwv duh udwlrqdo1 Iurp wklv shuvshfwlyh Iurrw dqg Revwihog +4<<4,
vxjjhvw lqwhusuhwlqj hpslulfdo whvwv iru exeeohv dv whvwv ri wkh udwlrqdo
h{shfwdwlrqv dvvxpswlrq1
Zh fdq sxw wklv dqrwkhu zd|1 UH prghov duh/ jhqhudoo|/ cryhu0
lghqwlhg*1 Wkdw lv/ hyhu|wklqj lq wkh uhgxfhg irup lv ghulyhg iurp
vwuxfwxudo uhodwlrqvklsv dqg pruh uhvwulfwlrqv h{lvw wkdq uhgxfhg0irup
sdudphwhuv> wkh uhgxfhg0irup huuruv duh frpelqdwlrqv ri wkh vwuxfwxudo
sdudphwhuv dqg vwuxfwxudo huuruv1 Huuruv vxfk dv exeeohv duh qrw suhvhqw
lq wkh vwuxfwxudo prgho dqg khqfh vkrxog qrw dsshdu lq wkh uhgxfhg irup1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6::
Rqo| ixqgdphqwdov duh lqfoxghg lq lw1
Vkrxog zh zulwh grzq d vwuxfwxudo prgho lq zklfk vrph exeeoh ydul0
deoh lv doorzhg wr hqwhu/ wkhq zh pxvw hqvxuh wkdw lw vdwlvhv erwk wkh
wudqvyhuvdolw| frqglwlrqv rq wkh prgho +lqfoxglqj sulydwh dqg exgjhw
frqvwudlqwv dv wkh| whqg wr lqqlw|, dqg wkh rswlpdolw| frqglwlrqv1 Iru
h{dpsoh/ dv zh vkrzhg lq fkdswhu 5/ ghwhuplqlvwlf h{sorvlyh exeeohv lq
sulfhv zloo ylrodwh uhdvrqdeoh jryhuqphqw olplwv rq lq dwlrq dqg vr zloo
eh uxohg rxw iurp wkh vwduw1 Lq wkh Eodqfkdug dqg Zdwvrq h{dpsoh uh0
ylhzhg deryh/ ri d vwrfkdvwlf exeeoh/ dw ohdvw rqh ri wkh sdwkv pxvw kdyh
d srvlwlyh suredelolw| ri frqwlqxlqj lqghqlwho| iru wkh exeeoh hyhq wr
ehjlq1 Lv vxfk d sdwk frqvlvwhqw zlwk vxfk olplwvB Deryh zh qrwhg wkh
remhfwlrqv ri Wluroh wr exeeohv lq rqh frqwh{w/ dqg wkh fdyhdwv h{suhvvhg
e| Revwihog dqg Urjr1 Lq hdfk fdvh ri d exeeoh zh pxvw dvn zkhwkhu
wkhru| dv deryh shuplwv lw1
Zkhq zh wxuq wr hpslulfdo whvwlqj/ zh fdq qrwh wkdw wkh ciruzdug
vroxwlrq* ri wkh prgho glvfxvvhg lq fkdswhu 5 kdv wkh irup ri d exeeoh1
Rqh fdq iru h{dpsoh zulwh grzq d prgho ri sulfhv lq zklfk d odujh
srvlwlyh ixwxuh vkrfn +wr wkh prqh| vxsso|/ vd|, lv h{shfwhg zlwk vrph
suredelolw|> dv wkh shulrg dssurdfkhv wkh sulfh ohyho ulvhv xqwlo dw wkh
shulrg lw hlwkhu mxpsv xszdugv li wkh vkrfn rffxuv ru froodsvhv li lw grhv
qrw1 Glvwlqjxlvklqj wklv qrq0exeeoh vroxwlrq iurp d vxssrvhg exeeoh
fohduo| lv gl!fxow1 Uhphpehu d exeeoh uhodwhv wr zkdw shrsoh lq pdunhwv
h{shfw iru wkh ixwxuh> wklv frxog zhoo uhodwh wr dq h{shfwdwlrq derxw d
ixqgdphqwdo1
Wkh gl!fxow| zlwk exeeohv lv wkhuhiruh erwk wkhruhwlfdo dqg hpslu0
lfdo1 Wklv fulwltxh ri exeeohv zlwklq udwlrqdo h{shfwdwlrqv prghov grhv
qrw ri frxuvh h{whqg wr prghov zkhuh h{shfwdwlrqv duh qrw udwlrqdo> khuh/
e| frqvwuxfwlrq/ shrsoh pd| eholhyh lq fxulrvd cidgv* dqg vr iruwk1 Wr
vxfk prghov zh qrz wxuq1
Idgv dqg Qrlvh Wudghuv
Vkloohu +4<;7,/ +4<;<, dqg +4<<:, kdv hpskdvlvhg wkh lpsruwdqfh ri pdvv
sv|fkrorj| lq qdqfldo pdunhwv zlwk wkh lpsolfdwlrq wkdw lqyhvwruv pd|
h{klelw idvklrqv ru idgv1 D idg lv d ghsdwxuh iurp ixqgdphqwdo ydoxhv
gxh wr csv|fkrorjlfdoo|* lqgxfhg fkdqjhv lq pdunhw vhqwlphqw1 Zh uvw
looxvwudwh wkh lpsolfdwlrqv iroorzlqj wkh prgho lq Idpd dqg Iuhqfk +4<;;,
dqg Fxwohu/ Srwhued dqg Vxpphuv +4<<3/ 4<<4,1
Ohw
Sw @ Iw . hw
+467,
zkhuh Sw lv wkh uhwxuq ru sulfh ri dq dvvhw/ Iw lv wkh ixqgdphqwdo ydoxh
6:;
Frqiurqwlqj Prghov zlwk Idfwv
dqg hw lv d idg1 Ixqgdphqwdov duh dvvxphg wr iroorz wkh surfhvv
Iw @ Iw4 . kw
+468,
Wkh idg lv ghvfulehg e| wkh vwdwlrqdu| surfhvv
hw @ hw4 . yw
+469,
zkhuh lv d srvlwlyh frqvwdqw ohvv wkdq xqlw| dqg yw lv d udqgrp huuru1
Fxwohu/ Srwhued dqg Vxpphuv +4<<4, frqvlghu wkh fdvh zkhuh d sur{|/
Iw / lv dydlodeoh wr phdvxuh ixqgdphqwdov/ jlyhq wkdw lq dssolhg zrun zh
w|slfdoo| gr qrw phdvxuh wkh ixqgdphqwdov zlwk suhflvlrq ru duh hpsor|0
lqj dssur{lpdwlrqv wr wkh wkhruhwlfdo prgho1 Ohw
Iw @ Iw . zw
+46:,
zkhuh zw lv wkh phdvxuhphqw huuru dvvxphg wr eh udqgrp1
Glhuhqflqj +467, dqg vxevwlwxwlqj iurp +468, dqg +469, zh rewdlq
wkh wuxh uhgxfhg irup iru Sw dv
Sw @ + 4,+Sw4 Iw4 , . kw . yw
+46;,
Frqvlghu wkh uhjuhvvlrq hvwlpdwh ri lq wkh uhodwlrqvkls
Sw @ . +Sw4 Iw4
, . w
+46<,
Wkh uhjuhvvlrq frh!flhqw zloo kdyh d suredelolw| olplw ri
@
+ 4,Fry+Sw4 Iw4 > Sw4 Iw4
,
ydu+Sw4 Iw4 ,
+ 4,Fry+hw4 > hw4 . zw4 ,
@
Y du+hw4 . zw4 ,
+ 4,ydu+h,
@
ydu+h, . ydu+z,
+473,
Wkh lpsolfdwlrq lv wkdw fkdqjhv lq dvvhw sulfhv zloo eh suhglfwhg e|
odjjhg ghyldwlrqv ri wkh sulfh iurp lwv sur{| ixqgdphqwdo +iru zklfk lq
wkhlu hpslulfdo zrun Fxwohu/ Srwhued dqg Vxpphuv +4<<4, hpsor| wkh
orj ri wkh uhdo glylghqg,1 Lq dgglwlrq sulfhv ru uhwxuqv fdq eh vkrzq wr
h{klelw d suhglfwdeoh sdwwhuq1
Ri frxuvh suhglfwdeoh sdwwhuqv lq wkh glhuhqfh ehwzhhq sulfhv dqg
ixqgdphqwdov udlvhv wkh lvvxh ri zk| udwlrqdo vshfxodwruv gr qrw duelwudjh
wkh surfhvv1 Lw lv lpsolflwo| dvvxphg wkdw wkh| idfh vrph vruw ri oltxlglw|
frqvwudlqw suhyhqwlqj wkhp iurp grplqdwlqj wkh pdunhw1 Krzhyhu/ lw lv
dovr srvvleoh wkdw wkh| gr vr duelwudjh/ vr wkdw wkh idg rffxuv exw lv wkhq
uhpryhg iurp wkh gdwd qh{w shulrg1 Wklv zrxog fruuhvsrqg wr @ 3
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6:<
lq rxu h{dpsoh1 Lq wklv fdvh/ sulfhv frxog glhu iurp wkhlu ixqgdphqwdo
ydoxh vr lpso|lqj lqh!flhqf|/ dqg |hw wkh glhuhqfh ehwzhhq wkh sulfh
dqg ixqgdphqwdov frxog vwloo eh xqsuhglfwdeoh1 Wklv glvwlqfwlrq srlqwv wr
d fdyhdw lq wkh lqwhusuhwdwlrq ri h!flhqf| jlyhq e| Idpd> Vkloohu pdnhv
wkh srlqw wkdw wkh devhqfh ri suhglfwdelolw| grhv qrw qhfhvvdulo| lpso|
pdunhw h!flhqf|1 Fohduo| wkh h{lvwhqfh ri v|vwhpdwlf deqrupdo surwv lv
vx!flhqw wr lqydolgdwh h!flhqf|1 Krzhyhu devhqfh ri suhglfwdeoh deqru0
pdo uhwxuqv frxog eh frqvlvwhqw zlwk pdunhw lqh!flhqf| li sulfhv zhuh
pruh yrodwloh wkdq zdv frqvlvwhqw zlwk wkh yrodwlolw| lq wkh ixqgdphqwdo1
Exeeohv ru vrph idg surfhvvhv duh frqvlvwhqw zlwk wklv lqwhusuhwdwlrq
ri lqh!flhqf|1 H!flhqf| wkhuhiruh uhtxluhv wkh frpsohwh holplqdwlrq ri
idgv iurp wkh sulfh surfhvv +hw @ 3,> rwkhuzlvh d wudglqj uxoh rshudwlqj
rq Sw @ Iw zrxog pdnh prqh| +wklv lv d sduwlfxoduo| vlpsoh uxoh/ yl}1 ri
fxuuhqw duelwudjh,1
Rwkhu prghov ri pdunhw lqh!flhqf| dvvxph wkh frh{lvwhqfh ri khwhur0
jhqhrxv wudghuv1 Lq wkhvh prghov wkhuh duh vpduw wudghuv/ wkh udwlrqdo
vshfxodwruv/ dv zhoo dv wkh qrlvh wudghuv1 Wkh qrlvh wudghu wudghv rq
idgv ru fkduwv ru rwkhu h{wudqhrxv lqirupdwlrq +cqrlvh*,1 Wkh nh| uh0
txluhphqw ri wkh prghov lv wkdw duelwudjh dfwlylw| e| wkh vpduw wudghuv
lv olplwhg1 Wklv fdq eh d frqvhtxhqfh ri wkh dvvxpswlrq ri ulvn0dyhuvh
udwlrqdo vshfxodwruv frpelqhg zlwk d plfur0vwuxfwxuh udwlrqdoh zkhuh
iru h{dpsoh duelwudjhxuv kdyh olplwhg fdsdflw| wr eruurz ixqgv gxh wr
vljqdoolqj sureohpv +vhh h1j1 Vkohlihu dqg Ylvkq|/ 4<<:,1
Wkh frqvhtxhqfh lv wkdw duelwudjh dfwlylw| e| wkh udwlrqdo wudghuv
pd| qrw holplqdwh wkh lq xhqfh ri wkh qrlvh wudghuv iurp wkh rxwfrph
iru uhwxuqv ru sulfhv vr wkdw wkh pdunhw lv lqh!flhqw lq djjuhjdwh +vhh
h1j1 Iljohzvnl/ 4<:<,1
Rwkhu vshflfdwlrqv ri wklv w|sh ri prgho kdyh wkh lqiruphg wudghuv dv
qrw kroglqj udwlrqdo h{shfwdwlrqv exw udwkhu iruplqj wkhlu h{shfwdwlrqv
rq wkh edvlv ri d uhodwlyho| zhoo0lqiruphg uxoh vxfk dv SSS ghyldwlrqv
iru h{fkdqjh udwhv +vhfwlrq ehorz, ru sulfh0glylghqg ghyldwlrqv iru vwrfn
sulfhv xvlqj vd| wkh Jrugrq +4<95, prgho 1 Li lw lv dvvxphg wkdw wkh sur0
sruwlrq ri wkh wzr w|shv ri djhqw lq wkh pdunhw ydulhv zlwk wkh h{whqw ri
wkh ghyldwlrq ri wkh dvvhw sulfh iurp lwv orqj uxq ydoxh/ wkhq wkhvh prg0
hov fdq h{klelw frpsoh{ qrqolqhdu g|qdplfv ri wkh dvvhw sulfh lqfoxglqj
fkdrwlf rxwfrphv +Wlph0Vhulhv Dqqh{> dqg h1j1 Gh Judxzh/ Ghzdfkwhu
dqg Hpeuhfkwv/ 4<<6,1
Vrph uhfhqw prghov ri dvvhw sulfh ghwhuplqdwlrq lqyhvwljdwh pruh iru0
pdoo| wkh plfur0vwuxfwxuh ghwhuplqlqj dvvhw sulfhv1 Dq lpsruwdqw fodvv
ri prghov frqvlghu wkh lpsolfdwlrqv ri dvvxplqj wkdw wudghuv gr qrw eh0
kdyh frpshwlwlyho| dqg wdnh sulfhv dv jlyhq1 Wkh prghov duh vroyhg xvlqj
wkh jdph0wkhruhwlf Ed|hvldq Qdvk htxloleulxp frqfhsw zkhuh wkh vwudwh0
6;3
Frqiurqwlqj Prghov zlwk Idfwv
jlhv ri rwkhu wudghuv udwkhu wkdq sulfhv duh wdnhq dv jlyhq1 Wkhvh prghov
doorz dqdo|vlv ri vwudwhjlf lqwhudfwlrqv lq zklfk wudghuv wdnh wkhlu sulfh
lpsdfw lqwr dffrxqw +vhh Euxqqhuphlhu/ 4<<</ iru d vxuyh|,1 Lq hvvhqfh
hdfk wudghu uhfrjqlvhv wkdw odujh wudghv zloo pryh sulfhv djdlqvw klp1
Wkh prghov lqfrusrudwh ihdwxuhv vxfk dv dv|pphwulf lqirupdwlrq/ qrlvh
wudghuv/ vshfxodwruv/ pdunhw0pdnhuv dqg h{solflw dvvxpswlrqv frqfhuqlqj
wkh phfkdqlvp e| zklfk wudghuv vxeplw rughuv wr wudgh1 Wkh| surgxfh
pdq| lqwhuhvwlqj lqvljkwv1
Zkhq lw lv dvvxphg wkdw wudghuv kdyh vkruw lqyhvwphqw krul}rqv/ shu0
kdsv gxh wr qdqfldo frqvwudlqwv zklfk pdnh duelwudjh dfwlylwlhv fkhdshu
iru vkruw0whup dvvhwv/ khug ehkdylrxu fdq uhvxow +vhh h1j1 Iurrw/ Vfkdui0
vwhlq dqg Vwhlq/ 4<<5,1 Lq dgglwlrq vwudwhjlf lqwhudfwlrq fdq jlyh ulvh wr
udwlrqdo surw0pd{lplvlqj vshfxodwruv/ dffhqwxdwlqj dvvhw sulfh yrodwlolw|
fdxvhg e| qrlvh wudghuv dv wkh| vwudwhjlfdoo| h{sorlw ihhgedfn hohphqwv
vxfk dv khuglqj +vhh h1j1 Gh Orqj/ Vkohlihu/ Vxpphuv dqg Zdogpdq/
4<<4,1
Wkh prghov fdq dovr jhqhudwh pdunhw fudvkhv dv lq h1j1 Urphu +4<<6,
zkhuh lqirupdwlrq lv khwhurjhqhrxv dqg lqyhvwruv duh xqfhuwdlq derxw wkh
txdolw| ri lqirupdwlrq rwkhu lqyhvwruv kdyh1
Dowkrxjk wklv olwhudwxuh lv ulfk lq lqvljkw dqg fdq surylgh d udwlrqdoh
iru pdunhw lqh!flhqf|/ lw uhpdlqv dq hpslulfdo lvvxh zkhwkhu h!flhqf|
dv ghqhg e| Idpd lv ylrodwhg1 Zh qrz wxuq wr vrph hpslulfdo whvwv1
HPSLULFDO WHVWV RI PDUNHW HIILFLHQF\
Wkhuh kdv ehhq dq hqruprxv dprxqw ri hpslulfdo zrun zklfk kdv h{0
dplqhg wkh h!flhqf| ri dvvhw pdunhwv lqfoxglqj jdpeolqj pdunhwv +vhh
h1j1 Vdxhu/ 4<<;> Wkdohu dqg ]lhped/ 4<;:,1 Ylrodwlrqv ri wkh h!flhqw
pdunhwv k|srwkhvlv duh w|slfdoo| uhihuuhg wr dv dqrpdolhv1 Wkh dqrpd0
olhv uhsruwhg pd| frqvwlwxwh hylghqfh ri deqrupdo uhwxuqv ru ylrodwlrqv
ri wkh udwlrqdo h{shfwdwlrqv dvvxpswlrq1 Ehiruh frqvlghulqj vrph ri wklv
hylghqfh lw lv lpsruwdqw wr qrwh wkdw wkh gdqjhuv ri gdwd0gulyhq lqihu0
hqfh ru gdwd0vqrrslqj zklovw dozd|v d frqvlghudwlrq lq dssolhg dqdo|vlv
+vhh Ohdphu/ 4<:;/ iru d glvfxvvlrq ri suhwhvw eldv, qhhgv wr eh jlyhq
irupdo zhljkw zkhq lqwhusuhwlqj vrph dqrpdolhv dv ghprqvwudwhg e|
Vxoolydq/ Wlpphupdqq dqg Zklwh +4<<</ 5334,1 Lq wkh olplwhg vdpsoh
vl}hv w|slfdoo| hqfrxqwhuhg lq vwxglhv/ vljqlfdqw uhodwlrqvklsv ru v|whp0
dwlf sdwwhuqv duh erxqg wr rffxu li wkh gdwd duh dqdo|}hg zlwk vx!flhqw
lqwhqvlw|1 Iru h{dpsoh wkh ruwkrjrqdolw| surshuw| ri udwlrqdo h{shfwd0
wlrqv zloo eh uhmhfwhg e| wkh xvxdo vwdwlvwlfdo fulwhuld yh shufhqw ri wkh
wlph1 Fohduo| li dxwkruv duh vhdufklqj lq wkh xqlyhuvh ri yduldeohv dqg
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6;4
vljqlfdqw uhmhfwlrqv duh uhsruwhg lw frxog dsshdu wkdw wkh k|srwkhvlv lv
lqydolg hyhq wkrxjk lw lv wuxh1
Dq dssduhqw uhmhfwlrq ri wkh pdunhw h!flhqf| k|srwkhvlv zklfk lv
gdwd0gulyhq lv wkh qglqj uhsruwhg e| qxphurxv uhvhdufkhuv wkdw vwrfn
uhwxuqv h{klelw vhdvrqdo uhjxodulwlhv +vhh Vxoolydq/ Wlpphupdqq dqg
Zklwh/ 5334/ iru nh| uhihuhqfhv,1 Deqrupdo uhwxuqv duh uhodwhg wr gd| ri
wkh zhhn/ zhhn ri wkh prqwk/ prqwk ri wkh |hdu/ krolgd|v hwf1 Li wkhvh
uhvxowv kdyh ydolglw| wkhq wkh h!flhqw pdunhwv k|srwkhvlv lv lq vhulrxv
wurxeoh jlyhq lw lv cvxfk d vlpsoh ylrodwlrq*1
Wkh lvvxh lv rqh ri cgdwd0vqrrslqj* lq rwkhu zrugv/ uhvhdufkhuv
fdqqrw dyrlg ehlqj lq xhqfhg e| wkhlu vdpsoh ri gdwd1 Wkh| cqg* uxohv
wkdw duh lq wkdw vhw ri gdwd e| uhshdwhg wuldov1 Wklv zloo rffxu hyhq
zkhq vrph gdwd lv nhsw dvlgh ehfdxvh wklv gdwd wrr lv nqrzq derxw wr
vrph ghjuhh dqg wuldov rq lw pd| ohdg wr uhvshflfdwlrq ri wkh uxoh rq
wkh xvhg gdwd ehiruh d uhqhzhg wuldo1 Wr frqwuro iru wklv sureohp wkh
uhvhdufkhu pxvw qrw doorz wkh gdwd wr lq xhqfh klv fkrlfh ri uxoh1 Wkxv
doo srvvleoh uxohv pxvw eh jlyhq dq htxdo fkdqfh wr zrun rq wkh gdwd>
vrph ri frxuvh zloo zrun zhoo iru sduwlfxodu vdpsohv hyhq li wkh| gr qrw
lq jhqhudo zrun dw doo1 E| grlqj wklv h{huflvh iru doo uxohv iru uhshdwhg
vdpsohv rqh pd| fdofxodwh wkh suredelolw| ri d uxoh cgrlqj zhoo* lq d jlyhq
vdpsoh1 Fohduo| iru d uxoh wr eh frqvlghuhg vljqlfdqw lw pxvw gr ehwwhu
wkdq wklv1 Wklv h{huflvh kdv ehhq fduulhg rxw e| Vxoolydq/ Wlpphupdqq
dqg Zklwh +5334, iru fdohqgdu hhfwv1
Wkh| frqvwuxfw d xqlyhuvh ri fdohqgdu wudglqj uxohv/ xvlqj shupxwd0
wlrqdo dujxphqwv wkdw gr qrw eldv lq idyrxu ru djdlqvw sduwlfxodu fdohqgdu
uxohv1Wkh| frqvlghu vrph <833 fdohqgdu hhfwv1 Wkh| lpdjlqh wkdw wklv
vhw ri fdohqgdu uxohv zdv qrw lqvshfwhg e| dq| rqh lqglylgxdo lqyhvwru
ru uhvhdufkhu1 Udwkhu wkh vhdufk iru fdohqgdu uxohv kdv rshudwhg vhtxhq0
wldoo| dfurvv wkh lqyhvwphqw frppxqlw| ru uhvhdufkhuv zlwk wkh uhvxowv
ri lqyhvwruv ru uhvhdufkhuv ehlqj uhsruwhg wkurxjk wkh vxuylydo ri wkh
cwwhvw* fdohqgdu uxohv1 Wkhlu qglqjv duh vwulnlqj dqg lpsruwdqw1 Wkh|
qg wkdw dowkrxjk pdq| fdohqgdu uxohv jhqhudwh deqrupdo uhwxuqv wkdw
duh kljko| vljqlfdqw zkhq frqvlghuhg lq lvrodwlrq/ rqfh wkh ghshqghqflhv
rshudwlqj dfurvv glhuhqw fdohqgdu uxohv duh doorzhg iru/ wkhq wkh ehvw
fdohqgdu uxoh lv qr orqjhu vljqlfdqw1 Lq dgglwlrq wkh| frqvlghu d vpdoohu
qxpehu ri 577 uxohv wkdw uhpryh dq| grxewv wkdw luuhohydqw uxohv duh
srrohg zlwkcjhqxlqh* uxohv lq wkh <833 h{shulphqw1 Wkhlu uhvxowv duh wkh
vdph1 Wkh dssduhqw vwdwlvwlfdo vljqlfdqfh ri wkh ehvw fdohqgdu hhfwv
lv qrw urexvw wr gdwd0vqrrslqj hhfwv1
6;5
Frqiurqwlqj Prghov zlwk Idfwv
Zhdn0irup h!flhqf| whvwv
Zhdn0irup h!flhqf| lq wkh vhqvh ri Idpd lpsolhv wkdw wkh h{shfwhg
uhwxuqv iurp d wudglqj uxoh edvhg rq sdvw sulfhv vkrxog eh ohvv wkdq wkrvh
jhqhudwhg e| d ex|0dqg0krog vwudwhj| htxdo wr wkh qxpehu ri wudghv wlphv
wkh wudqvdfwlrqv frvwv1 Lq idfw dffruglqj wr Wd|oru dqg Doohq +4<<5, dqg
Eurfn/ Odnrqlvkrn dqg OhEdurq +4<<5, fkduwlvw ru whfkqlfdo whfkqltxhv
duh zlgho| hpsor|hg lq dvvhw pdunhwv dv hlwkhu d edvlv iru sxeolvkhg
whfkqlfdo frpphqwdu| ru iru gluhfw xvh1 Wkh whup whfkqlfdo dqdo|vlv ru
fkduwlvp lv jlyhq dv d jhqhulf wlwoh iru dq| wudglqj uxoh edvhg rq wkh sdvw
klvwru| ri sulfhv ru uhwxuqv1 Hvvhqwldoo| pruh ru ohvv vrsklvwlfdwhg uxohv
duh hpsor|hg wr h{wudsrodwh sdvw fkdqjhv lq dvvhw sulfhv ru uhwxuqv1 Rqh
frpprq h{dpsoh lv wr hpsor| wzr prylqj dyhudjhv ri sdvw uhwxuqv rqh
edvhg rq d vkruw krul}rq/ vd| 4 wr 43 gd|v/ dqg wkh rwkhu rq d orqj prylqj
dyhudjh/ vd| 483 wr 833 gd|v1 Zkhq wkh vkruw prylqj dyhudjh ulvhv deryh
+ru idoov ehorz, wkh orqj prylqj dyhudjh wklv lv d vljqdo wr ex| +ru vhoo,1
D yduldqw ri wklv surfhgxuh lv wr prgli| wkh uxoh e| lqwurgxfwlrq ri d
edqg durxqg wkh prylqj dyhudjh wr holplqdwh czklsodvk* vljqdov zkhq
wkh wzr prylqj dyhudjhv duh forvh1
Wkhuh lv d sohwkrud ri hpslulfdo zrun wkdw kdv lqyhvwljdwhg wkh h!fdf|
ri whfkqlfdo dqdo|vlv h1j1 Doh{dqghu +4<94,> Idpd dqg Eoxph +4<99,>
Wd|oru dqg Doohq +4<<5,> Eurfn/ Odnrqlvkrn dqg OhEdurq +4<<5,> Qhho|/
Zhoohu dqg Glwwpdu +4<<:,> Vxoolydq/ Wlpphupdqq dqg Zklwh +4<<<,1
Zklovw pdq| uxohv ghyhorshg e| fkduwlvwv kdyh ehhq uhsruwhg wr jhq0
hudwh deqrupdo uhwxuqv/ dqg vrph wkhruhwlfdo dqdo|vlv h1j1 Eurzq
dqg Mhqqlqjv +4<;<,> Eoxph/ Hdvoh| dqg R*Kdud +4<<7, ghprqvwudwhv
wkdw whfkqlfdo dqdo|vlv fdq kdyh ydoxh sduwlfxoduo| iru vpdoo ohvv zlgho|0
iroorzhg dvvhwv/ wkh hpslulfdo uhvxowv ri Vxoolydq/ Wlpphupdqq dqg Zklwh
+4<<<, dsshdu wr eh ri lpsruwdqfh lq wkh lqwhusuhwdwlrq ri fkduwlvp1 Iro0
orzlqj d vlplodu phwkrgroj| wr wkh rqh rxwolqhg deryh wkh| frqvwuxfw
:;79 sdudphwhul}dwlrqv ri wudglqj uxohv zklfk duh dssolhg wr wkh Grz
Mrqhv Lqgxvwuldo Dyhudjh ryhu wkh ixoo shulrg iurp 4;<: wr4<;9 dv zhoo
dv irxu vxeshulrgv1 Wkh shulrg 4<;:04<<9 lv hpsor|hg wr hydoxdwh wkh
uxohv rq d krog0rxw vdpsoh1 Wkhlu lghd lv wr ghyhors d whvw vwdwlvwlf
zklfk hydoxdwhv wkh shuirupdqfh ri d fkduwlvw uxoh uhodwlyh wr wkh ixoo vhw
ri prghov wkdw jdyh ulvh wr wkh uxoh/ vr wkdw wkh hhfw ri gdwd0vqrrslqj
lv h{solflwo| doorzhg iru1 Rqh lpsruwdqw frqfoxvlrq lv wkdw hyhq wkrxjk
d sduwlfxodu wudglqj uxoh lv fdsdeoh ri surgxflqj vxshulru shuirupdqfh
ri doprvw 43( gxulqj wkh vdpsoh shulrg/ zklfk lv vljqlfdqw dw 7( ohyho
zkhq frqvlghuhg lq lvrodwlrq/ wkh idfw wkdw wkh wudglqj uxoh lv gudzq iurp
d zlgh xqlyhuvh ri uxohv phdqv wkdw lwv hhfwlyh gdwd0vqrrslqj0dgmxvwhg
suredelolw| ydoxh lv 31</ l1h1 wrwdoo| lqvljqlfdqw1 Wkh| dovr qg wkdw
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6;6
wkh ehvw0shuiruplqj wudglqj uxoh lq vdpsoh lv wrwdoo| lqvljqlfdqw rxw ri
vdpsoh hyhq dw frqyhqwlrqdo vwdwlvwlfdo ohyhov1 Fohduo| fkduwlvw uxohv pd|
h{lvw zklfk jhqhudwh deqrupdo uhwxuqv> krzhyhu wkh dqdo|vlv ri Vxoolydq/
Wlpphupdqq dqg Zklwh +4<<<, ghprqvwudwhv wkdw juhdw fduh kdv wr eh
wdnhq lq wkhlu lqwhusuhwdwlrq1
Dqrwkhu lvvxh frqfhuqlqj fkduwlvwv ru whfkqlfdo uxohv lv wkdw wkh| fdq
eh lqwhusuhwhg dv vshfldo fdvhv ri xqlyduldwh wlph vhulhv ri d olqhdu ru
qrqolqhdu irup +Wlph0Vhulhv Dqqh{,1 Li wkhuh lv suhglfwdelolw| lq sdvw uh0
wxuqv lw zrxog dsshdu wkdw hpsor|phqw ri dq h{solflw xqlyduldwh prgho
zrxog lq jhqhudo grplqdwh1 Vwxglhv vxfk dv Glherog dqg Qdvrq +4<<3,/
Odqh/ Shho dqg Udhexuq +4<<9, duh vxjjhvwlyh wkdw qrqolqhdu xqlyduldwh
prghov iru prghoolqj wkh frqglwlrqdo phdq ri d vhulhv surylgh lpsuryh0
phqw ryhu olqhdu prghov exw gr qrw dsshdu wr iruhfdvw ehwwhu wkdq olqhdu
prghov/ zlwk qhlwkhu ri wkhp jhqhudwlqj deqrupdo uhwxuqv +wkrxjk vhh
Judqjhu dqg Shvdudq/ 4<<<,1Wkh uhvxowv ri Kvlhk +4<<4, zkr irxqg wkdw
vwrfn uhwxuqv h{klelw qrqolqhdu vwuxfwxuh exw wkdw wklv lv sduvlprqlrxvo|
fdswxuhg e| d prgho olqhdu lq phdq exw qrqolqhdu lq yduldqfh +Wlph0
Vhulhv Dqqh{, lv suredeo| wkh prgho wkdw lv fxuuhqwo| wkrxjkw ri dv
prvw dssolfdeoh wr dvvhw sulfhv ru uhwxuqv1 Fohduo| fkduwlvw phwkrgv pd|
frqfhlydeo| ghdo pruh sduvlprqlrxvo| zlwk qrqvwdwlrqdulwlhv ru uhjlph
fkdqjhv wkdq h{solflw qrqolqhdu prghov wkrxjk lw lv qrw dowrjhwkhu fohdu
zk| wklv vkrxog eh wkh fdvh1
Ryhudoo lw zrxog dsshdu wkdw wkh pdunhw h!flhqf| k|srwkhvlv kdv qrw
dv |hw ehhq lqydolgdwhg dw wkh zhdn0irup ohyho1
Vhpl0Vwurqj Pdunhw H!flhqf|
Qxphurxv hpslulfdo whvwv ri vhpl0vwurqj h!flhqf| h{lvw1 Vrph ri wkh
zrun uhsruwlqj dqrpdolhv zrxog dsshdu wr eh vxemhfw wr wkh gdwd0vqrrs0
lqj fdyhdw1 Rwkhuv duh kdughu wr h{sodlq1 Lq wkh odwwhu frqwh{w zh phq0
wlrq wkh hpslulfdo zrun wkdw ghprqvwudwhv wkdw prqwko| ru txduwhuo|
vwrfn uhwxuqv duh suhglfwdeoh +h1j1 Fdpsehoo/ 4<;:> Idpd dqg Iuhqfk/
4<;<> Shvdudq dqg Wlpphupdqq/ 4<<7/ 4<<8/ 5333,1 Frqvlghudwlrq ri
wkh orjolqhdu irup ri wkh suhvhqw ydoxh prgho looxvwudwhv wkdw li h{shf0
wdwlrqv ri ixwxuh glylghqg fkdqjh duh qrw wrr qrlv| wkhq vwrfn uhwxuqv
pd| kdyh suhglfwdelolw| lq wkdw yduldeohv wkdw khos suhglfw ixwxuh uhwxuqv
pd| h{lvw1 Wklv zrxog qrw lqydolgdwh pdunhw h!flhqf|1 Wkh txhvwlrq lv
zkhwkhu wkh suhglfwdelolw| kdv hfrqrplf ydoxh lq wkh vhqvh ri jhqhudwlqj
deqrupdo uhwxuqv1 Iurp wklv shuvshfwlyh vwdqgdug phdvxuhv ri iruhfdvw0
lqj dffxudf| kdyh prghvw lqirupdwlrq frqwhqw/ vlqfh wkh| gr qrw doorz
iru wudqvdfwlrqv frvwv ru pds lqwr wkh qdwxuh ri wkh surw ghflvlrq1 Iru
h{dpsoh d iruhfdvw wkdw wkh pdunhw zrxog ulvh e| 45( zkhq lw ulvhv e|
6;7
Frqiurqwlqj Prghov zlwk Idfwv
8( lv lqihulru iurp wkh shuvshfwlyh ri d vtxduhg huuru phdvxuh ri iruhfdvw
dffxudf| wr d iruhfdvw ri 3(1 Fohduo| d ghflvlrq uxoh ri wkh ex|0vhoo w|sh
zrxog ydoxh wkh iruhfdvw glhuhqwo| +vhh h1j1 Shvdudq/ 4<<5> Judqjhu dqg
Shvdudq/ 4<<<,1 Lw lv qrw fohdu dv |hw zkhwkhu wkh suhglfwdelolw| lq dvvhw
uhwxuqv jhqhudwhv deqrupdo uhwxuqv sduwlfxoduo| zkhq wkh srwhqwldo iru
fkdqjlqj ulvn lv doorzhg iru1
Vwurqj0Irup Pdunhw H!flhqf|
Lqvlghu wudglqj lq wkh pdmru dvvhw pdunhwv lv vxemhfw wr ohjdo uhvwulfwlrqv
lq d qxpehu ri frxqwulhv dqg kdv ehhq loohjdo iru d qxpehu ri |hduv1 Iru
lqvwdqfh lq wkh XV wkh Vhfxulwlhv dqg H{fkdqjh Dfw ri 4<67 surklelwv
djhqwv iurp wudglqj vhfxulwlhv zkloh lq srvvhvvlrq ri pdwhuldo lqvlgh lqiru0
pdwlrq1 Lqvlghuv duh ghqhg wr lqfoxgh qrw rqo| frusrudwh lqvlghuv exw
dovr dq|rqh zkr rewdlqv pdwhuldo/ qrq0sxeolf lqirupdwlrq iurp d fru0
srudwh lqvlghu/ ru iurp wkh lvvxhu/ ru zkr vwhdov vxfk lqirupdwlrq iurp
dqrwkhu vrxufh1 Lq vxevhtxhqw |hduv ixuwkhu dfwv zhuh sdvvhg dqg lq 4<;;
lqvlghu wudglqj vdqfwlrqv zhuh ixuwkhu lqfuhdvhg vr wkdw lqiulqjhphqw fdq
uhvxow lq qhv ri xs wr rqh ploolrq grooduv ru yh wr whq |hduv lq mdlo1 Lq
wkh XN lqvlghu wudglqj lv d fulplqdo rhqvh dqg fdq uhvxow lq xs wr wzr
|hduv lq mdlo1 Dv d frqvhtxhqfh ri wklv/ lqgluhfw phwkrgv kdyh wr eh hp0
sor|hg wr dvfhuwdlq zkhwkhu lqvlghuv fdq pdnh deqrupdo uhwxuqv rq wkh
edvlv ri wkhlu sulydwh lqirupdwlrq1 Wkdw lqvlghu wudglqj grhv wdnh sodfh
lv sulpd idflh vxssruwhg e| vwxglhv vxfk dv Nhrzq dqg Slqnhuwrq+4<;4,
zkr irxqg wkdw rq dyhudjh 73083( ri wkh sulfh jdlq h{shulhqfhg e| d
wdujhw up*v vwrfn rffxuv ehiruh wkh dfwxdo wdnhryhu dqqrxqfhphqw1
Lqvlghuv zkr reh| wkh odz duh lq d qxpehu ri frxqwulhv/ vxfk dv wkh
XV dqg XN/ uhtxluhg wr qrwli| wkh vwrfn h{fkdqjhv ri wkhlu curxwlqh
wudghv* lq wkh vkduhv ri wkhlu frpsdqlhv1 Wklv lqirupdwlrq lv sxeolvkhg
uhdvrqdeo| txlfno|1 D qxpehu ri uhvhdufkhuv kdyh lqyhvwljdwhg zkhwkhu
wklv ohjdo wudglqj e| frusrudwh lqvlghuv fdq suhglfw ixwxuh vwrfn pdunhw
uhwxuqv h1j1 Ilqqhuw| +4<;9,/ Iulhghulfk hw do1 +5333,/ Juhjru| hw do1
+4<<7,/ Srsh hw do1 +4<<3,/ Vh|kxq +4<;9/ 4<<5,1Wkhvh whvwv duh vhpl0
vwurqj whvwv/ vlqfh wkh lqirupdwlrq vhw xqghuslqqlqj wkh whvwv lv sxeolf1
Wkh hylghqfh lv vxjjhvwlyh wkdw h{fhvv uhwxuqv fdq eh rewdlqhg wkrxjk
wkh phdvxuhphqw ri qrupdo uhwxuqv lv dq lvvxh lq vrph vwxglhv1
Mhqj hw do1 +4<<<, h{dplqh wkh uhwxuqv wr frusrudwh lqvlghuv wkhp0
vhoyhv +edvhg rq ohjdo wudghv, xvlqj d frpsuhkhqvlyh vdpsoh ri uhsruwhg
lqvlghu wudqvdfwlrqv iurp 4<:8 wr4<<91 Wkhlu fduhixoo|0frqgxfwhg vwxg|
surylghv hylghqfh ri vljqlfdqw deqrupdo uhwxuqv1
Phxoeurhn +4<<5, lqyhvwljdwhv wkh uhwxuqv wr loohjdo wudghv e| lqvlghuv1
Khu vdpsoh frqvlvwhg ri lqglylgxdov fkdujhg zlwk lqvlghu wudglqj gxulqj
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6;8
4<;304<;<1 Khu gdwd edvh lqfoxghg lqirupdwlrq rq wkh fkdujhv eurxjkw/
shqdowlhv lqfxuuhg/ surwv hduqhg/ qxpehu ri vhfxulwlhv wudghg/ w|sh dqg
vrxufh ri wkh lqvlgh lqirupdwlrq1 Dprqj wkh ghihqgdqwv zkr wudghg/ wkh
phgldq ghihqgdqw wudqvdfwhg lq rqh vhfxulw| dqg uhdshg '4:/95; lq surw
shu vhfxulw|1 E| dqdo|vlqj vhfxulw| sulfhv rq wkh gd|v lqvlghuv wudghg
dqg glg qrw sulru wr sxeolf dqqrxqfhphqwv Phxoeurhfn frxog lqyhvwljdwh
wkh lpsdfw ri lqvlghuv rq vwrfn sulfh pryhphqwv1 Vkh frqfoxghg wkdw
lqvlghu wudglqj lqfuhdvhv vwrfn sulfh dffxudf| e| prylqj vwrfn sulfhv
vljqlfdqwo|1 Vkh irxqg wkdw wkh deqrupdo sulfh pryhphqw rq lqvlghu
wudglqj gd|v lv 73083( ri wkh vxevhtxhqw sulfh uhdfwlrq wr wkh sxeolf
dqqrxqfhphqw ri wkh lqvlgh lqirupdwlrq1 Lw zrxog dsshdu wkhuhiruh iurp
wkh vwxglhv frqgxfwhg vr idu wkdw vwurqj0irup h!flhqf| lv qrw hpslulfdoo|
vxssruwhg1
Wkh dujxphqwv iru dqg djdlqvw wkh ohjdolw| ri lqvlghu wudglqj duh
glvfxvvhg lq h1j1 Dxvxeho +4<<3,/ Ehqderx dqg Odurtxh +4<<5,/ Ohodqg
+4<<5,/ Nkdqqd/ Voh}dn/ dqg Eudgoh| +4<<7,/ Wljkh dqg Plfkhqhu +4<<7,/
Zh qrz frqvlghu lq pruh ghwdlo vrph whvwv iru vhpl0vwurqj pdunhw
h!flhqf|1
Yduldqfh Erxqgv Whvwv
Dq lpsruwdqw whvw ri wkh h!flhqf| ri suhvhqw ydoxh prghov ri dvvhw sulfhv
zdv ruljlqdoo| ghyhorshg e| OhUr| dqg Sruwhu +4<;4, dqg Vkloohu +4<:</
4<;4,1 Wklv h{sorlwv wkh surshuw| ri udwlrqdo h{shfwdwlrqv wkdw wkh ydul0
dqfh ri wkh rxwfrph lv juhdwhu wkdq wkh yduldqfh ri wkh udwlrqdo h{shf0
wdwlrq iruhfdvw1
Iru d yduldeoh \w
ydu+\w , @ ydu+Hw \w.l , . ydu+y,
+474,
zkhuh yw lv wkh iruhfdvw huuru1 Wklv iroorzv vlqfh udwlrqdo h{shfwdwlrqv
lpsolhv wkdw wkh fryduldqfh ehwzhhq wkh iruhfdvw dqg wkh iruhfdvw huuru
lv }hur1
Lq prghov ri dvvhw sulfhv udwlrqdo h{shfwdwlrqv lpsolhv wkdw wkh dfwxdo
yduldqfh ri dq dvvhw sulfh/ Sw / zklfk lv d zhljkwhg dyhudjh ri ixwxuh
h{shfwdwlrqv ri ixqgdphqwdov vkrxog eh ohvv wkdq wkh yduldqfh ri wkh
dvvhw sulfh frpsxwhg rq wkh edvlv ri wkh dfwxdo vwrfkdvwlf surfhvv ri wkh
ixqgdphqwdov1
Iru vlpsolflw| zh frqvlghu wkh prgho lq ohyho udwkhu wkdq orj ohyho
irup1 Wkh dvvhw sulfh kdv wkh ydoxh/ dv zh kdyh vhhq lq h1j1 +66,/ ri=
%"
&
[ Gw.l
Sw @ Hw
+475,
+4 . U,l
l@4
6;9
Frqiurqwlqj Prghov zlwk Idfwv
zkhuh Gw.l duh wkh ixqgdphqwdo uhwxuqv dqg 4 . U lv wkh frqvwdqw glv0
frxqw udwh1
Dvvxplqj udwlrqdo h{shfwdwlrqv zh fdq ghfrpsrvh wkh suhvhqw ydoxh
uhodwlrqvkls lqwr wzr frpsrqhqwv rqh ri zklfk lv wkh shuihfw iruhvljkw
sdwk dqg wkh rwkhu wkh vhtxhqfh ri udwlrqdo h{shfwdwlrqv suhglfwlrq hu0
uruv=
"
"
[
[
Gw.l
xw.l
Sw @
+476,
l
+4
.
U,
+4
. U,l
l@4
l@4
zkhuh wkh xw.l duh wkh iruhfdvw huuruv1 Frqvhtxhqwo|
Sw @ Sw . yw
zkhuh
"
[
Sw @
l@4
+477,
"
[
Gw.l
xw.l
@
y
dqg
w
l
+4 . U,
+4 . U,l
l@4
Iurp +477,
ydu+Sw , @ ydu+Sw , . ydu+yw ,
+478,
Khqfh sodlqo| ydu Sw ydu Sw =
Wkh vwulnlqj qglqj ri wkh hduo| olwhudwxuh zdv wkdw vwrfn sulfhv ds0
shdu wr pryh wrr pxfk wr eh frqvlvwhqw zlwk vxevhtxhqw fkdqjhv lq
dfwxdo ixqgdphqwdov1 Iru h{dpsoh Vkloohu +4<;4/ e, hpsor|lqj dqqxdo
gdwd rq sulfh dqg glylghqgv iurp 4;:4 wr4<:< iru wkh XV vwrfn pdunhw/
vr wkdw wkh dfwxdo glylghqg vwuhdp lv hpsor|hg xs wr 4<:< soxv d sur{|
iru 4<;3 wr lqqlw| +w|slfdoo| wkh dfwxdo odvw ydoxh ri wkh dvvhw sulfh,
wr frpsxwh wkh yduldqfh lq 4;:4/ irxqg wkdw wkh dfwxdo dvvhw sulfh zdv
vrph yh wlphv kljkhu wkdq wkh shuihfw iruhvljkw dvvhw sulfh1 Ohur| dqg
Sruwhu +4<;4, rewdlqhg vlplodu uhvxowv hpsor|lqj phdvxuhv ri hduqlqjv
dv ixqgdphqwdov1
Wzr vwdwlvwlfdo sureohpv zlwk wklv dqdo|vlv zhuh srlqwhg rxw e| Iodylq
+4<;6,/ Nohlgrq +4<;9, dqg Pduvk dqg Phuwrq +4<;9,1 Frpsxwdwlrq ri
wkh vdpsoh yduldqfhv lq +477, lqyroyhv xvh ri wkh vdpsoh phdq1 Lq vpdoo
vdpsohv Iodylq dqg Nohlgrq ghprqvwudwh wkdw xvh ri wkh vdpsoh udwkhu
wkdq srsxodwlrq phdq ohdgv wr d eldv wrzdugv uhmhfwlqj h!flhqf|1
Pduvk dqg Phuwrq ghprqvwudwh wkdw wkh yduldqfh erxqgv whvwv sur0
srvhg e| Vkloohu duh qrw dssursuldwh li wkh surfhvv jhqhudwlqj wkh ixqgd0
phqwdov duh qrq vwdwlrqdu|1
Pdqnlz/ Urphu dqg Vkdslur +4<;8, ghyhors d yduldqfh erxqgv whvw
zklfk lv urexvw wr wkhvh wzr srlqwv1 Frqvlghu d qdlyh iruhfdvw ri wkh
dvvhw sulfh/ Swq
"
[
Gw.l
Swq @ Hwq
+479,
+4 . U,l
l@4
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6;:
zkhuh Hwq lv wkh qdlyh h{shfwdwlrq ri ixwxuh ixqgdphqwdov +vd| d frqvwdqw
jurzwk ri glylghqgv prgho,1
Zh fdq zulwh
Sw Swq @ +Sw Sw , . +Sw Swq ,
+47:,
Wdnlqj wkh vtxduh ri wkh ohiw0 dqg uljkw0kdqg vlghv ri +47:, dqg wdnlqj
xqfrqglwlrqdo h{shfwdwlrqv ri w lqirupdwlrq
H+Sw Swq ,5 @ H+Sw Sw ,5 . H+Sw Swq ,5
+47;,
vlqfh e| udwlrqdo h{shfwdwlrqv H+Sw Sw ,+Sw Swq , @ 3 +l1h1 wkh udwlrqdo
h{shfwdwlrqv iruhfdvw huuru/ wkh uvw whup lq eudfhv/ lv ruwkrjrqdo wr wkh
vhfrqg whup,
Htxdwlrq +474, lpsolhv wkdw
H+Sw Swq ,5 H+Sw Sw ,5
+47<,
H+Sw Swq ,5 H+Sw Swq ,5
+483,
dqg
Wkh uvw h{suhvvlrq vwdwhv wkdw wkh shuihfw iruhvljkw sdwk lv ehwwhu
iruhfdvw e| wkh dfwxdo dvvhw sulfh wkdq wkh qdlyh iruhfdvw> wkh vhfrqg
h{suhvvlrq vwdwhv wkdw wkh shuihfw iruhvljkw sdwk xfwxdwhv pruh durxqg
wkh qdlyh sdwk wkdq grhv wkh dfwxdo sulfh1
Rq wkh edvlv ri whvwv wkdw h{sorlw wkhvh erxqgv Pduvk dqg Phuwrq
dovr uhmhfw h!flhqf| wkrxjk wkh ylrodwlrq lv vpdoohu wkdq lq wkh ruljlqdo
whvwv1 Lq ghihqvh ri wkh h!flhqf| k|srwkhvlv zh vkrxog qrwh/ dv srlqwhg
rxw e| ydulrxv dxwkruv +h1j1 OhUr| dqg OdFlylwd/ 4<;4,/ wkdw wkh prghov
dvvxph d frqvwdqw glvfrxqw udwh ru frqvwdqw uhdol}hg uhwxuqv li wkh prgho
lv fdvw lq orjolqhdu irup +vhh h1j1 Fdpsehoo/ Or dqg Pdfnlqod|/ 4<<:,/
dqg wkh dvvxpswlrq wkdw djhqwv duh ulvn qhxwudo1
Hpslulfdo Whvwv= Wkh Iruhljq H{fkdqjh Pdunhw
Wkh iruhljq h{fkdqjh pdunhw lq sduwlfxodu kdv ehhq orrnhg dw sduwlfxoduo|
h{kdxvwlyho| dqg lv fohduo| ri juhdw lqwhuhvw iru pdfurhfrqrplfv1 Zh
h{dplqh vrph ri wkh hpslulfdo lvvxhv lq ghwdlo lq wklv vhfwlrq vlqfh pdq|
ri wkh lvvxhv duh uhohydqw wr rwkhu hpslulfdo zrun1 D exloglqj eorfn
ri pdq| dqdo|vhv lv wkdw fryhuhg lqwhuhvw duelwudjh krogv1 Vlqfh wklv lv
dq duelwudjh frqglwlrq/ ylrodwlrq ri lw zrxog eh kljko| gdpdjlqj wr wkh
h!flhqw pdunhwv k|srwkhvlv1 Qxphurxv hpslulfdo vwxglhv kdyh h{dplqhg
wkh fryhuhg frqglwlrq1 Lw dsshduv wkdw/ rqfh dssursuldwh doorzdqfh lv
pdgh iru wudqvdfwlrqv frvwv dqg fduh lv wdnhq wr hqvxuh wkdw gdwd lq
hpslulfdo whvwv lv vdpsohg dw wkh vdph prphqw lq wlph/ wkh frqglwlrq
6;;
Frqiurqwlqj Prghov zlwk Idfwv
lv ylrodwhg yhu| lqiuhtxhqwo| +vhh h1j1 Wd|oru/ 4<;:/ 4<;<,1 Iurp wkh
xqfryhuhg dqg fryhuhg duelwudjh frqglwlrqv zh rewdlq wkdw wkh orjdulwkp
ri wkh iruzdug h{fkdqjh udwh lv htxdo wr wkh pdunhw h{shfwdwlrq ri wkh
orjdulwkp ri wkh ixwxuh vsrw udwh ghqhg e| wkh pdwxulw| gdwh ri wkh
lqwhuhvw udwh1 Lq wkh fdvh ri d rqh shulrg krul}rq zh rewdlq
Iww @ Hw Vw.4
dv
+484,
Xqghu wkh dvvxpswlrq ri udwlrqdo h{shfwdwlrqv zh fdq uhzulwh +484,
Vw.4 @ Iww . xw.4
+485,
zkhuh xw.4 lv wkh udwlrqdo h{shfwdwlrqv iruhfdvw huuru vr wkdw xw.4 @
Vw.4 Hw Vw.4 =
Vlqfh Vw.4 > dqg Iww duh revhuydeoh/ hduo| whvwv ri wkh udwlrqdolw| ri Iww
zhuh frqgxfwhg e| hvwlpdwlqj wkh uhodwlrqvkls
Vw.4 @ 3 . 4 Iww . w.4
=
+486,
=
dqg whvwlqj zkhwkhu 3 @ 3> 4 @ 4 dqg wkh huuru whup h{klelwhg vhuldo
fruuhodwlrq1
Wkhvh whvwv zhuh vxssruwlyh lq jhqhudo ri wkh h!flhqf| ri wkh iruzdug
udwh dv d suhglfwru1 Krzhyhu lw zdv uhdol}hg vxevhtxhqwo| wkdw ehfdxvh
wkh vsrw dqg iruzdug udwh dsshdu wr eh qrqvwdwlrqdu| L+4, surfhvvhv
+Wlph0Vhulhv Dqqh{,/ d frh!flhqw ri xqlw| rewdlqhg lq hvwlpdwhv ri 4
zrxog eh d zhdn whvw/ hyhq li dssursuldwh vwdqgdug huuruv zhuh hpsor|hg/
vlqfh qrq0udwlrqdo suhglfwruv zklfk vkduhg d frpprq wuhqg zrxog h{klelw
wklv surshuw|1
Idpd +4<;7, uhsruwhg uhjuhvvlrq hvwlpdwhv iru qxphurxv fxuuhqflhv
lq wkh srvw0zdu rdwlqj shulrg ri wkh irup
Vw.4 Vw @ 3 . 4 +Iww Vw , . w.4
+487,
Frpsdulvrq ri +485, dqg +487, vkrzv wkdw li wkh hvwlpdwhv ri 4 dqg
3 duh qrw vljqlfdqwo| glhuhqw iurp rqh dqg }hur uhvshfwlyho| wkhq wkh
iruzdug udwh lv dq h!flhqw suhglfwru1
Lq idfw Idpd rewdlqhg wkh vwulnlqj uhvxow wkdw wkh hvwlpdwhv ri 4 zhuh
qhjdwlyh dqg vljqlfdqwo| glhuhqw iurp xqlw| +ru }hur,1 Txdolwdwlyho|
wkh vdph uhvxowv kdyh ehhq rewdlqhg iru qxphurxv glhuhqw gdwd vhwv dqg
wlph shulrgv1
Iru h{dpsoh xvlqj prqwko| gdwd iurp 4<:;134 wr 4<<313: PfFdo0
oxp+4<<7, uhsruwv wkh iroorzlqj uhvxowv iru wkh Groodu2Srxqg
5
Vw.4 @ 3=346: . 3=<::Iw U @ 3=<9
+488,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
5
Vw.4 Vw @ 3=33:; 7=:736+Iww Vw , U @ 3=444
6;<
+489,
Zklovw hpsor|lqj zhhno| gdwd iru wkh Groodu2Srxqg ehwzhhq Ghfhp0
ehu43wk 4<54Pd| 53wk4<58 zh rewdlq
5
Vw.7 @ 3=454 . 3=<53Iw U @ 3=;9
5
Vw.7 Vw @ 3=3389 6=6499+Iww Vw , U @ 3=376
+48:,
+48;,
Wkh frh!flhqwv rq wkh iruzdug suhplxp duh vljqlfdqwo| glhuhqw
iurp xqlw| lq wkhvh uhjuhvvlrqv dqg dovr vljqlfdqwo| ohvv wkdq }hur1 Wkh
qhjdwlyh frh!flhqw lpsolhv wkdw qrw rqo| lv wkh iruzdug udwh dq lqh!0
flhqw suhglfwru/ lw lv dovr lqihulru wr wkh vsrw udwh dv d suhglfwru ri wkh
ixwxuh vsrw udwh1 Wkhvh uhvxowv dsshdu wr lpso| wkdw wkh iruhljq h{fkdqjh
pdunhw lv lqh!flhqw1 Zklovw wklv pd| eh wkh fdvh/ dqg gluhfwo| revhuyhg
vxuyh| gdwd ri h{fkdqjh udwh h{shfwdwlrqv vxssruw vxfk dq lqwhusuhwdwlrq
vlqfh wkh| w|slfdoo| h{klelw eldv dqg lqh!flhqf| +vhh h1j1 Iudqnho dqg
Iurrw/ 4<;:/ dqg Iurrw dqg Iudqnho/ 4<;<, d qxpehu ri srvvleoh uhvrox0
wlrqv ri wklv dssduhqw dqrpdo| kdyh ehhq vxjjhvwhg zklfk duh frqvlvwhqw
zlwk pdunhw h!flhqf|1 Zh pljkw dovr qrwh lq sdvvlqj wkdw li wkh rxw0
frph zdv gxh wr lqh!flhqw h{shfwdwlrqv lw srvhv d pdmru sureohp iru
hfrqrplf dqdo|vlv jlyhq wkh v|vwhpdwlf qdwxuh ri wkh eldv ri wkh iruzdug
suhplxp ryhu :3 ru vr |hduv1
Wkh uvw h{sodqdwlrq ri wkh dqrpdorxv qglqj zdv vhw rxw e| Idpd
+4<;7,= wkh h{lvwhqfh ri d yduldeoh ulvn suhplxp1
D wlph0ydu|lqj ulvn suhplxp
Li zh uhod{ wkh dvvxpswlrq wkdw lqyhvwruv duh ulvn dyhuvh wkhq djhqwv
zloo uhtxluh d ulvn suhplxp iru xqghuwdnlqj wkh xqfryhuhg srvlwlrq1 D
ydulhw| ri dowhuqdwlyh wkhruhwlfdo ghulydwlrqv ri wkh ulvn suhplxp kdyh
ehhq sursrvhg +vhh h1j1 Ehndhuw/ 4<<9> Kdqvhq dqg Krgulfn/ +4<;6,/
Krgulfn/ 4<;<> Vlehuw/ 4<;<,1 Lq zkdw iroorzv zh xvh wkh Frqvxpswlrq
FDSP prgho1
Wkh h{0srvw qrplqdo uhwxuq/ +4 . Uwv ,/ wr dq xqfryhuhg grphvwlf
wudqvdfwlrq/ lv
4 . Uwv @
+4 . Uwi ,vw.4
vw
+48<,
zkhuh Uwi lv wkh ulvn0iuhh qrplqdo iruhljq lqwhuhvw udwh dqg vw lv wkh h{0
fkdqjh udwh phdvxuhg dv grphvwlf fxuuhqf| shu xqlw ri iruhljq fxuuhqf|1
Fryhuhg lqwhuhvw duelwudjh iru wkh ulvn0iuhh qrplqdo grphvwlf lqwhuhvw
6<3
Frqiurqwlqj Prghov zlwk Idfwv
udwh/ +4 . Uw , lv jlyhq e|
4 . Uw @
+4 . Uwi ,iw
vw
+493,
zkhuh iw lv wkh iruzdug h{fkdqjh udwh lq ohyhov1 +Qrwh wkdw +9;, lv dq
dssur{lpdwlrq wr +493, zkhuh orj+4 . Ui , * Ui > dqg V @ oq v,
Vxevwlwxwlrq rxw ri wkh iruhljq lqwhuhvw udwh iurp wkh fryhuhg frqgl0
wlrq dqg vxevwlwxwlrq lq +48<, jlyhv
4 . Uwv @
+4 . Uw ,vw.4
iw
+494,
Wkh uhdo h{0srvw grphvwlf uhwxuq iurp dq xqfryhuhg vshfxodwlrq +4 .
v
Uw , lv jlyhq e|
v
4 . Uw @
+4 . Uw ,vw.4 Sw
+4 . Uw ,vw.4 vw Sw
@
iw Sw.4
vw iw Sw.4
+495,
zkhuh S lv wkh sulfh ohyho1 Wkh uhdo h{0srvw grphvwlf uhwxuq iurp wkh
ulvn0iuhh qrplqdo erqg +4 . Uw , lv
4 . Uw @
+4 . Uw ,Sw
Sw.4
+496,
Iurp wkh frqvxpswlrq FDSP
v
Hw +4 . Uw ,x! +fw.4 ,
Hw +4 . Uw ,x! +fw.4 ,
@
x! +fw ,
x! +fw ,
+497,
Dvvxplqj frqglwlrqdo orj qrupdolw| dqg wkh irup ri xwlolw| ixqfwlrq
lq +54, zh fdq hpsor| +497, dqg +495, wr rewdlq
Hw Vw.4 @ Iw 3=8ydu+orj
vw.4
, . fry+^orj vw.4 @vw ` orj Sw.4 @Sw , .
vw
fry+^orj vw.4 @vw ` orj fw.4 @fw , +498,
+qrwh djdlq= I @ orj i> V @ orj v,1
Wkh ulvn suhplxp uh hfwv wkh fryduldwlrq ehwzhhq fkdqjhv lq wkh
orjdulwkp ri wkh h{fkdqjh udwh dqg wkh orjdulwkp ri fkdqjhv lq frq0
vxpswlrq1 Lw lv lqwhuhvwlqj wr qrwh wkdw li @ 3/ vr wkdw wkh frqvxphu lv
ulvn qhxwudo wkhuh duh vwloo whupv gulylqj d zhgjh ehwzhhq wkh h{shfwhg
ixwxuh vsrw udwh dqg wkh iruzdug udwh1 Wkhvh whupv rffxu e| yluwxh ri
zkdw lv nqrzq dv Mhqvhq*v lqhtxdolw|1
Li
iw @ Hw vw.4
+499,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6<4
zkhuh zh phdvxuh vd| lq srxqgv wr grooduv zh dovr uhtxluh phdvxulqj lq
grooduv wr srxqgv wkdw=
4
4
@ Hw
iw
vw.4
Krzhyhu e| Mhqvhq*v lqhtxdolw|
4
4
Hw
A
vw.4
Hw vw.4
+49:,
+49;,
Zh fdqqrw vlpxowdqhrxvo| kdyh +499, dqg +49:, kroglqj zklfk zrxog
dsshdu wr lpso| wkdw rqh sduw| fdq pdnh h{shfwhg qrplqdo surwv/
nqrzq dv Vlhjho*v +4<:5, sdudgr{1 Wkh uhvroxwlrq lv wkdw wudghuv duh
lqwhuhvwhg lq h{shfwhg uhdo surwv1 Jlyhq wkh dvvxpswlrq ri orjqrupdo
glvwulexwlrqv +498, vkrzv wkh dssursuldwh frqglwlrq1 Wkh whupv uhvxowlqj
iurp Mhqvhq*v lqhtxdolw| duh jhqhudoo| suhvxphg wr eh vpdoo exw Vlehuw
+4<;<, vkrzv wkdw wklv dvvxpswlrq pd| eh lqdssursuldwh1
Dgglwlrq ri d ulvn suhplxp wr wkh xqfryhuhg duelwudjh frqglwlrq
lpsolhv/ lq frqmxqfwlrq zlwk wkh fryhuhg frqglwlrq zklfk uhpdlqv xq0
fkdqjhg +lw lv ulvnohvv,/ wkdw
Iw @ Hw Vw.4 . sw
+49<,
zkhuh sw lv wkh culvn suhplxp* +lqfoxvlyh ri doo wkh whupv,1
Vxewudfwlqj wkh vsrw udwh iurp erwk vlghv ri +49<, zh rewdlq wkdw
Iw Vw @ Hw Vw.4 Vw . sw
+4:3,
Qrz frqvlghu wkh uhjuhvvlrqv
Iw Vw.4 @ 4 . 4 +Iw Vw , . 4w.4
+4:4,
Vw.4 Vw @ 5 . 5 +Iw Vw , . 5w.4
+4:5,
dqg
zkhuh wkh dqg duh frqvwdqwv dqg wkh duh wkh huuru whupv1
Wkh ohdvw vtxduhv hvwlpdwhv ri 4 dqg 5 duh jlyhq e|
4 @
fry+Iw Vw.4 > Iw Vw ,
@
ydu+Iw Vw ,
5 +s, . fry+sw > Hw Vw.4 Vw ,
5
+s, . 5fry+sw > Hw Vw.4 Vw , . 5 +Hw Vw.4 Vw ,
dqg
5 @
fry+Vw.4 Vw > Iw Vw ,
@
ydu+Iw Vw ,
+4:6,
6<5
Frqiurqwlqj Prghov zlwk Idfwv
5 +Hw Vw.4 Vw , . fry+sw > Hw Vw.4 Vw ,
+4:7,
5 +s, . 5fry+sw > Hw Vw.4 Vw , . 5 +Hw Vw.4 Vw ,
+uhfdoo lq wkh ohdvw vtxduhv uhjuhvvlrq \ @ . [> wkh hvwlpdwh ri
@ fry+\>[,
ydu+[, , dqg wkdw xqghu udwlrqdo h{shfwdwlrqv fry+w.4 > dq| yduldeoh
dw w, @ 3= Qrwh dovr wkdw Vw.4 Vw @ Hw Vw.4 Vw . w.4 > zkhuh w.4 lv
wkh udwlrqdo h{shfwdwlrqv iruhfdvw huuru1
Zh qrwh wkh uhjuhvvlrqv duh pluuru lpdjhv ri hdfk rwkhu zlwk wkh
surshuw| wkdw 4 . 5 @ 4> 4 . 5 @ 3 dqg 4w.4 . 5w.4 @ 3=
Zh qrwh iurp wkh qxphudwru ri +4:7, wkdw li wkh hvwlpdwh ri 5 lv
qhjdwlyh wklv lv frqvlvwhqw zlwk d wlph0ydu|lqj ulvn suhplxp li wkh frydul0
dwlrq ehwzhhq h{shfwhg fkdqjhv lq wkh vsrw udwh dqg wkh ulvn suhplxp lv
qhjdwlyh dqg juhdwhu wkdq wkh yduldqfh ri h{shfwhg fkdqjhv lq wkh vsrw
udwh1
Ehfdxvh 4 lv srvlwlyh wklv lpsolhv wkdw wkh yduldqfh ri wkh ulvn suh0
plxp lv juhdwhu wkdq wkh fryduldwlrq ehwzhhq wkh h{shfwhg fkdqjh lq
wkh vsrw udwh dqg wkh ulvn suhplxp1 Wrjhwkhu wkhvh wzr frqglwlrqv dovr
lpso| wkdw wkh ulvn suhplxp pxvw kdyh d yduldqfh juhdwhu wkdq wkh ydul0
dqfh ri h{shfwhg fkdqjhv lq wkh vsrw udwh1 Krgulfn dqg Vulydwdyd +4<;9,
vkrzhg krz d qhjdwlyh fryduldwlrq ehwzhhq h{shfwhg fkdqjhv lq wkh vsrw
udwh dqg wkh ulvn suhplxp frxog eh rewdlqhg lq d surshuo| irupxodwhg
wkhruhwlfdo prgho1 Frqvhtxhqwo| wkh qhjdwlyh frh!flhqw rewdlqhg lq wkh
uhjuhvvlrqv ri fkdqjhv lq vsrw rq wkh iruzdug suhplxp frxog eh frqvlv0
whqw zlwk udwlrqdo h{shfwdwlrqv dqg d wlph ydu|lqj ulvn suhplxp1 Xq0
iruwxqdwho| gdwd zklfk zrxog doorz gluhfw sur{|lqj ri wkh ulvn suhplxp
duh qrw dydlodeoh dw wkh iuhtxhqflhv uhtxluhg1 Wlph vhulhv phwkrgv edvhg
rq prghoolqj wkh ulvn suhplxp dv d ixqfwlrq ri wkh yduldqfh ri wkh vsrw
udwh ru iruhfdvw huuruv kdyh qrw surylghg pxfk gluhfw vxssruw iru d ydul0
deoh ulvn suhplxp/ wkrxjk wkh prghov ri wkh ulvn suhpld duh srvvleo|
srru dssur{lpdwlrqv vhh h1j1 Edloolh dqg Eroohuvohy +4<<3, dqg Gr0
przlw} dqg Kdnnlr +4<;8,/ Eroohuvohy dqg Krgulfn +4<<9, Hqjho +4<<9,1
Lq dgglwlrq wkh pdjqlwxgh ri wkh yduldqfh ri wkh ulvn suhpld lpsolhg e|
wkh hvwlpdwhv ri 5 duh uhjdughg e| vrph dv shukdsv lpsodxvleo| kljk
+wkrxjk vhh Sdjdq/ 4<;;,1 Iru h{dpsoh dq hvwlpdwh ri 5 ri 7 lpsolhv
5
w Vw.4 Vw ,. +sw ,
wkdw ydu+sw , A 8ydu+Iw Vw , +qrwh 5 @ 7 @ 4 +fry+sw >Hydu+I
w Vw ,
lv d glhuhqw zd| ri zulwlqj wkh hvwlpdwh ri 5 ,=
D vhfrqg srwhqwldo h{sodqdwlrq ri wkh hpslulfdo qglqjv lv fdoohg wkh
cshvr sureohp*/ wr zklfk zh qrz wxuq1
Shvr Sureohpv
Wkh h{suhvvlrq cshvr sureohp* zdv suredeo| uvw hpsor|hg e| Plowrq
Iulhgpdq lq klv dqdo|vlv ri wkh ehkdylrxu ri wkh Ph{lfdq fxuuhqf|/ wkh
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6<6
shvr/ lq wkh hduo| 4<:3v1 Dowkrxjk wkh Ph{lfdq h{fkdqjh udwh zdv {hg
ehwzhhq Dsulo 4<87 dqg Dxjxvw 4<:9 dw 313; grooduv shu shvr/ Ph{lfdq
lqwhuhvw udwhv zhuh vxevwdqwldoo| deryh XV lqwhuhvw udwhv1 Wklv qglqj
suhvhqwhg dq dssduhqw sx}}oh/ vlqfh udwlrqdo h{shfwdwlrqv ri d frqwlqxlqj
{hg h{fkdqjh udwh uhjlph zrxog/ jlyhq qhjoljleoh ulvn suhpld/ lpso| yld
xqfryhuhg lqwhuhvw duelwudjh qhdu htxdolw| ri Ph{lfdq dqg XV lqwhuhvw
udwhv ri vlplodu w|sh1 Iulhgpdq udwlrqdol}hg wkh glhuhqwldo lq whupv ri
d pdunhw h{shfwdwlrq ri d ghydoxdwlrq ri wkh shvr1 Lq idfw lq Dxjxvw
4<:9 wkhvh h{shfwdwlrqv zhuh vxevhtxhqwo| uhdol}hg zkhq wkh shvr zdv
doorzhg wr rdw dqg ihoo vrph 79( lq ydoxh1 Wkh shvr sureohp wkxv uhihuv
wr d vlwxdwlrq zkhuh udwlrqdo djhqwv dqwlflsdwh wkh srvvlelolw| ri ixwxuh
fkdqjhv lq wkh gdwd0jhqhudwlqj phfkdqlvp ri hfrqrplf yduldeohv1
Wr looxvwudwh wkh qdwxuh ri wkh shvr sureohp vxssrvh wkhuh duh wzr
srvvleoh hfrqrplf uhjlphv lq zklfk wkh yduldeoh |w iroorzv wkh surfhvv
|w @ |4 . |w4 . xw lq uhjlph 4
+4:8,
|w @ |5 . xw lq uhjlph 5
+4:9,
ru
zkhuh | 4 / |5 dqg duh frqvwdqwv dqg xw lv d udqgrp yduldeoh zklfk lv
dvvxphg wr eh wkh vdph/ iru vlpsolflw|/ lq hdfk uhjlph1
Frqvlghu wkh udwlrqdo h{shfwdwlrq ri |w.4 iruphg rq wkh edvlv ri wkh
lqirupdwlrq vhw dydlodeoh dw wlph w> H+|w.4 m w ,1 Zh dvvxph wkdw
udwlrqdo djhqwv gr qrw nqrz zklfk uhjlph zloo suhydlo lq wkh qh{w shulrg>
udwkhu wkh| kdyh suredelolwlhv tw wkdw uhjlph 4 zloo rffxu dqg 4 tw wkdw
uhjlph 5 zloo rffxu1
Wkh udwlrqdo h{shfwdwlrq ri |w.4 lv wkhq jlyhq e|
H+|w.4 m w , @ tw +|4 . |w , . +4 tw ,|5
+4::,
Wkh shvr sureohp lq lwv vwdunhvw irup fdq eh looxvwudwhg e| dvvxp0
lqj wkdw rqo| rqh ri wkh uhjlphv lv hyhu revhuyhg lq d vdpsoh ri gdwd1
Frqvlghu wkh fdvh zkhuh rqo| uhjlph 4 rffxuv1 Wkh udwlrqdo h{shfwdwlrqv
iruhfdvw huuru iru wklv fdvh lv jlyhq e|
| 4 . |w . xw.4 H+|w.4 m w , @ xw.4 . +4 tw ,^|4 |5 ` . +4 tw ,|w
+4:;,
Zh revhuyh iurp +4:;, wkdw wkh h{0srvw iruhfdvw huuru lq wklv fdvh zloo
h{klelw eldv dqg h{klelw fruuhodwlrq zlwk yduldeohv lq wkh lqirupdwlrq vhw
zkhq h{shfwdwlrqv duh iruphg +|w ,= Lq idfw li @ 4/ lq rxu h{dpsoh wkh
iruhfdvw huuru zloo eh qrq0vwdwlrqdu|/ d srvvlelolw| srlqwhg rxw e| Hydqv
dqg Ohzlv +4<<6/ 4<<7,1
6<7
Frqiurqwlqj Prghov zlwk Idfwv
Lw lv dovr lqwhuhvwlqj wr qrwh zkdw kdsshqv zkhq d vwdqgdug ruwkrjr0
qdolw| ru h!flhqf| whvw iru udwlrqdo h{shfwlrqv lv uxq rq gdwd jhqhudwhg
e| wklv rxwfrph/ dvvxplqj tw lv frqvwdqw= ohw wkh uhvhdufkhu hvwlpdwh wkh
uhodwlrqvkls
|w.4 H+|w.4 m w , @ . +H+|w.4 m w , |w , . w.4
+4:<,
zkhuh dqg duh frqvwdqwv dqg w.4 lv wkh huuru whup1
e @fry+\> [,@ydu+[, zkhuh \ lv wkh
Wkh ohdvw vtxduhv hvwlpdwh ri ghshqghqw yduldeoh dqg [ wkh lqghshqghqw, lv jlyhq e|
e @ +4 t,
+4 t,
+4;3,
Frqvhtxhqwo| iru A 3 wkh uhjuhvvlrq hvwlpdwh ri > zklfk lv qhjd0
wlyh/ zrxog lpso| wkdw wkh udwlrqdo h{shfwdwlrq ri |w.4 zdv dq lqihulru
suhglfwru wr wkh fxuuhqw ohyho ri |w 1
Qrq0rffxuuhqfh ri d uhjlph +hyhqw, lq d sduwlfxodu vdpsoh surylghv
wkh h{wuhph h{dpsoh ri wkh shvr sureohp1 Vxfk srvvleolwlhv frxog rffxu
zkhq iru lqvwdqfh wkhuh lv d vpdoo suredelolw| ri d fkdqjh lq uhjlph1
Vxfk fdvhv zrxog qrw eh dphqdeoh wr glhuhqwldwlqj hpslulfdoo| ehwzhhq
udwlrqdo h{shfwdwlrqv dqg dq dowhuqdwlyh xqohvv wkh suredelolw| t dqg wkh
dowhuqdwlyh uhjlphv frxog eh ghvfulehg1
Hydqv +4<<9, kdv vhw rxw d jhqhudo phwkrg ri dqdo|vlv ri wkh shvr
sureohp vrph ri zklfk zh qrz vhw rxw1 Zh fdq ghqh wkh udwlrqdo
h{shfwdwlrqv iruhfdvw huuru/ hw.4 / dv
hw.4 |w.4 H+|w.4 m w ,
+4;4,
Wr h{dplqh krz wkh surshuwlhv ri wkh iruhfdvw huuru duh dhfwhg e|
wkh suhvhqfh ri glvfuhwh fkdqjhv lq uhjlph Hydqv vxssrvhv wkdw |w.4
fdq vzlwfk ehwzhhq wzr surfhvvhv zklfk duh lqglfdwhg e| fkdqjhv lq d
glvfuhwh0ydoxhg yduldeoh/ ]w @ i3> 4j/ vr wkdw ]w rqo| wdnhv wkh ydoxh
}hur ru xqlw|1 Ohw |w.4 +}, ghqrwh uhdol}hg uhwxuqv lq uhjlph ]w.4 @ }=
Wkh shvr sureohp lv wr frqvlghu wkh ehkdylrxu ri iruhfdvw huuruv/ |w.4 H+|w.4 m w ,= Hydqv grhv wklv e| ghfrpsrvlqj uhdol}hg uhwxuqv lqwr wkh
frqglwlrqdoo| h{shfwhg uhwxuq lq uhjlph }> zklfk lv ghqrwhg H+|w.4 +}, m
w , dqg d uhvlgxdo xw.4 +zklfk iru vlpsolflw| lv dvvxphg wr eh wkh vdph
lq erwk uhjlphv,1
Wkh ghfrpsrvlwlrq kdv wkh iroorzlqj irup=
|w.4 @ H+|w.4 +3, m w , . H+|w.4 m w ,]w.4 . xw.4
+4;5,
zkhuh
H+|w.4 m w , H+|w.4 +4, m w , H+|w.4 +3, m w ,
+4;6,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6<8
Vxevwlwxwlrq ri +4;6, lqwr +4;5, dqg vhwwlqj ]w.4 dv hlwkhu rqh ru }hur
lqirupv xv/ dv srlqwhg rxw e| Hydqv/ wkdw lw zloo dozd|v eh srvvleoh wr
ghfrpsrvh |w.4 lq wklv zd| uhjdugohvv ri wkh surfhvv |w.4 iroorzv lq hlwkhu
uhjlph ru wkh suhflvh vshflfdwlrq ri wkh lqirupdwlrq vhw w = Jlyhq wkh
dvvxpswlrq ri udwlrqdo h{shfwdwlrqv H+xw.4 m w , @ 3/ vr wkdw wkh huuru
whup xw.4 kdv wkh frqyhqwlrqdo udwlrqdo h{shfwdwlrqv surshuwlhv/ Hydqv
ghqhv wklv huuru dv wkh zlwklq0uhjlph iruhfdvw huuru/ vlqfh lw uhsuhvhqwv
wkh huuru zkhq wkh w . 4 uhjlph lv nqrzq1
Zkhq djhqwv duh xqdzduh ri wkh uhjlph lq w . 4 wkhlu iruhfdvw huuruv
zloo glhu iurp wkh zlwklq0uhjlph huuruv1 Wdnlqj h{shfwdwlrqv ri erwk
vlghv ri +4;5, rq wkh edvlv ri w lqirupdwlrq/ zh kdyh iurp wkh surshuwlhv
ri udwlrqdo h{shfwdwlrqv wkdw
H+|w.4 m w , @ H+|w.4 +3, m w , . H+|w.4 m w ,H+]w.4 m w ,
+4;7,
zkhuh zh qrwh wkdw wkh uljkw0kdqg vlgh qrz frqwdlqv wkh h{shfwhg uhjlph
lq w . 4=
Vxewudfwlrq ri +4;7, iurp +4;5, dqg vxevwlwxwlrq lq +4;4, jlyhv wkh
h{0srvw iruhfdvw huuru dv
hw.4 @ xw.4 . H+|w.4 m w ,^]w.4 H+]w.4 m w ,`
+4;8,
Zh revhuyh iurp +4;8, wkdw zkhq wkh uhjlph dw w . 4 lv nqrzq/
vr wkdw ]w.4 H+]w.4 m w , @ 3/ wkh iruhfdvw huuru lv wkh vwdqgdug
zlwklq0uhjlph huuru vr wkdw wkhuh lv qr shvr sureohp1 Zkhq wkh ixwxuh
uhjlph lv xqnqrzq wkh vhfrqg whup lq +4;8, dggv d frpsrqhqw wr wkh
zlwklq0uhjlph huuru +jlyhq wkdw wkh zlwklq0uhjlph iruhfdvwv glhu vr wkdw
H+|w.4 m w , 9@ 3,=
Hydqv looxvwudwhv wklv srlqw pruh fohduo| e| vxssrvlqj wkdw lq shulrg
w . 4 uhjlph 4 rffxuv1 Iurp +4;8, zlwk ]w.4 @ 4 zh rewdlq wkdw
hw.4 +4, @ xw.4 . H+|w.4 m w ,^4 H+]w.4 m w ,`
+4;9,
ru
hw.4 +4, @ xw.4 . H+|w.4 m w , Sui+]w.4 @ 3 m w ,j
+4;:,
zkhuh Su ghqrwhv suredelolw|1
Zkhq wkh suredelolw| ri uhjlph 3 lv qrq0}hur wkh udwlrqdo h{shfwdwlrq
huuru frqwdlqv dq dgglwlrqdo frpsrqhqw zklfk lv wkh glhuhqfh ehwzhhq
wkh zlwklq0uhjlph iruhfdvwv pxowlsolhg e| wkh suredelolw| wkdw uhjlph
3 rffxuv1 Dv zh looxvwudwhg zlwk rxu h{dpsoh deryh zkhq wkh zlwklq0
uhjlph iruhfdvwv glhu wklv fdq jhqhudwh dq h{0srvw udwlrqdo huuruv zklfk
pd| kdyh d qrq0}hur phdq/ vr wkdw wkh| dsshdu wr eh eldvhg ru vhuldoo|
fruuhodwhg/ dqg vr lqh!flhqw1 Lw lv dovr dssduhqw iurp +4;8, wkdw wkh
h{whqw wr zklfk wklv lvvxh zloo eh lpsruwdqw lq hpslulfdo hydoxdwlrqv ri
6<9
Frqiurqwlqj Prghov zlwk Idfwv
iruhfdvwv zloo eh ghshqghqw xsrq wkh iuhtxhqf| ri uhjlph vkliwv lq wkh
gdwd vdpsoh1 Lq wkh h{wuhph fdvh zkhq rqo| rqh uhjlph rffxuv wkh huuruv
zloo pdwfk wkrvh lq +4;9,1 Zkhq wkhuh duh d qxpehu ri uhjlph fkdqjhv
wkh iruhfdvw huuru zloo eh d zhljkwhg dyhudjh ri hw.4 +4, dqg hw.4 +3, +gh0
qhg dqrorjrxvo| wr +4;9,,1Wkh hhfw lq wklv fdvh rq wkh surshuwlhv ri
wkh iruhfdvw huuru zloo ghshqg rq wkh vdpsoh surshuwlhv ri wkh iruhfdvw
huuru iru uhjlphv ]w.4 H+]w.4 m w ,1 Zkhq wkh qxpehu ri uhjlph
fkdqjhv lq wkh vdpsoh lv uhsuhvhqwdwlyh ri wkh xqghuo|lqj glvwulexwlrq
ri uhjlph fkdqjhv/ iurp zklfk wkh udwlrqdo h{shfwdwlrqv ri djhqwv duh
jhqhudwhg/ wkhq wkh iruhfdvw huuru iru uhjlphv zloo h{klelw d }hur phdq
dqg wkh iruhfdvw huuruv zloo h{klelw wkh vwdqgdug udwlrqdo h{shfwdwlrqv
huuruv1
Dq lqwhuhvwlqj h{whqvlrq ri wkh shvr lvvxh frqvlghuhg e| Hydqv dqg
uhihuuhg wr e| Ndplqvn| +4<<6, dv wkh jhqhudolvhg shvr sureohp lv zkhq
djhqwv gr qrw nqrz zklfk fxuuhqw uhjlph wkh| duh lq +iru lqvwdqfh
wkh fhqwudo edqn*v suhihuhqfhv,1 Hydqv looxvwudwhv vrph lpsolfdwlrqv
e| dvvxplqj wkdw wkh ghjuhh ri xqfhuwdlqw| ri wkh uhjlph lv jlyhq e|
Su+]w m w ,= Zkhq wkh uhjlph lv nqrzq dqg wkhuh lv qr xqfhuwdlqw|
]w @ } dqg Su+]w m w , @ 4= Zkhq Su+]w m w , 9@ 4 Hydqv glvfxvvhv vrph
lpsolfdwlrqv lq wkh iroorzlqj pdqqhu1 Iluvw zh hpsor| wkh lghqwlw|
Su+]w.4 @ 3, Su+]w.4 @ 3 m ]w @ 4, Su+]w.4 @ 3 m ]w @ 4,
. Su+]w.4 @ 3, +4;;,
zkhuh iru vlpsolflw| zh kdyh gursshg wkh qrwdwlrq iru wkh lqirupdwlrq
vhw w 1 +Zh uhdg h1j1 wkh vhfrqg whup ri +4;;,/ Su+]w.4 @ 3 m ]w @ 4,> dv
iroorzv= wkh suredelolw| ri ]w.4 @ 3 jlyhq ]w @ 4/ doo frqglwlrqdo rq wkh
lqirupdwlrq vhw w ,=
Zh vxevwlwxwh +4;;, lqwr +4;:, wr rewdlq
hw.4 +4, @ xw.4 . H+|w.4 , Su+]w.4 @ 3 m ]w @ 4, Su+]w.4 @ 3 m ]w @ 4, . Su+]w.4 @ 3,` +4;<,
ru
hw.4 +4, @ xw.4 . H+|w.4 ,iSu+]w.4 @ 3 m ]w @ 4j H+|w.4 ,^Su+]w.4 @ 3 m ]w @ 4, Su+]w.4 @ 3,` +4<3,
Qrz
^Su+]w.4 @ 3 m ]w @ 4, Su+]w.4 @ 3,`
+4<4,
wkh odvw whup lq +4<3,/ lv htxdo wr
^Su+]w.4 @ 3 m ]w @ 4, Su+]w.4 @ 3 m ]w @ 3,` Su+]w @ 3,
+4<5,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6<:
vlqfh htxdwlqj +4<4, dqg +4<5, dqg uhduudqjlqj jlyhv wkh ghqlwlrqdo
vwdwhphqw
Su+]w.4 @ 3, @ Su+]w.4 @ 3 m ]w @ 4,^4 Su+]w @ 3,` .
Su+]w.4 @ 3 m ]w @ 3,^Su+]w @ 3,` +4<6,
+qrwh 4 Su+]w @ 3, @ Su+]w @ 4,,=
Frqvhtxhqwo| wkh odvw whup lq +4<3, fdq eh uhzulwwhq dv
H+|w.4 ,^Su+]w.4 @ 3 m ]w @ 4, Su+]w.4 @ 3 m ]w @ 3,` Su+]w @ 3,
+4<7,
Uhfrjql}lqj wkdw wkh uvw wzr whupv rq wkh uljkw0kdqg vlgh ri +4<3,
duh wkh vdph dv wkrvh lq +4;:, zh vhh wkdw xqfhuwdlqw| derxw wkh fxuuhqw
uhjlph pdqlihvwv lwvhoi lq dq dgglwlrqdo whup lq wkh iruhfdvw huuru jlyhq
e| +4<7,1 Zh revhuyh wkdw wklv whup zloo rqo| eh }hur li wkh suredelolw|
ri uhjlph 3 dw wlph w . 4 lv lqghshqghqw ri wkh fxuuhqw uhjlph ru ri
frxuvh wkdw zlwklq0uhjlph iruhfdvwv duh htxdo1 Lq dgglwlrq zh revhuyh
wkdw fkdqjhv lq Su+]w @ 3, +shukdsv gxh wr ohduqlqj derxw wkh uhjlph,
zloo frqwulexwh wr wkh vwuxfwxuh ri wkh huuru whup1
Hydqv +4<<9, grfxphqwv d qxpehu ri rwkhu lqwhuhvwlqj lpsolfdwlrqv
ri wkh shvr sureohp1 Lq wkh frqwh{w ri dvvhw sulfhv/ zkhuh fxuuhqw ydoxhv
ghshqg rq h{shfwhg ydoxhv ri ixqgdphqwdov lqwr wkh lqghqlwh ixwxuh/
kh ghprqvwudwhv krz qhzv derxw ixwxuh ixqgdphqwdov fdq lq xhqfh fxu0
uhqw sulfhv wkurxjk wkh qrupdo fkdqqho ri fkdqjhg iruhfdvwv ri ixwxuh
ixqgdphqwdov dv zhoo dv wkh dgglwlrqdo fkdqqhov ri wkh glhuhqfh lq ixq0
gdphqwdov lq wkh glhuhqw uhjlphv dv zhoo dv wkh g|qdplfv ri uhjlph
vzlwfklqj1 Lq wkh frqwh{w ri hpslulfdo zrun kh ghprqvwudwhv krz shvr
sureohpv fdq frqwulexwh wr wkh h{sodqdwlrq ri vrph ri wkh sx}}ohv irxqg
lq dvvhw pdunhwv/ vxfk dv wkh eldv ri wkh iruzdug suhplxp1 Ehndhuw/
Krgulfn dqg Pduvkdoo +5334, vkrz krz shvr sureohpv fdq frqwulexwh wr
dq h{sodqdwlrq ri dqrpdolhv lq wkh whup vwuxfwxuh ri lqwhuhvw udwh +vhh
ehorz,1
Wkh shvr sureohp lv wkdw vdpsoh vwdwlvwlfv duh qrw uhsuhvhqwdwlyh ri
wkh srsxodwlrq vr wkdw vwdwlvwlfdo lqihuhqfhv duh srwhqwldoo| plvohdglqj1
Lw lv fohduo| ri frqvlghudeoh lpsruwdqfh lq hydoxdwlqj prghov ri h{shfwd0
wlrqv1
Vwdwlvwlfdo Udwlrqdoh
Dq dowhuqdwlyh vwdwlvwlfdo udwlrqdoh iru wkh iruzdug suhplxp eldv lv vxj0
jhvwhg e| Edloolh dqg Eroohuvohy +4<<:,1 Dq hpslulfdo uhjxodulw| lq wkh
h{fkdqjh pdunhw lv wkdw wkh vwdwlvwlfdo surshuwlhv ri fkdqjhv lq wkh vsrw
udwh dqg wkh iruzdug suhplxp duh pdunhgo| glhuhqw1 Lq prqwko| gdwd
6<;
Frqiurqwlqj Prghov zlwk Idfwv
wkh yduldqfh ri fkdqjhv lq wkh vsrw udwh lv w|slfdoo| vrph 433 ru vr wlphv
juhdwhu wkdq wkh yduldqfh ri wkh iruzdug suhplxp1 Lq dgglwlrq zklovw
fkdqjhv lq wkh vsrw udwh w|slfdoo| h{klelw olwwoh hylghqfh ri vhuldo fruuh0
odwlrq dqg fdq riwhq eh zhoo dssur{lpdwhg e| pduwlqjdohv wkh iruzdug
suhplxp w|slfdoo| glvsod|v yhu| shuvlvwhqw vorzo| ghfd|lqj dxwrfruuhod0
wlrqv zklfk pd| eh ghvfulehg dv d iudfwlrqdo surfhvv vhh h1j1 Edloolh dqg
Eroohuvohy +4<<7,/ E|huv dqg Shho +4<<9,/ dovr wkh Wlph0Vhulhv Dqqh{,1
Wkh vwdwlvwlfdo surshuwlhv ri fkdqjhv lq wkh vsrw udwh dqg wkh iruzdug
suhplxp lpso| wkdw hvwlpdwhv ri 5 lq vpdoo vdpsohv pd| eh iudjloh1
Edloolh dqg Eroohuvohy vkrz wklv lv wkh fdvh1 Wkh| uhsruw hvwlpdwhv ri 5
lq wkh Idpd uhjuhvvlrq rewdlqhg iurp d uroolqj vdpsoh ri gdwd1 Hpsor|0
lqj prqwko| gdwd iru wkh GP2groodu wkh| frqvlghu 53; yh0|hdu uroolqj
uhjuhvvlrq hvwlpdwhv iru 5 rewdlqhg e| ehjlqqlqj lq Pdufk 4<:6 dqg
xvlqj d wrwdo ri vl{w| revhuydwlrqv wkurxjk wr Iheuxdu| 4<:;1 Wkhq wkh
qh{w hvwlpdwh zdv rewdlqhg e| xvlqj gdwd iurp Dsulo 4<:6 wkurxjk wr
Pdufk 4<:;/ xqwlo wkh qdo hvwlpdwh zdv edvhg rq gdwd iurp Ghfhpehu
4<<3 wkurxjk wr Qryhpehu 4<<81 Wkh hvwlpdwhv ri 5 glvsod| pdunhg
lqvwdelolw| ydu|lqj ehwzhhq 46 dqg 61851 Wkhvh qglqjv vxjjhvw wkdw
iru pdq| vdpsoh vl}hv hqfrxqwhuhg lq sudfwlfh/ wkh hvwlpdwh ri 5 lv olnho|
wr eh xqlqirupdwlyh derxw wkh wuxh ydoxh ri 5 =
Wkh| surylgh ixuwkhu vxssruw iru wklv ylhz e| vlpxodwlqj gdwd iurp
d nqrzq vwuxfwxuh zklfk lv fdoleudwhg wr surgxfh wkh vw|ol}hg vwdwlvwlfdo
surshuwlhv h{klelwhg e| wkh vsrw dqg iruzdug suhplxpv1 Wkh h{shfwd0
wlrq hperglhg lq wkh iruzdug suhplxp lv udwlrqdo e| frqvwuxfwlrq1 Wkhlu
vlpxodwhg prghov duh irxqg wr jhqhudwh uhvxowv zklfk duh vlplodu wr wkrvh
uhsruwhg lq wkh olwhudwxuh1 Lq sduwlfxodu/ wkh iruzdug suhplxp h{klelwv
shuvlvwhqw dxwrfruuhodwlrq dqg wkh hvwlpdwhv ri 5 duh zlgho| glvshuvhg
ehwzhhq qhjdwlyh dqg srvlwlyh qxpehuv1Wkhvh hpslulfdo qglqjv duh ri
vrph lqwhuhvw vlqfh wkh vwdwlvwlfdo surshuwlhv h{klelwhg e| wkh ghshqghqw
dqg lqghshqghqw yduldeohv lq wkh Idpd uhjuhvvlrq duh txdolwdwlyho| vlp0
lodu wr wkrvh h{klelwhg lq rwkhu whvwv ri pdunhw h!flhqf| vxfk dv erqg
udwhv +vhh ehorz,1
PfFdooxp +4<<7, looxvwudwhv dqrwkhu lpsruwdqw srlqw wkdw qhhgv wr eh
jlyhq frqvlghudwlrq zkhq whvwlqj pdunhw h!flhqf| uhodwlrqvklsv1 Wklv lv
zkhwkhu wkh uhodwlrqvkls lv lqyduldqw wr jryhuqphqw srolf|1 Wr looxvwudwh/
PfFdooxp dvvxphv wkdw wkh prqhwdu| dxwkrulwlhv pdqdjh lqwhuhvw udwhv
vr dv wr vprrwk wkhlu pryhphqwv zkloh dovr uhvlvwlqj fkdqjhv lq h{fkdqjh
udwhv1
Ghqlqj {w @ Uw UwI kh frqvlghuv wkh srolf| uxoh
{w @ +Vw Vw4 , . {w4 . hw
zkhuh > duh frqvwdqwv dqg hw lv d udqgrp huuru1
+4<8,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
6<<
Xqfryhuhg duelwudjh lv jlyhq e|
{w @ Hw Vw.4 Vw . yw
+4<9,
PfFdooxp lqwhusuhwv yw dv phdvxuhphqw huuru wkrxjk lw frxog eh
lqwhusuhwhg dv d ulvn suhplxp1 Kh dvvxphv wkdw
yw @ yw4 . xw
+4<:,
zkhuh lv d frqvwdqw dqg xw lv d udqgrp huuru1 Dvvxplqj d vroxwlrq iru
fkdqjhv lq wkh vsrw udwh ri wkh irup
Vw Vw4 @ D{w4 . Exw . Fhw
+4<;,
dqg htxdwlqj frh!flhqwv zh rewdlq wkh vroxwlrq dv
Vw Vw4 @
xw
hw
+ ,
{w4 .
+ . ,
+4<<,
i
@ Iw4 Vw4 ,1 Wkh vroxwlrq iru {w fdq
+qrwh wkdw {w4 @ Uw4 Uw4
dovr eh rewdlqhg dv
xw
{w @ {w4 .
+533,
+ . ,
Wkh lpsruwdqw srlqw wr qrwh lv wkdw wkh frh!flhqw rq {w4 +@ Iw4 Vw4 yld wkh fryhuhg duelwudjh frqglwlrq, lq +4<<, frqwdlqv wkh srolf|
sdudphwhuv dqg = Li lv forvh wr xqlw| dqg juhdwhu wkdq dqg lv
srvlwlyh/ wkhq wkh frh!flhqw fdq eh qhjdwlyh1 Lq dgglwlrq/ wkh iruzdug
suhplxp zloo h{klelw shuvlvwhqw vhuldo fruuhodwlrq iru forvh wr xqlw| dqg
li wkh yduldqfh ri xw lv yhu| vpdoo uhodwlyh wr wkh yduldqfh ri hw wkhq
fkdqjhv lq wkh vsrw udwh zloo eh forvho| dssur{lpdwhg e| d zklwh qrlvh
huuru surfhvv hyhq wkrxjk wkh wuxh surfhvv lv d shuvlvwhqw DUPD+4/ 4,
surfhvv1
Wklv frqfoxghv rxu glvfxvvlrq ri whvwv ri pdunhw h!flhqf| lq wkh h{0
fkdqjh pdunhw1 Lw kdv hpskdvlvhg wkh frqvlghudeoh gl!fxowlhv lq ghwhu0
plqlqj iurp h{0srvw uhjuhvvlrq zkhwkhu wklv pdunhw lv h!flhqw1
Zh qh{w frqvlghu/ pruh eulh |/ wkh lpsolfdwlrqv ri pdunhw h!flhqf|
iru wkh ehkdylrxu ri lqwhuhvw udwhv1
ORQJ0UXQ LQWHUHVW UDWHV XQGHU PDUNHW
HIILFLHQF\
Wkhuh duh d juhdw ydulhw| ri erqgv1 Erqgv duh lvvxhg e| sulydwh frpsd0
qlhv/ orfdo dxwkrulwlhv dqg e| wkh jryhuqphqw1 Qdwxudoo| wkh suredelolw|
733
Frqiurqwlqj Prghov zlwk Idfwv
ri ghidxow fdq ydu| dfurvv erqgv zlwk glhuhqw lvvxhuv1 Exw jryhuqphqw
erqgv duh qrupdoo| uhjdughg e| lqyhvwruv dv iuhh ri ghidxow ulvn dqg lw
lv wkh sulflqj ri wkhvh w|shv ri erqgv lq dq h!flhqw pdunhw wkdw zh
zloo frqvlghu khuh1 Jryhuqphqw erqgv fdq eh eurdgo| fdwhjrul}hg dv
ehlqj ri wzr pdlq w|shv1 Wkh uvw w|sh duh erqgv zklfk sd| d uhjxodu
frxsrq/ xvxdoo| vhpl0dqqxdo/ zklfk lv d {hg iudfwlrq ri wkh idfh ydoxh
ri wkh erqg xs wr wkh pdwxulw| gdwh zkhq wkh idfh ydoxh lv dovr uhsdlg1
Wkh rwkhu w|sh duh }hur0frxsrq erqgv/ dovr fdoohg glvfrxqw erqgv zklfk
pdnh rqh sd|phqw dw wkh pdwxulw| gdwh ri wkh erqg1 Erqgv duh rhuhg
zlwk d juhdw ydulhw| ri pdwxulw| gdwhv/ iurp vkruw xs wr xqgdwhg +frq0
vrov lq wkh XN,1 Erqgv sd|lqj frxsrqv fdq eh lqwhusuhwhg dv sdfndjhv
ri }hur0frxsrq erqgv zlwk rqh fruuhvsrqglqj wr hdfk frxsrq sd|phqw
dqg rqh fruuhvsrqglqj wr wkh uhsd|phqw ri wkh sulqflsdo dqg frxsrq dw
wkh pdwxulw| gdwh1 Iru dq lqglylgxdo wudghu/ orqj0uxq erqgv duh vxevwl0
wxwdeoh iru vkruw0uxq erqgv/ vlqfh lw lv srvvleoh iru klp wr krog d vhulhv
ri vkruw erqgv udwkhu wkdq d orqj erqg ryhu wkh vdph kroglqj shulrg/ ru
frqyhuvho| wr krog d orqj erqg iru d vkruw shulrg dqg wkhq vhoo lw udwkhu
wkdq krog d vkruw erqg wr pdwxulw|1
Pruh irupdoo|/ dvvxph lqlwldoo| wkdw wudghuv duh frpsohwho| fhuwdlq ri
wkh ixwxuh1 Lqlwldoo| zh frqvlghu sxuh glvfrxqw erqgv vr wkdw wkh uhwxuq lv
vlpso| wkh glvfrxqw iurp sdu dw zklfk wkh erqg lv vrog dw wkh ehjlqqlqj
ri wkh shulrg uhodwlyh wr wkh uhghhphg sdu sulfh dw wkh hqg ri wkh shulrg1
Wkh sulfh/ Sqw / dw wlph w ri d erqg zklfk pdnhv d sd|phqw ri Â4 dw
wlph w . q lv jlyhq e|
Sqw @
4
+4 . Uqw ,q
+534,
zkhuh Uqw lv wkh erqg*v |lhog wr pdwxulw|1
Lq d zruog ri fhuwdlqw| wkh sxuh h{shfwdwlrqv k|srwkhvlv dvvxphv wkdw
wkh wkh iroorzlqj frqglwlrq pxvw krog=
+4 . Uqw ,q @ +4 . U4w ,+4 . U4w.4 ,+4 . U4w.5 , = = = +4 . U4w.q4 , +535,
Wkh ohiw kdqg vlgh ri +535, lv vlpso| wkh udwh ri uhwxuq rq kroglqj dq
q0shulrg erqg xqwlo pdwxulw|1 Wkh uljkw vlgh ri +535, lv wkh udwh ri uhwxuq
lpsolhg e| kroglqj d rqh0shulrg erqg iru rqh shulrg/ wkhq uhlqyhvwlqj
wkh surfhhgv +sulqflsdo soxv lqwhuhvw, lq d rqh0shulrg erqg iru wkh qh{w
shulrg dqg vr rq1
E| wdnlqj orjdulwkpv dqg uhfdoolqj wkdw iru vpdoo ydoxhv ri wkh iudf0
wlrq ]/ oq+4 . ], fdq eh dssur{lpdwhg e| ]/ zh fdq uhzulwh +535, dv=
Uqw @
4
+U4w . U4w.4 . U4w.5 . = = = . U4w.q4 ,
q
+536,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
734
Lq rwkhu zrugv +536, lqirupv xv wkdw orqj0uxq lqwhuhvw udwhv duh vlpso|
dyhudjhv ri ixwxuh lqwhuhvw udwhv ryhu wkh wlph shulrg wr pdwxulw|1 Lw whoov
xv/ iru lqvwdqfh/ wkdw li vkruw0uxq lqwhuhvw udwhv uhpdlq frqvwdqw iru wkh
lqghqlwh ixwxuh/ wkhq orqj udwhv zloo eh htxdo wr vkruw udwhv1 Frqyhuvho|/
li ixwxuh vkruw udwhv duh h{shfwhg wr idoo/ wkh fxuuhqw orqj udwh zloo eh
ehorz wkh vkruw udwh1 Wkh uhodwlrqvkls lv nqrzq dv wkh h{shfwdwlrqv
wkhru| ri wkh whup vwuxfwxuh ri lqwhuhvw udwhv1
Zkhq zh uhod{ wkh dvvxpswlrq ri shuihfw nqrzohgjh ri wkh ixwxuh
dqg uhfrjql}h wkdw wudghuv fdq revhuyh wkh fxuuhqw orqj udwh Uqw dqg wkh
fxuuhqw rqh0shulrg vkruw udwh U4w / lw iroorzv li wudghuv duh ulvn qhxwudo
wkdw=
4
Uqw @ ^U4w . Hw U4w.4 . Hw U4w.5 . = = = Hw U4w.q4 `
+537,
q
Li wudghuv duh ulvn dyhuvh/ wkhq zh pxvw dgg d ulvn suhplxp wr wkh
uljkw0kdqg vlgh ri +537, exw wklv grhv qrw dhfw wkh dujxphqw/ surylghg
lw lv frqvwdqw1
Li zh frqvlghu d erqg zlwk d wzr0shulrg pdwxulw| gdwh wkhq
U5w @
4
^U4w . Hw U4w.4 `
5
+538,
Zh fdq uhzulwh +538, dv
Hw U4w.4 U4w @ 5+U5w U4w ,
+539,
ru dvvxplqj udwlrqdo h{shfwdwlrqv wkdw
U4w.4 U4w @ 5+U5w U4w , . yw.4
+53:,
zkhuh yw.4 lv wkh h{shfwdwlrqv iruhfdvw huuru1 Dffruglqjo| zh revhuyh
wkdw wkh h{shfwhg fkdqjh lq |lhog rq d rqh0shulrg erqg lv uhodwhg wr
wzlfh wkh glhuhqfh ehwzhhq wkh |lhog rq d wzr0shulrg erqg dqg wkh
|lhog rq d rqh0shulrg erqg dqg wkdw wkh dfwxdo fkdqjh glhuv iurp wklv
whup e| dq h{shfwdwlrqdo huuru1 Wkh h{shfwdwlrqv k|srwkhvlv lpsolhv
whvwdeoh lpsolfdwlrqv vxfk dv wklv iru erqgv ri glhuhqw pdwxulwlhv1 Zh
qrz h{dplqh vrph ri wkhvh lpsolfdwlrqv ixuwkhu1
Zh uvw lqwurgxfh wkh frqfhsw ri wkh kroglqj0shulrg uhwxuq/ Kqw.4 /
zklfk iru vlpsolflw| zh dvvxph lv rqh0shulrg1 Wklv lv wkh uhwxuq iurp
sxufkdvlqj dq q0shulrg erqg dw wlph w dqg vhoolqj lw dw wlph w . 41 Dw
wlph w . 4 wkh erqg zloo ehfrph dq q 4 shulrg erqg dqg zloo eh vrog
iru d sulfh Sq4w.4 1
Iurp +534,
4 . Kqw.4 @
Sq4w.4
+4 . Uqw ,q
@
Sqw
+4 . Uq4w.4 ,q4
+53;,
735
Frqiurqwlqj Prghov zlwk Idfwv
Wkh h{shfwdwlrqv k|srwkhvlv dovr lpsolhv wkdw
U4w @ Hw ^Kqw.4 `
+53<,
Htxdwlrq +53<, whoov xv wkdw wkh |lhog rq d rqh0shulrg erqg vkrxog
htxdo/ ljqrulqj dq| ulvn suhplxp/ wkh h{shfwhg kroglqj0shulrg uhwxuq
rq dq q0shulrg erqg khog iru rqh shulrg1 Wdnlqj orjv ri +53;, dqg
dssur{lpdwlqj dv deryh/ dqg wkhq wdnlqj h{shfwdwlrqv/ zh fdq xvh +53<,
wr holplqdwh wkh h{shfwhg kroglqj0shulrg |lhog1 Pdqlsxodwlrq ri wkh
uhvxowdqw jlyhv=
Hw +Uq4w.4 Uqw , @
+Uqw U4w ,
q4
+543,
Htxdwlrq +543, lqirupv xv wkdw zkhq wkh vsuhdg +wkh glhuhqfh lq
|lhog ehwzhhq wkh orqj0pdwxulw| dqg vkruw0pdwxulw| erqg, lv srvlwlyh/
fkdqjhv lq |lhogv rq orqj0pdwxulw| erqgv duh h{shfwhg wr lqfuhdvh1 Fdps0
ehoo/ Or dqg Pdfnlqod| +4<<:, dqg rwkhuv kdyh whvwhg +543, e| hvwlpdwlqj
wkh htxdwlrq
Uq4w.4 Uqw @ . +Uqw U4w ,
. %w
q4
+544,
zkhuh %w lv wkh iruhfdvw huuru1
Wkh hvwlpdwh ri vkrxog qrw glhu iurp xqlw| zkhq htxdwlrq +544,
lv hvwlpdwhg iru glvfrxqw erqgv ri glhuhqw pdwxulw| gdwhv1 Wkh uhvxowv
duh qrw vxssruwlyh ri wkh h{shfwdwlrqv k|srwkhvlv1 Hvwlpdwhv ri duh
riwhq qhjdwlyh/ sduwlfxoduo| iru erqgv ri orqj pdwxulw| gdwhv1
D ydulhw| ri h{sodqdwlrqv h{lvw iru wklv dqrpdorxv qglqj dqg wkh|
plplf wkh h{sodqdwlrqv iru wkh dqrprorxv hpslulfdo qglqj lq wkh h{0
fkdqjh pdunhw glvfxvvhg deryh zkhuhe| wkh iruzdug suhplxp lv qhjd0
wlyho| uhodwhg wr ixwxuh fkdqjhv lq wkh vsrw udwhv1 Wkhvh duh wlph0ydu|lqj
ulvn suhpld/ shvr sureohpv/ jryhuphqw srolf| uxohv dqg orz vwdwlvwlfdo
srzhu ri wkh uhjuhvvlrqv1
Zh fdq dovr zulwh +537, lq wkh irup
Uqw U4w @
4
^U4w . +Hw U4w.4 U4w . U4w , . +Hw U4w.5 U4w . U4w , .
q
= = = . +Hw U4w.q4 U4w . U4w ,` U4w +545,
ru
Uqw U4w @ Hw
q
[
U4w.l4
l@4
q
+546,
Htxdwlrq +546, lqirupv xv wkdw wkh vsuhdg lv d suhglfwru ri ixwxuh
fkdqjhv lq vkruw uxq lqwhuhvw udwhv1 Wklv lpsolfdwlrq fdq dovr eh hpslu0
lfdoo| whvwhg1 Rqh phwkrg lv wr uhsodfh wkh h{shfwhg whupv rq wkh uljkw
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
736
kdqg vlgh ri +546, e| wkhlu dfwxdo ydoxhv dqg uhjuhvv wkh uhvxowdqw rq wkh
vsuhdg/ hpsor|lqj dq hvwlpdwru wkdw fruuhfwv iru wkh ryhuodsslqj h{shf0
wdwlrqdo huuruv1 Wkh frh!flhqw vkrxog qrw glhu iurp xqlw|1 Fdpsehoo/
Or dqg Pdfnlqod| +4<<:, uhsruw vxfk hvwlpdwhv1 Wkhlu uhvxowv vxjjhvw
d X0vkdshg sdwwhuq lq wkh hvwlpdwhv iru glhuhqw pdwxulwlhv1 Iru vkruw
pdwxulwlhv/ wkh vsuhdg kdv d srvlwlyh frh!flhqw zklfk lv ohvv wkdq xqlw|
dqg ghfolqhv lqlwldoo|/ ehfrplqj lqvljqlfdqw zlwk wkh pdwxulw| krul}rq1
Dw orqjhu krul}rqv d vljqlfdqw srvlwlyh frh!flhqw lv rewdlqhg vrphwlphv
juhdwhu wkdq xqlw|1 Lw wkxv dsshduv wkdw wkh vsuhdg kdv delolw| wr suhglfw
vkruw0uxq lqwhuhvw udwhv fkdqjhv iru erwk uhodwlyho| vkruw dqg orqj krul0
}rqv exw qrw lqwhuphgldwh krul}rqv1 Wkh hpslulfdo uhvxowv fdq djdlq eh
h{sodlqhg e| uhihuhqfh wr wkh dujxphqwv h{sodlqlqj suhylrxv dqrpdorxv
qglqjv deryh1
Zh qrz wxuq wr dqdo|vlv ri frxsrq0sd|lqj erqgv1
Wkh |lhog wr pdwxulw|/ ru orqj0uxq lqwhuhvw udwh \wq / rq dq q0shulrg
erqg lv ghwhuplqhg e| wkh idfw wkdw wkh sulfh Swq ri wkh erqg lv wkh
suhvhqw ydoxh ri d frxsrq +F,/ dvvxphg sdlg dw wkh hqg ri hdfk shulrg/
dqg uhsd|phqw ri wkh sulqflsdo +qrupdol}hg khuh wr xqlw|, dw wkh whuplqdo
gdwh glvfrxqwhg e| \wq 1 Zh kdyh wkdw
Swq @
F
F
F
F
.
.
. === .
q
q
q
5
6
4 . \w
+4 . \w ,
+4 . \w ,
+4 . \wq ,q
4
.
+4 . \wq ,q
Ohwwlqj x @
4
4.\wq
+547,
zh fdq pxowlso| +547, e| x wr rewdlq
xSwq @ Fx5 . Fx6 . Fx7 . = = = . Fxq.4 . xq.4
+548,
Vxewudfwlqj +548, iurp +547, zh rewdlq
Swq +4 x, @ Fx Fxq.4 . xq xq.4
+549,
Glylglqj +549, e| 4x/ qrwlqj wkdw Swq +4x, @ Fx+4xq ,.xq +4x,/
x
@ \4q / uhduudqjlqj diwhu vxevwlwxwlrq edfn iru x zh rewdlq wkdw
dqg 4x
w
Swq @
F
\qF
. q w
q
\w
\w +4 . \wq ,q
+54:,
Li \wq @ F> Swq @ 4/ dqg frqyhuvho|1 Erqgv zlwk wklv fkdudfwhulvwlf/
zkrvh sulfh wrgd| htxdov wkh sulqflsdo sdlg dw pdwxulw| duh vhoolqj cdw
sdu*1
Zh dovr qrwlfh iurp +54:, wkdw iru dq xqgdwhg vhfxulw| ru shushwxlw|/
q @ 4 dqg vr Sw @ \Fw / vr wkdw wkh sulfh lv wkh frxsrq glylghg e| wkh
yhu| orqj0whup lqwhuhvw udwh/ \w 1
737
Frqiurqwlqj Prghov zlwk Idfwv
Wkh rqh0shulrg kroglqj |lhog/ Kwq / rq d q0shulrg frxsrq0sd|lqj erqg
frqvlvwv ri erwk wkh |lhog lq wkh kroglqj shulrg soxv wkh fdslwdo jdlq ru
orvv1 Wklv jlyhv xv
Kwq @
q4
Sw.4
Swq . F
Swq
+54;,
uhfdoolqj djdlq wkdw dq q0shulrg erqg dw wlph w ehfrphv dq q 4 erqg
dw wlph w . 41
Zh qrwh iurp +54;, wkdw iru d shushwxlw|/ q @ 4/
Kw @ \w +\w.4 \w ,
\w.4
+54<,
Htxdwlqj wkh h{shfwhg kroglqj0shulrg |lhog iurp wkh shushwxlw| wr
wkh |lhog rq d vkruw0whup frxsrq0sd|lqj erqg zh kdyh wkdw
\w4 @ \w +\w.4 \w ,
\w.4
+553,
Li zh wdnh d uvw0rughu Wd|oru h{sdqvlrq ri wkh odvw whup rq wkh uljkw0
kdqg vlgh ri +553, durxqg \ zh rewdlq
\w4 @ \w 4 . 4 .
+\w \ , \ +\w.4 \ ,
5
\
\
+554,
Uhduudqjlqj +554, zh rewdlq lq h{shfwdwlrqdo irup
Hw \w.4 \w @ \ +\w \w4 ,
+555,
Zh revhuyh wkdw h{shfwhg fkdqjhv lq wkh |lhog rq wkh shushwxlw| duh
srvlwlyho| uhodwhg wr wkh vsuhdg ehwzhhq wkh |lhog rq wkh shushwxlw| dqg
wkh |lhog rq wkh vkruw pdwxulw| erqg1
Htxdwlrq +555, fdq dovr eh vroyhg iruzdug wr rewdlq wkh vroxwlrq iru
\w dv
"
4
[
\w.l
\
Hw
\w @
4.\
+4 . \ ,l
l@3
+556,
vr wkdw wkh shushwxlw| lv d zhljkwhg dyhudjh ri h{shfwhg ixwxuh vkruw
|lhogv1 Fohduo| +556, lv d olqhdu dssur{lpdwlrq wr d qrqolqhdu vwuxfwxudo
htxdwlrq dqg lwv delolw| wr dssur{lpdwh zloo qdwxudoo| eh ghshqghqw rq
lqwhuhvw udwhv qrw ehlqj wrr yrodwloh vr wkdw wkh dssur{lpdwlrq uhpdlqv
uhdvrqdeo| dffxudwh
Lq wkh pruh jhqhudo fdvh ri frxsrq0sd|lqj erqgv zlwk dq q0shulrg
krul}rq wkh ghulydwlrq fdq surfhhg lq d vlplodu pdqqhu1 Zh vxevwlwxwh
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
738
iru wkh sulfh ri wkh erqg iurp +54:, lqwr wkh kroglqj shulrg |lhog +54;,1
Wklv jlyhv xv wkh udwkhu phvv| uhodwlrq
4 F
\wq.4
F
F . \ q4 . \ q4 +4.\ q4 ,q4
w.4
w.4
w.4
Kwq @
+557,
4
\wq F
F
.
q
q
q
q
\
\ +4.\ ,
w
w
w
q4
@\ @
+Lq Vkloohu/ 4<:</ htxdwlrq +557, lv olqhdul}hg durxqg \wq @ \w.4
F=, Wklv jlyhv wkh uhodwlrqvkls
Kwq @
q4
\wq q \w.4
4 q
+558,
zkhuh q @ i4 . \ ^4 4@+4 . \ ,q4 `4 j4
Htxdwlqj +558, wr d vkruw |lhog/ \w4 +soxv dq| frqvwdqw ulvn suhplxp,
zh fdq vroyh wkh uhvxowlqj glhuhqfh htxdwlrq iru \wq / jlyhq wkh whuplqdo
frqglwlrq wkdw wkh sulfh ri wkh erqg lv 4 dw w . q/ dv
\wq
"
[
4
4
@
Hw
l \w.l
. !q
4 q l@3
+559,
zkhuh !q lv dq| frqvwdqw ulvn ru oltxlglw| suhplxp1 Htxdwlrq +559,
lqirupv xv wkdw wkh q0shulrg |lhog rq d frxsrq0sd|lqj erqg lv d zhljkwhg
dyhudjh ri wkh h{shfwhg |lhogv rq wkh rqh0shulrg erqg1
Vkloohu h{sorlwv wkhvh surshuwlhv wr hydoxdwh hpslulfdoo| zkhwkhu orqj0
uxq erqg |lhogv duh wrr yrodwloh wr eh frqvlvwhqw zlwk udwlrqdo h{shfwd0
wlrqv dqg wkh revhuyhg yrodwlolw| lq vkruw udwhv +d whvw dqdorjrxv wr wkh
yduldqfh erxqgv whvw iru vwrfn sulfhv rxwolqhg deryh,1 Klv hpslulfdo qg0
lqjv duh lqfrqvlvwhqw zlwk udwlrqdo h{shfwdwlrqv1 Djdlq wkh uhvxowv frxog
eh uhfrqflohg e| dsshdo wr wkh vruw ri lvvxhv udlvhg deryh1
OHDUQLQJ DV DQ DOWHUQDWLYH WR UDWLRQDO
H[SHFWDWLRQV=
Wkh whvwv zh kdyh ehhq h{dplqlqj doo dvvxph udwlrqdo h{shfwdwlrqv1 Rqh
srvvleoh fdxvh ri idloxuh frxog wkhuhiruh eh wkdw h{shfwdwlrqv duh qrw ud0
wlrqdo exw udwkhu wkh uhvxow ri d ohduqlqj surfhgxuh ri zklfk dgds0
wlyh h{shfwdwlrqv lv dq dssur{lpdwlrq/ dv vkrzq e| Ehqmdplq Iulhgpdq
+4<:<,1
Zh revhuyhg lq wkh dsshqgl{ wr fkdswhu 5 wkdw li d vhulhv iroorzhg dq
DULPD+3/4/4, surfhvv wkhq dq dgdswlyh h{shfwdwlrqv vfkhph frxog lq
idfw eh d uhsuhvhqwdwlrq ri wkh udwlrqdo h{shfwdwlrq1 Hduolhu lq wklv fkds0
whu zh dovr irxqg wkdw d uhjuhvvlyh h{shfwdwlrqv vfkhph frxog eh udwlrqdo
739
Frqiurqwlqj Prghov zlwk Idfwv
lq wkh Gruqexvfk ryhuvkrrwlqj prgho1 Lq jhqhudo ri frxuvh wkhvh phwk0
rgv ri h{shfwdwlrqv irupdwlrq zloo qrw kdyh wkh udwlrqdo h{shfwdwlrqv
surshuw|1 Wkh| looxvwudwh krzhyhu wkdw lq dq| sduwlfxodu prgho vwuxf0
wxuh/ vlqfh wkh udwlrqdo h{shfwdwlrq vroxwlrq fdq eh zulwwhq lq whupv ri
revhuydeoh yduldeohv/ wkhuh lv d cphfkdqlvwlf* phwkrg ri h{shfwdwlrq iru0
pxodwlrq zklfk zloo kdyh wkh surshuwlhv ri udwlrqdo h{shfwdwlrqv iru wkh
sduwlfxodu prgho vwuxfwxuh1 Lq wklv errn zh duh sulpdulo| frqfhuqhg zlwk
lqyhvwljdwlqj wkh surshuwlhv ri prghov zkhq lw lv dvvxphg wkdw djhqwv*
h{shfwdwlrqv/ ru wkh djjuhjdwh ri djhqwv* h{shfwdwlrqv/ kdyh wkh udwlrqdo
h{shfwdwlrqv surshuw|1 Pdlqwdlqhg dvvxpswlrqv ri wklv dssurdfk duh wkdw
djhqwv nqrz wkh wuxh prgho/ ru dfw dv wkrxjk wkh| glg/ dqg dovr dvvxph
wkdw rwkhu djhqwv dovr srvvhvv wklv lqirupdwlrq +vhh Wrzqvhqg/ 4<:;,1 Lq
wklv dssurdfk hfrqrphwulfldqv zloo frph dfurvv fdvhv ri cdgdswlyh* h{shf0
wdwlrqv zklfk e| fkdqfh uhsuhvhqw wkh udwlrqdo iruhfdvwlqj phfkdqlvp1
Krzhyhu wkhuh lv dq lpsruwdqw olwhudwxuh wkdw uhod{hv wkhvh dvvxps0
wlrqv dqg dvvxphv wkdw h{shfwdwlrqv iroorz d ohduqlqj uxoh/zklfk kdv
wkh srwhqwldo wr frqyhujh wr udwlrqdo h{shfwdwlrqv +vhh h1j1 Hydqv dqg
Krqndsrkmd/ 4<<</ 5334/ dqg Vdujhqw/ 4<<6,1 Iurp wklv shuvshfwlyh wkh
dgdswlyh ru uhjuhvvlyh h{shfwdwlrqv vfkhphv fdq eh lqwhusuhwhg dv vlpsoh
ohduqlqj uxohv zklfk kdyh frqyhujhg rq wkh udwlrqdo h{shfwdwlrqv vrox0
wlrq1 D ydulhw| ri ohduqlqj uxohv/ lq jhqhudo srvvleo| pruh lqwhooljhqw/
kdyh ehhq vwxglhg1 Rqh vxfk h{dpsoh lv ohdvw0vtxduhv ohduqlqj zkhuhe|
djhqwv hpsor| ohdvw0vtxduhv uhjuhvvlrq wr hvwlpdwh wkh sdudphwhuv ri d
prgho dqg hpsor| wkh uhvxowlqj prgho wr iruhfdvw wkh yduldeoh ri lqwhuhvw1
Wklv dssurdfk wr ohduqlqj prghov djhqwv lq wkh vdph zd| dv hfrqrplvwv
zkr hpsor| hfrqrphwulfv dqg vwdwlvwlfdo lqihuhqfh lq ghflglqj ehwzhhq
frpshwlqj prghov1 Hpsor|lqj wklv dssurdfk dovr kljkoljkwv dqrwkhu dv0
shfw ri udwlrqdo h{shfwdwlrqv wkdw djhqwv lq wkh prgho duh dvvxphg wr
srvvhvv pruh lqirupdwlrq wkdq wkh rxwvlgh revhuyhu1
Rqh prwlydwlrq iru vwxg|lqj ohduqlqj uxohv lv wr dvfhuwdlq zkhwkhu
wkh| frqyhujh rq udwlrqdo h{shfwdwlrqv dqg krz idvw wklv surfhvv lv1
Dqvzhuv wr vxfk txhvwlrqv pd| eh uhohydqw iru vrph lq ghflglqj krz
sodxvleoh wkh udwlrqdo h{shfwdwlrqv dvvxpswlrq lv1
Zklovw wkh lpsodxvlelolw| ri wkh dvvxpswlrq wkdw djhqwv nqrz wkh
wuxh prgho pd| eh vx!flhqw mxvwlfdwlrq iru vrph iru vwxg|lqj wkh lp0
solfdwlrqv ri wkh ohduqlqj dvvxpswlrq wkhuh duh rwkhu surshuwlhv ri wkh
udwlrqdo h{shfwdwlrqv dvvxpswlrq wkdw kdyh ehhq xvhg wr mxvwli| vxfk dq
dssurdfk1
Wkh uvw ri wkhvh lv wkh lvvxh ri cqrq0xqltxhqhvv* +dv vhw rxw lq fkdswhu
5,1 Zkhq wkh prgho h{klelwv pruh wkdq rqh vwdwlrqdu| vroxwlrq phwkrgv
wkdw duh qrw sduw ri wkh irupdo prgho vwuxfwxuh kdyh wr eh hpsor|hg wr
ghwhuplqh zklfk vroxwlrq lv fkrvhq e| djhqwv1 Wkh vroxwlrq fkrvhq e|
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
73:
glhuhqw ohduqlqj uxohv kdv ehhq zlgho| vwxglhg1 Krzhyhu/ dv zh dujxhg
lq fkdswhu 5/ cqrq0xqltxhqhvv* vkrxog uhdoo| eh vhhq dv wkh idloxuh ri d
vwdelolw| frqglwlrq iru wkh ciruzdug* urrw1 Wkh xowlpdwh uhphg| lv wr qg
d vshflfdwlrq zklfk vdwlvhv vwdelolw| frqglwlrqv wkdw suhvxpdeo| krog
lq wkh uhdo zruog1
Wkh vhfrqg mxvwlfdwlrq lv zkhq d prgho kdv pruh wkdq rqh htxlole0
ulxp vroxwlrq1 Iru h{dpsoh/ wdnh wkh zlgho| xvhg prgho ri k|shulq dwlrq
zklfk kdv ehhq h{whqvlyho| vwxglhg lq wklv frqwh{w1 Li zh dvvxph d jry0
huqphqw sulqwv prqh| wr qdqfh d frqvwdqw exgjhw ghflw/ wkhq
Sw Jw @ Pw Pw4
+55:,
zkhuh Sw lv wkh sulfh ohyho/ Jw @ J lv wkh frqvwdqw uhdo ghflw/ dqg Pw
lv wkh prqh| vwrfn1
Dvvxph d ghpdqg ixqfwlrq iru prqh| ri wkh irup
Pw
@ i+Hw sw.4 ,
Sw
+55;,
zkhuh Hw sw.4 @ Hw +orj+Sw.4 @Sw , lv wkh h{shfwhg udwh ri lq dwlrq=
w
CP
Sw
CHw sw.4
?3
+55<,
dqg uhdo rxwsxw kdv ehhq dvvxphg frqvwdqw1
Dvvxplqj prqh| ghpdqg lv htxdo wr prqh| vxssolhg zh fdq vxevwl0
wxwh +5, lqwr +4, wr rewdlq
J @ i+Hw sw.4 , i+Hw4 sw ,hsw
+563,
+vlqfh orj SSww 4 @ sw> SSww 4 @ hsw ,
Htxdwlrq +6, kdv wzr htxloleulxp vroxwlrqv li J lv qrw wrr odujh +zkhuh
Hw sw.4 @ Hw4 sw @ sw @ s,1 Lqwxlwlyho| li lq dwlrq lv }hur lq htxloleulxp
wkh dxwkrulwlhv jhqhudwh qr uhfhlswv/ zklovw li lq dwlrq lv lqqlwh djhqwv
krog qr prqh| vr uhfhlswv duh dovr }hur1 Frqvhtxhqwo| wkh htxloleulxp
vxuidfh J @ j+s, h{klelwv wzr htxloleuld iru J ? Jpd{ = Dvvxplqj ud0
wlrqdo h{shfwdwlrqv wkh htxloleulxp h{klelwlqj kljkhu lq dwlrq lv pdwkh0
pdwlfdoo| vwdeoh dqg wkh orzhu rqh xqvwdeoh1 Wkhvh udqnlqjv duh uhyhuvhg
dvvxplqj dgdswlyh h{shfwdwlrqv1 Li wkh ylhz lv wdnhq wkdw pdwkhpdwlfdo
vwdelolw| lv qrw wkh dssursuldwh vhohfwlrq fulwhuld lq d udwlrqdo h{shfwd0
wlrqv prgho wkhq wkhuh lv qr phfkdqlvp/ zklfk lv sduw ri wkh irupdo
prgho vwuxfwxuh/ wr fkrrvh ehwzhhq wkh wzr htxloleuld1 Pruh jhqhudoo| d
udwlrqdo h{shfwdwlrqv prghov frxog h{klelw pxowlsoh vwdeoh htxloleuld1
D wklug mxvwlfdwlrq lv vwuxfwxudo fkdqjh1 Vd| d qhz jryhuqphqw ru
d qhz fhqwudo edqnhu dsshduv/ lw lv qdwxudo wr prgho djhqwv dv ohduqlqj
73;
Frqiurqwlqj Prghov zlwk Idfwv
derxw wkh qhz uhjlph1 Krz czhw* ru cgu|* wkh qhz jryhuqru lv dqg krz
wkh suredelolwlhv fkdqjh dv qhz lqirupdwlrq dffuxhv vhhp ehvw dqdo|}hg
lq d ohduqlqj iudphzrun1
Zh zloo qrz eulh | looxvwudwh vrph h{dpsohv ri ohduqlqj phfkdqlvpv1
Hydqv dqg Krqndsrkmd +4<<<, kdyh glylghg wkh dssurdfkhv lqwr wkuhh
jurxsv dqg zh iroorz wkhlu wd{rqrp|1
Hgxfwlyh dssurdfkhv
Lq wklv olwhudwxuh uhvhdufkhuv lqyhvwljdwh zkhwkhu wkh frruglqdwlrq ri h{0
shfwdwlrqv rq dq udwlrqdo h{shfwdwlrqv htxloleulxp fdq eh dwwdlqhg e|
d phqwdo surfhvv ri uhdvrqlqj1 Zh looxvwudwh zlwk dq h{dpsoh edvhg rq
GhFdqlr+4<:<,1
Vxssrvh wkh ghpdqg dqg vxsso| lq d pdunhw duh jlyhq e|
tw @ d esw . zw
+564,
tw @ f . gHw4 sw . yw
+565,
zkhuh sw lv wkh sulfh ohyho/ zw dqg yw duh udqgrp glvwxuedqfhv dqg d> e> f
dqg g duh frqvwdqwv1
Wkh uhgxfhg irup iru sulfhv lq wklv v|vwhp dvvxplqj ghpdqg lv htxdo
wr vxsso| lv jlyhq e|
sw @
df g
zw yw
@ D EHw4 sw . xw
Hw4 sw .
e
e
e
+566,
zkhuh wkh ghqlwlrqv ri D/E/xw duh reylrxv1
Vxssrvh djhqwv irup wkhlu h{shfwdwlrqv lqlwldoo| lq dq duelwudu| pdq0
qhu1Wkh txhvwlrq lv zkhwkhu wkh| fdq prgli| wkhlu ehkdylrx u lq vxfk
D
d zd| dv wr ohdg wkhp forvhu wr udwlrqdo h{shfwdwlrqv/ jlyhq e| 4.E
1
Vxssrvh wkh lqlwldo/ duelwudu| h{shfwdwlrq ri doo djhqwv lv jlyhq e|
3
sw @ sw4
Hw4
+567,
Iurp +565, jlyhq wklv h{shfwdwlrq wkh dfwxdo hyroxwlrq ri sulfhv zloo
eh jlyhq e|
sw @ D Esw4 . xw
+568,
GhFdqlr dvvxphv wkdw diwhu vrph sdvvdjh ri wlph djhqwv uhdol}h +uhd0
vrq ru ghgxfh, wkdw sulfhv duh hyroylqj dffruglqj wr +567, dqg irup wkh
qhz h{shfwdwlrq
4
Hw4
sw @ D Esw4
+569,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
73<
Krzhyhu wklv qhz h{shfwdwlrq fkdqjhv wkh hyroxwlrq ri wkh v|vwhp wr
sw @ D E+D Esw4 , . xw @ D ED . E 5 sw4 . xw
+56:,
Djhqwv revhuylqj wkh qhz hyroxwlrq ri sulfhv lq wkh pdunhw djhqwv
uhylvh h{shfwdwlrqv wr
5
Hw4
sw @ D ED . E 5 sw4
+56;,
vr wkdw dfwxdo sulfhv hyroyh dv
sw @ D E+D ED . E 5 sw4 , . xw @ D ED . E 5 D E 6 sw4 . xw
+56<,
Frqwlqxlqj lq wklv pdqqhu diwhu q lwhudwlrqv zh zloo kdyh
q
sw @ D ED . E 5 D E 6 D . . DE q . E q sw4
Hw4
+573,
Li mEm ? 4/ iru odujh q h{shfwdwlrqv zloo frqyhujh wr wkh udwlrqdo h{shf0
wdwlrq
q
sw @
Hw4
D
4.E
+574,
4
@ 4 E . E 5 E 6 . . iru mEm ? 4=
vlqfh 4.E
Fohduo| frqyhujhqfh wr wkh udwlrqdo h{shfwdwlrq lv qrw jxdudqwhhg
hyhq lq wkh vlpsoh h{dpsoh li mEm A 4= Zkhq lwhudwlyh h{shfwdwlrqv frq0
yhujh rq wkh udwlrqdo h{shfwdwlrqv vroxwlrq wkh udwlrqdo h{shfwdwlrq lv
vdlg wr eh lwhudwlyho| H0vwdeoh1 Wkh lwhudwlyh h{shfwdwlrqv ri djhqwv zhuh
dvvxphg wr eh krprjhqrxv lq wkh deryh h{dpsoh1 Zkhq frqyhujhqfh
rffxuv dqg wkh lwhudwlyh h{shfwdwlrqv ri djhqwv duh khwhurjhqhrxv dv lq
Jxhvqhulh +4<<5, wkh udwlrqdo h{shfwdwlrqv prgho lv vdlg wr eh vwurqjo|
udwlrqdo1 Hydqv +4<;8/ 4<;9, hpsor| wkh lwhudwlyh h{shfwdwlrqv phwkrg
lq prghov hperg|lqj pxowlsoh vroxwlrqv/ Shho dqg Fkdsshoo +4<;9, lq d
prgho hperg|lqj pxowlsoh htxloleuld dqg Exoodug dqg Plwud +5333, lq d
prgho zkhuh djhqwv ohduq derxw prqhwdu| srolf| uxohv1
Dgdswlyh Dssurdfkhv
Hduo| rq Ehqmdplq Iulhgpdq +4<:<, dujxhg wkdw djhqwv zrxog ohduq
iurp gdwd yld uhjuhvvlrq derxw wkh prgho dqg wkh srolf| uhjlph1 Wklv/
kh srlqwhg rxw/ zrxog surgxfh h{shfwdwlrqv irupdwlrq yhu| vlplodu wr
dgdswlyh h{shfwdwlrqv zlwkrxw qhfhvvdulo| hyhu ohdglqj wr udwlrqdo h{0
shfwdwlrqv1 Vxfk vwdwlvwlfdo ohduqlqj zdv vxevhtxhqwo| h{dplqhg wr vhh
zkhwkhu lw zrxog frqyhujh rq udwlrqdo h{shfwdwlrqv1 Zh frqvlghu wkh
ohdvw0vtxduhv ohduqlqj phfkdqlvp lqlwldoo| dqdo|}hg e| Eud| dqg Vdylq
+4<;9, dqg Irxujhdxg/ Jrxulhurx{ dqg Sudgho +4<;9,/ wkrxjk zh qrwh
743
Frqiurqwlqj Prghov zlwk Idfwv
wkdw pruh frpsolfdwhg hvwlpdwlrq surfhgxuhv vxfk dv qhxudo qhwv dqg
jhqhwlf dojrulwkpv kdyh ehhq hpsor|hg +vhh Vdujhqw/ 4<<6,1
Vxssrvh iru vlpsolflw| wkdw wkh uhgxfhg irup iru sulfhv iroorzv wkh
surfhvv
sw @ D . EHw4 sw . F}w4 . xw
Iurp +574, wkh udwlrqdo h{shfwdwlrq lv jlyhq e| Hw4 sw @
vr wkdw sulfhv hyroyh dv
sw @ D .
+DE . F,
}w4 . xw @ D . J}w4 . xw
4E
+575,
D.F}w4
4E
+576,
Vxssrvh djhqwv eholhyh wkdw sulfhv iroorz wkh surfhvv jlyhq e| +575,
exw duh xqdzduh ri wkh ydoxhv ri wkh sdudphwhuv D dqg J1 Lq wkh ohdvw0
vtxduhv dssurdfk wr ohduqlqj djhqwv duh dvvxphg wr uxq ohdvw0vtxduhv
uhjuhvvlrqv ri sw rq }w4 dqg dq lqwhufhsw xvlqj suhylrxv gdwd rq wkh
yduldeohv1 H{shfwdwlrqv duh wkhq jhqhudwhg iurp wkh hvwlpdwhg prgho1
Dv pruh gdwd ehfrphv dydlodeoh wkh prgho lv wkhq uhhvwlpdwhg/ h{shf0
wdwlrqv iruphg dqg vr rq1 Uhvhdufkhuv kdyh ghprqvwudwhg wkdw wkh frq0
glwlrqv iru frqyhujhqfh ri uhfxuvlyh ohdvw0vtxduhv h{shfwdwlrqv fdq eh
zhdnhu wkdq wkrvh xqghu lwhudwlyh h{shfwdwlrqv1 +E ? 4 lq wklv prgho dv
rssrvhg wr mEm ? 4 zlwk lwhudwlyh h{shfwdwlrqv1,
Lq wkh fdvh ri ohdvw0vtxduhv ohduqlqj zkhuh djhqwv shufhlyh wkh uhgxfhg
irup dv
3
|w @ {w . hw
+577,
3
zkhuh lv d yhfwru ri frh!flhqwv/ {w d yhfwru ri h{sodqdwru| yduldeohv
dqg hw dq huuru/ wkh ohdvw0vtxduhv hvwlpdwhg frh!flhqwv duh jlyhq e|
# w4
$4 # w4
$
[
[
3
{l {l
{l |l
w @
+578,
l@3
l@3
Lw fdq eh ghprqvwudwhg wkdw wkh uhfxuvlyh ohdvw0vtxduhv hvwlpdwhv duh
jhqhudwhg dv
3
w @ w4 . w Uw4 {w4 +|w4 w4 {w4 ,
dqg
3
Uw @ Uw4 . w +{w4 {w4 Uw4 ,
+579,
+57:,
zlwk w @ 4w = dqg zkhuh Uw lv dq hvwlpdwh ri wkh prphqw pdwul{ iru {w =
w4
S
3
{l {l =
Iru vxlwdeoh lqlwldo frqglwlrqv Uw @ w4
l@3
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
744
Zh qrwh wkdw wkh whup w @ 4w lv nqrzq dv wkh cjdlq*1 Lw lv lpsruwdqw
lq ghwhuplqlqj wkh vshhg ri frqyhujhqfh wr wkh wuxh sdudphwhu1 Vrph
lqwxlwlrq ri wkh ohdvw0vtxduhv xsgdwlqj irupxodh fdq eh rewdlqhg e| frq0
vlghulqj wkh uhfxuvlyh ohdvw0vtxduhv hvwlpdwh ri wkh phdq H}w @ = Wkh
w
S
}q =
ohdvw0vtxduhv hvwlpdwh lv wkh vdpsoh phdq } w @ 4w
q@4
Vxewudfwlqj wkh vdpsoh phdq dw w 4 iurp erwk vlghv ri } w dqg
uhduudqjlqj jlyhv
4
} w @ } w4 . +}w } w4 ,
w
vlqfh w} w @
w
S
}q @ }w .
q@4
w4
S
} w4 , @ }w .
q@4
w4
S
q@4
+57;,
}q dqg +w 4,} w4 @
w4
S
q@4
}q vr wkdw w+} w }q w} w4
@ }w . +w 4,} w4 w} w4 @ }w } w4 =
Dgdswlyh phwkrgv ri ohduqlqj kdyh wkh vdph jhqhudo w|sh ri vwuxfwxuh
zklfk lv jlyhq e|
w @ w4 . w T+w> w4 > [w ,
+57<,
zkhuh w lv d yhfwru ri sdudphwhuv/ w lv wkh jdlq sdudphwhu htxdo wr 4w lq
wkh fdvh ri ohdvw0vtxduhv/ T lv d ixqfwlrq dqg [w lv wkh yhfwru ri yduldeohv
lq wkh vwuxfwxudo prgho1 Zh qrwh wkdw dgdswlyh h{shfwdwlrqv lv d vshfldo
fdvh ri +57;, zkhuh wkh jdlq sdudphwhu lv frqvwdqw11
Wkh hyroxwlrq ri [w zloo ghshqg rq w4 > lq wkh fdvh ri d olqhdu v|vwhp
[w @ D+w4 ,[w4 . E+w4 ,Zw
+583,
zkhuh Zw lv d yhfwru ri glvwxuedqfh whupv1
Pdufhw dqg Vdujhqw +4<;<d/e,/ Hydqv dqg Krqndsrkmd +4<<;, ghulyh
vwdelolw| uhvxowv iru olqhdu +dqg qrqolqhdu, v|vwhpv1 Vdujhqw +4<<<, dv0
vxphv wkh XV dxwkrulwlhv xvhg frqvwdqw0jdlq ohdvw0vtxduhv ohduqlqj derxw
wkh Skloolsv Fxuyh dqg pd{lplvhg d vrfldo remhfwlyh ixqfwlrq wr slfn lq0
dwlrq> kh dujxhv wklv wv XV srvw0zdu gdwd/ dffrxqwlqj iru wkh cjuhdw
lq dwlrq* zkhuh udwlrqdo h{shfwdwlrqv fdqqrw1
Udwlrqdo Ohduqlqj
Udwlrqdo ohduqlqj kdv wr eh lqwhusuhwhg iurp d shuvshfwlyh wkdw dfnqrzo0
hgjhv wkh ehqhwv dqg frvwv ri pruh dffxudwh iruhfdvwv iru dq djhqw dv
lq Ihljh dqg Slhufh +4<:9, ru Hydqv dqg Udpvh| +4<<5,/ vr wkdw udwlr0
qdo h{shfwdwlrqv pd| qrw eh dwwdlqhg xqohvv fdofxodwlrq frvwv duh }hur1
Krzhyhu wkh phwkrg prvw zlgho| hpsor|hg wr prgho udwlrqdo ohduqlqj
745
Frqiurqwlqj Prghov zlwk Idfwv
kdv ehhq wr hpsor| Ed|hv* uxoh1 Ed|hv* uxoh lv wkh edvlf surshuw| ri frq0
glwlrqdo suredelolw| dqg lv d phwkrg ri xsgdwlqj rxu eholhi ru suredelolw|
ri hyhqw ru k|srwkhvlv D jlyhq qhz hylghqfh E1 Vshflfdoo|/ rxu srvwh0
ulru eholhi S +D@E, lv fdofxodwhg e| pxowlso|lqj rxu sulru eholhi S +D, e|
wkh olnholkrrg S +E@D, wkdw E zloo rffxu li D lv wuxh1 Wr vhh wklv zh fdq
uhduudqjh wkh frqglwlrqdo suredelolw| irupxod wr jhw=
S +D@E,S +E, @ S +D> E,
+584,
zkhuh S +D> E, lv wkh mrlqw suredelolw| ri D dqg E=
E| v|pphwu| zh dovr kdyh
S +E@D,S +D, @ S +D> E,
+585,
Lw iroorzv iurp +583, dqg +584, wkdw 1
S +D@E, @
S +E@D,=S +D,
S +E,
+586,
Htxdwlrq +585, lv fdoohg Ed|hv* uxoh ru Ed|hv* wkhruhp1 Wkh irupxodwlrq
fduulhv wkh lpsolfdwlrq wkdw eholhiv fkdqjh e| ohduqlqj= djhqwv frph wr
nqrz ri d qhz idfw dqg irup wkhlu srvwhulru eholhi e| frqglwlrqlqj wkhlu
sulru eholhi rq wkhvh idfwv1
Dq dowhuqdwlyh irup ri Ed|hv* uxoh lv jlyhq e|
S +E@D,=S +D,
S +D@E, @ S
S +E@Dl ,=S +Dl ,
+587,
l
S
vlqfh S +E, @ S +E@Dl ,=S +Dl , zkhuh Dl uhihuv wr wkh hyhqw vsdfh1
l
Iurp +585, zh revhuyh wkdw wkh gdwd ru qhz idfwv E rqo| lq xhqfh wkh
srvwhulru lqihuhqfh wkurxjk wkh ixqfwlrq ru suredelolw| S +E@D, zklfk lv
fdoohg wkh olnholkrrg ixqfwlrq1
Wkh udwlr ri wkh srvwhulru suredelolw| hydoxdwhg dw wkh wzr hyhqwv D4
dqg D5 lv fdoohg wkh srvwhulru rggv1
Zh kdyh
S +D4 @E,
S +D4 ,S +E@D4 ,@S +E,
S +D4 ,S +E@D4 ,
@
@
S +D5 @E,
S +D5 ,S +E@D5 ,@S +E,
S +D5 ,S +E@D5 ,
+588,
Lq zrugv/ wkh srvwhulru rggv duh htxdo wr wkh sulru rggv pxowlsolhg
e| wkh olnholkrrg udwlr1
Ed|hv* uxoh kdv ehhq zlgho| hpsor|hg wr prgho ohduqlqj lq wkh hfr0
qrplfv olwhudwxuh vhh h1j1 F|huw dqg GhJurrw +4<:7,/ Edfnxv dqg Guli0
oo +4<;8,/ Hoolvrq dqg Ydood +5333,/ Ohzlv +4<<;, Vloo dqg Zudvh +4<<<,/
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
746
Wrzqvhqg +4<:;,1 W|shv ri sureohp dqdo|}hg hpsor|lqj Ed|hv*uxoh lq0
foxgh ohduqlqj derxw d qhz uhjlph1 Zh zloo jlyh rqh h{dpsoh zklfk lv d
voljkw prglfdwlrq ri Ohzlv +4<;;,1 Vkh ghprqvwudwhv krz eholhiv wkdw
d srolf| surfhvv pd| kdyh vzlwfkhg fdq lqgxfh dssduhqw h{0srvw eldvhg
iruhfdvwv ri h{fkdqjh udwhv hyhq diwhu wkh vzlwfk kdv rffxuuhg1 Lq dggl0
wlrq/ lq khu prgho/ h{fkdqjh udwhv pd| dsshdu wr frqwdlq d vshfxodwlyh
exeeoh frpsrqhqw vlqfh wkh| zloo v|vwhpdwlfdoo| ghyldwh iurp wkh ohyhov
lpsolhg e| revhuylqj ixqgdphqwdov h{0srvw1
Dvvxph wkh uhgxfhg irup iru wkh h{fkdqjh udwh +vhh fkdswhu 47, lv
jlyhq e|
vw @ pw . +Hw vw.4 vw ,
+589,
zkhuh vw lv wkh prqh| vxsso| dw wlph w/ vw lv wkh h{fkdqjh udwh dqg lv
d srvlwlyh frqvwdqw1
Dvvxph wkh prqh| vxsso| surfhvv lv jlyhq e|
pw @ 3 . %3w
+58:,
%3w
zkhuh 3 lv d frqvwdqw dqg
lv d qrupdoo| glvwulexwhg udqgrp yduldeoh
zlwk phdq }hur dqg yduldqfh 53 =
Vxssrvh dw d sduwlfxodu srlqw lq wlph/ vd| w @ 3/ djhqwv frph wr
eholhyh wkdw wkh prqh| vxsso| surfhvv pd| kdyh fkdqjhg/ gxh wr dq
h{rjhqrxv surfhvv vxfk dv d fkdqjh lq jryhuqphqw ru d vwdwhphqw e|
r!fldov1 Wkh qhz surfhvv kdv iru vlpsolflw| wkh vdph irup dv wkh rog
h{fhsw zlwk d glhuhqw phdq dqg yduldqfh=
pw @ 4 . %4w iru w 3
+58;,
Lw lv dvvxphg wkdw 4 ? 3 dqg 4 @ 3> vr wkdw wkh surfhvv fdq eh
lqwhusuhwhg dv jrlqj iurp corrvh* wr cwljkw* prqh|1 Djhqwv duh qrw vxuh
zklfk prqh| vxsso| surfhvv lv lq rshudwlrq1 Lw lv ixuwkhu dvvxphg iru
vlpsolflw| wkdw djhqwv eholhyh wkdw li srolf| kdv fkdqjhg lw zloo qrw eh
fkdqjhg edfn dqg wkh| dovr nqrz wkh sdudphwhuv ri wkh srwhqwldo qhz
surfhvv1
Vroylqj +5<, iruzdug zh rewdlq wkh vroxwlrq
vw @ +4 ,
"
[
l Hw pw.l
+58<,
l@3
4
=
zkhuh @ 4.
H{shfwhg prqh| vxsso| jlyhq wkh dvvxpswlrqv deryh lv htxdo wr
Hw pw.l @ 3 +4 S4w , iru dq| l A 3> w 3
+593,
zkhuh S4w lv djhqwv* dvvhvvhg suredelolw| dw wlph w wkdw wkh surfhvv
fkdqjhg dw wlph 31
747
Frqiurqwlqj Prghov zlwk Idfwv
Frqvhtxhqwo| wkh h{fkdqjh udwh lv jlyhq e|
vw @ +4 ,pw . +4 S4w ,3
+594,
4
@ 4 . 4 . 5 .
+qrwh Hw vw.4 @ 3 +4 S4w , iurp +63, dqg +64, vlqfh 4
4
@ 4 ,
. " dqg 4.
Wr rewdlq wkh ehvw hvwlpdwh ri S4w > djhqwv frpelqh wkhlu sulru eholhiv
derxw wkh suredelolw| wrjhwkhu zlwk wkhlu revhuydwlrqv ri prqh| rxw0
frphv hdfk shulrg wr xsgdwh wkhlu srvwhulru suredelolwlhv dffruglqj wr
Ed|hv* Uxoh1
S4w @
S4w4 i+Lw @4 ,
S4w4 i+Lw @4 , . S3w4 i+Lw @3 ,
+595,
zkhuh S3w lv wkh suredelolw| ri qr fkdqjh dw w @ 3/ i+Lw @l , lv wkh sure0
delolw| ri revhuylqj wkh lqirupdwlrq vhw Lw jlyhq wkdw pw iroorzv wkh lwk
surfhvv1 Wkh srvwhulru suredelolwlhv ri hdfk surfhvv/ wkh srvwhulru rggv
udwlr/ lv jlyhq e|
5
6
5
+ p5 ,
4
54
S4w4 9 + 4 ,h
S4w4 i+pw @4 ,
S4w
:
@
@
+596,
7
+p3 ,5 8
S3w
S3w4 i+pw @3 ,
S3w4
+
,
4
55
3
+ 3 ,h
Wkh uvw whup rq wkh uljkw0kdqg vlgh ri +595, vkrzv wkdw wkh fkdqjh
iurp w 4 wr w lq wkh uhodwlyh frqglwlrqdo suredelolwlhv ghshqgv xsrq
wkh revhuydwlrq ri wkh fxuuhqw prqh| vxsso| dw wlph w1 Iru h{dpsoh iru
vrph revhuydwlrq ri fxuuhqw prqh| vxsso|/ vd| p> wkh suredelolw| ri eh0
4w
lqj xqghu hlwkhu uhjlph lv htxdo/ vr wkdw wkh srvwhulru suredelolwlhv/ S
S3w >
4w4
duh htxdo wr wkh sulru suredelolwlhv S
S3w4 = Revhuydwlrqv ri prqh| vxs0
so| glhuhqw iurp p frqyh| lqirupdwlrq derxw wkh uhjlphv fdxvlqj wkh
suredelolwlhv wr eh uhylvhg1 Wkh odvw whup rq wkh uljkw0kdqg vlgh ri
+595, txdqwlhv wklv lqirupdwlrq1 Iru d qrupdoo| glvwulexwhg huuru wkh
vdpsolqj glvwulexwlrq iru d vlqjoh vfdodu revhuydwlrq/ p> iurp d qrupdo
glvwulexwlrq sdudphwhul}hg e| d phdq dqg d yduldqfh 5 lv jlyhq e|
,5
+p
h 55
S +p@, @ s
5
+597,
Fohduo| wkh lqirupdwlrq dffuxlqj iurp dq revhuydwlrq zloo ghshqg rq
wkh sduwlfxodu glvwulexwlrq ri wkh huuru whup1
Li iru vlpsolflw| zh dvvxph wkdw 3 @ 4 @ wkhq zh fdq zulwh +69,
dv
oq+
S4w
S4w4
i+pw @4 ,
, @ oq
. oq
S3w
S3w4
i+pw @3 ,
+598,
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
vr wkdw iru dq| uhdol}dwlrq ri wkh prqh| surfhvv pn
+pn 3 ,5 p5n
i+pn @4 ,
oq
@
i+pn @3 ,
55
748
+599,
Vxevwlwxwlqj +598, lqwr +599, zh rewdlq d olqhdu glhuhqfh htxdwlrq
wkdw fdq eh vroyhg dv
w S4w
S4>3 [ +pn 3 ,5 p5n
oq+
, @ oq
.
+59:,
S3w
S3>3
55
n@4
Lq htxdwlrq +599, wkh hyroxwlrq ri suredelolwlhv lv vhhq wr ghshqg rq
wkh dfwxdo revhuydwlrqv ri wkh surfhvv1 Zkhq wkh prqh| vxsso| revhuyhg
wrgd| lv yhu| qhjdwlyh djhqwv zloo wklqn lw pruh olnho| wkdw srolf| kdv
fkdqjhg dqg ylfh yhuvd1
Wdnlqj h{shfwdwlrqv ri +599,/ ghqlqj l dv wkh wuxh jlyhv
S4w
+l 3 ,5 5l
S4>3
, @ oq
.w
H oq+
S3w
S3>3
55
5
S4>3
5l 3
@ oq
.w 3
S3>3
55
+59;,
+59<,
Htxdwlrq +59;, looxvwudwhv wkdw wkh h{shfwhg ydoxh ri wkh cwuxh* surfhvv
ulvhv ryhu wlph1 Iru lqvwdqfh zkhq srolf| kdv fkdqjhg vr l @ 4 @
3> iurp +59;, zh revhuyh wkdw wkh orj suredelolw| lqfuhdvhv wr lqqlw|
w 5
ehfdxvh ri wkh wuhqg whup 535 +ru wkdw S3w jrhv wr }hur,1 Zkhq srolf|
kdv qrw fkdqjhg vr wkdw l @ r wkh uhyhuvh lv wuxh gxh wr wkh wuhqg
w5
whup 553 = Frqvhtxhqwo| djhqwv* dvvhvvhg suredelolwlhv ri d srolf| fkdqjh
duh udqgrp yduldeohv ghwhuplqhg e| udqgrp revhuydwlrqv ri wkh prqh|
vxsso|1
Jlyhq wklv dqdo|vlv ri wkh hyroxwlrq ri suredelolwlhv Ohzlv lv deoh wr
lqyhvwljdwh wkh hhfwv rq wkh h{fkdqjh udwh dqg iruhfdvw huuruv1Wdnlqj
h{shfwdwlrqv dw w 4 ri +593, dqg vxewudfwlqj iurp +593, zh rewdlq wkh
iruhfdvw huuruv fruuhvsrqglqj wr hdfk srwhqwldo surfhvv dv=
vw Hw4 vw @ +4 ,%3w . 3 +S4>w4 S4w , li l @ 3
+5:3,
vw Hw4 vw @ +4 ,%4w 3 +S3>w4 S3w , li l @ 4
+5:4,
dqg
+qrwh Hw4 S4w @ S4>w4 > Hw4 S3w @ S3>w4 ,1
Htxdwlrqv +59<, dqg +5:3, looxvwudwh wkdw zklovw djhqwv duh ohduqlqj/
h{shfwdwlrq huuruv zloo h{klelw d v|vwhpdwlf frpsrqhqw dqg cdsshdu qrq0
udwlrqdo h{0srvw*1
749
Frqiurqwlqj Prghov zlwk Idfwv
Wdnlqj h{shfwdwlrqv ri wkh iruhfdvw huuruv lq +5:3, frqglwlrqdo xsrq
d fkdqjh lq srolf| wr 4 wkh h{shfwhg hyroxwlrq ri wkh iruhfdvw huuruv/ iru
d odujh qxpehu ri pn lv jlyhq e|
H+vw Hw4 vw @4 , @ 3 ^H+S3>w4 @4 , H+S3w @4 ,` ? 3
+5:5,
Wkh lqhtxdolw| lv qhjdwlyh vlqfh lv ohvv wkdq rqh dqg HS3w @4 ?
H+S3>w4 @4 ,=
Iurp +5:4, zh revhuyh wkdw li djhqwv gr qrw ixoo| uhdol}h wkdw srolf|
kdv fkdqjhg wkh h{fkdqjh zloo eh h{shfwhg wr eh zhdnhu wkdq vxevh0
txhqwo| rffxuv1 H{0srvw h{shfwdwlrqv zloo dsshdu wr eh luudwlrqdo1
Ohzlv* prgho qlfho| looxvwudwhv krz ohduqlqj derxw d uhjlph fkdqjh
xvlqj Ed|hv* uxoh fdq plplf wkh rxwfrphv ri wkh Shvr sureohp glvfxvvhg
hduolhu lq wklv fkdswhu1
FRQFOXVLRQV
Lq wklv fkdswhu zh kdyh glvfxvvhg vrph ri wkh lpsolfdwlrqv ri wkh sursr0
vlwlrq wkdw qdqfldo pdunhwv duh h!flhqw1 Dv sduwlfxodu dssolfdwlrqv ri
wkh sursrvlwlrq zh h{dplqhg wkh ehkdylrxu ri vwrfnv/ h{fkdqjh udwhv
dqg erqg pdunhwv1 Wkh wkhruhwlfdo zrun rq h!flhqf| lq dvvhw pdunhwv
lv odujh dqg lv h{sdqglqj gudpdwlfdoo| vr wkdw zh kdyh rqo| ehhq deoh
wr surylgh d dyrxu ri wkh ghedwh1 Zklovw wkh qhz jhqhudwlrq ri prg0
hov zklfk vwuhvv plfur0vwuxfwxuh dujxphqwv dsshdu ri juhdw lqwhuhvw/ rxu
rzq ylhz lv wkdw wkh hpslulfdo hylghqfh iru +dssur{lpdwh, vhpl0vwurqj
h!flhqf| lq wkh fdslwdo pdunhw lv vx!flhqwo| srzhuixo dqg frqylqflqj iru
lw wr eh uhjdughg dv d cvw|ol}hg idfw*1 Lq idfw lw lv qrz vwdqgdug sudfwlfh
iru pdfurhfrqrplf prgho exloghuv wr vlpxodwh ru iruhfdvw wkh lpsdfw ri
fkdqjhv lq jryhuqphqw srolf| zlwklq prghov wkdw dvvxph fdslwdo pdunhw
h!flhqf|1
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
74:
DSSHQGL[ 47D4 LQFRPSOHWH FXUUHQW LQIRU0
PDWLRQ
Wkh sxusrvh ri wklv dsshqgl{ lv wr vkrz wkh pdqqhu lq zklfk wkh vwdq0
gdug whvw ri h!flhqf| edvhg rq htxdwlrq +;4, ri wklv fkdswhu kdv wr eh
prglhg li djhqwv kdyh dq lqfrpsohwh fxuuhqw lqirupdwlrq vhw1 Iru dqd0
o|wlfdo vlpsolflw| +wkrxjk wkh dujxphqw lv txlwh jhqhudo,/ zh dvvxph wkh
vhulhv |w lv wkh vxppdwlrq ri wzr lqqlwh prylqj dyhudjh huuru surfhvvhv
lq wkh wzr zklwh qrlvh huuruv/ dqg }=
|w @ | .
"
[
l wl .
l@3
"
[
l }wl
+4,
l@3
=zkhuh | lv wkh phdq ri wkh vhulhv/ dqg wkh l dqg l duh frqvwdqw frh!0
flhqwv1
Frqvlghu wkh udwlrqdo h{shfwdwlrq ri |w.4 iruphg dw wlph w1 Li wkhuh
lv ixoo fxuuhqw lqirupdwlrq dw wlph w wkh h{shfwdwlrq zloo eh jlyhq e|=
"
"
[
[
l wl.4 .
l }w4.l
Hw |w.4 @ | .
+5,
l@3
l@3
Lq wklv fdvh/ jlyhq wkh rqh0shulrg iruhfdvw krul}rq/ wkh h{0srvw iruh0
fdvw huuru zloo eh jlyhq e| d zklwh qrlvh huuru=
|w.4 Hw |w.4 @ 3 w.4 . 3 }w.4
+6,
Frqvhtxhqwo| wkh vwdqgdug whvwv edvhg rq +;4, duh fruuhfw lq wkhvh
flufxpvwdqfhv1
Vxssrvh qh{w wkdw djhqwv kdyh lqfrpsohwh fxuuhqw lqirupdwlrq dw
wlph w/ dqg lqvwhdg revhuyh vrph fxuuhqw joredo lqirupdwlrq +iru lqvwdqfh
yld dvvhw pdunhwv,/ exw rwkhu joredo lqirupdwlrq zlwk d rqh0shulrg odj1 Lq
sduwlfxodu zh zloo dvvxph iru vlpsolflw| +wkrxjk wkh dujxphqw lv hdvlo|
jhqhudol}hg, wkdw wkhuh lv rqh joredo lqglfdwru +vd| wkh lqwhuhvw udwh,
zklfk lv jlyhq wkh uhsuhvhqwdwlrq=
"
[
Uw @ u .
+gl wl . kl }wl ,
+7,
l@3
zkhuh u lv wkh phdq ri wkh vhulhv dqg wkh gl dqg kl duh frqvwdqw frh!0
flhqwv1
Lq wklv lqfrpsohwh fxuuhqw lqirupdwlrq fdvh wkh rqh0shulrg0dkhdg h{0
shfwdwlrq ri |w lv jlyhq e|=
"
"
[
[
Hw |w.4 @ | . 4 Hw w .
l wl.4 . 4 Hw }w .
l }wl.4 +8,
l@5
l@5
74;
Frqiurqwlqj Prghov zlwk Idfwv
Frqvhtxhqwo| wkh iruhfdvw huuru lv jlyhq e|=
|w.4 Hw |w.4 @ 3 w.4 . 3 }w.4 . 4 ^w Hw w ` . 4 ^}w Hw }w `
+9,
Jlyhq fxuuhqw revhuydwlrq ri wkh joredo lqglfdwru Uw> dqg xvlqj wkh
xvxdo vljqdo h{wudfwlrq irupxodh +dv glvfxvvhg lq fkdswhu 6, zh rewdlq=
4
! +g3 w . k3 }w ,
gr +:,
4
+4 ! ,+g3 w . k3 }w ,
k3
+;,
Hw w @
Hw }w @
zkhuh
! @
g5r 5
g5r 5 . k5r 5}
dqg 5 dqg 5} duh wkh yduldqfhv ri wkh wzr huuruv/ dqg } uhvshfwlyho|1
Frqvhtxhqwo| wkh iruhfdvw huuru +Nw.4 , lv jlyhq e|=
kr
Nw.4 @ 3 w.4 . 3 }w.4 . 4 +4 ! ,w ! }w .
gr
g3
4 ! }w +4 ! ,w
+<,
kr
Vhuldo Fruuhodwlrq ri Iruhfdvw Huuruv
Li zh wdnh h{shfwdwlrqv ri wzr vxffhvvlyh huuruv zh qg=
5} 5
^4 k3 g4 g3 `=^3 k3 3 g3 `
H+Nw.4 > Nw , @
g5r 5 . k53 5}
+43,
Frqvhtxhqwo|/ lq jhqhudo/ lqfrpsohwh fxuuhqw lqirupdwlrq zloo jlyh
ulvh wr d prylqj0dyhudjh huuru surfhvv1 Wklv zloo qrw eh wkh fdvh li/
uvw/ zh kdyh lpsolflwo| ixoo fxuuhqw lqirupdwlrq +iru h{dpsoh/ li wkhuh
duh dv pdq| joredo lqglfdwruv lq wkh hfrqrp| dv udqgrp vkrfnv> vhh
Nduql/ 4<;3,/ ru li/ vhfrqgo|/ zh revhuyh wkh fxuuhqw ydoxh ri wkh yduldeoh
wr eh iruhfdvw +lw ehlqj lwvhoi d joredo lqglfdwru,1 Lq wklv odwwhu fdvh
3 @ g3 > 3 @ k3 +dovr 4 @ g4 / 4 @k4 , dqg wkh h{shfwhg fruuhodwlrq
lq +43, lv htxdo wr }hur1
Lw zrxog dsshdu iurp wklv uhvxow wkdw vwdqgdug whvwv ri h!flhqf| edvhg
rq +;4, zloo kdyh wkh xvxdo surshuwlhv iru dvvhw sulfhv lq sduwlfxodu +zklfk
lw fdq eh dvvxphg duh revhuyhg fxuuhqwo|,/ hyhq xqghu lqfrpsohwh fxu0
uhqw lqirupdwlrq1 Krzhyhu/ wkh jhqhudo srlqw uhpdlqv wkdw yduldeohv qrw
fxuuhqwo| revhuyhg +l1h1 wkh pdmrulw|, zloo xqghu lqfrpsohwh lqirupdwlrq
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
74<
eh lqdssursuldwho| whvwhg iru h!flhqf| e| wkhvh phwkrgv1 Ixuwkhupruh/
rqh qhhgv wr vfuxwlql}h fduhixoo| wkh dvvxpswlrq wkdw wkh dvvhw sulfhv lq
txhvwlrq duh frqwhpsrudqhrxvo| revhuyhg1 Lq yhu| kljk iuhtxhqf| gdwd
+h1j1 krxuo|, wklv zloo reylrxvo| qrw eh vr h{fhsw iru d ihz frqwlqxrxvo|0
eurdgfdvw dvvhw sulfhv> lw zloo dovr qrw eh vr lq orzhu iuhtxhqf| gdwd
iru dyhudjhv ri yduldeohv +h1j1 wkh ohyho ri doo vkruw0whup lqwhuhvw udwhv,/
zklfk duh riwhq h{dplqhg lq wkhvh vwxglhv1
Lq jhqhudo/ lq flufxpvwdqfhv ri lqfrpsohwh fxuuhqw lqirupdwlrq wkh
prylqj0dyhudjh huuru lq htxdwlrq +<, zloo eh jlyhq e| v . m 4/ zkhuh
v lv wkh wlph krul}rq ri wkh iruhfdvw dqg m lv wkh orqjhvw odj rq joredo
lqirupdwlrq uhohydqw iru iruhfdvwlqj |w 1 Fohduo| wkhuh pd| eh vrph d
sulrul grxew dv wr wkh pdjqlwxgh ri m/ zklfk pd| fdxvh vrph sureohpv
lq lqwhusuhwdwlrq ri whvwv edvhg rq +;4,1 Dv d frqvhtxhqfh ri wkh prylqj0
dyhudjh huuru surfhvv/ ohdvw vtxduhv hvwlpdwhv ri +;4, xqghu lqfrpsohwh
fxuuhqw lqirupdwlrq zloo eh lqh!flhqw exw xqeldvhg/ vlqfh
H+Nw.4 > H+|w.4 ,, @ 3
+44,
dqg ohdvw vtxduhv hvwlpdwhv kdyh wkh surshuw| ri xqeldvhgqhvv hyhq lq
wkh suhvhqfh ri prylqj0dyhudjh huuru surfhvvhv1 Wklv vlwxdwlrq lv wkh
vdph dv wkdw ri ryhuodsslqj lqirupdwlrq lq wkh xvxdoo| dvvxphg fdvh ri
ixoo fxuuhqw lqirupdwlrq> ryhuodsslqj lqirupdwlrq khuh rffxuv zlwk v A 4/
idplolduo| lqwurgxflqj d prylqj0dyhudjh surfhvv zlwk wkh vdph hhfwv1
Wkhvh uhvxowv kdyh srwhqwldo lpsolfdwlrqv iru d qxpehu ri hpslul0
fdo vwxglhv +vhh h1j1 Kroghq dqg Shho/ 4<::> Wxuqryvn|/ 4<:3, lq zklfk
dq lpsolflw dvvxpswlrq ri ixoo fxuuhqw lqirupdwlrq kdv ehhq pdgh zkhq
vwxg|lqj gluhfwo|0revhuyhg frqvxphu sulfh h{shfwdwlrqv gdwd zklfk fdq0
qrw uhdglo| eh dvvxphg wr eh sduw ri wkh fxuuhqw lqirupdwlrq vhw1 Wkh
srlqw khuh lv wkdw sulfh gdwd duh qrw fxuuhqwo| revhuydeoh rq dq| uhd0
vrqdeoh dvvxpswlrqv1 Frqvhtxhqwo|/ wkh ch{shfwdwlrqv huuruv* vkrxog eh
vhuldoo| fruuhodwhg/ dv lqghhg kdv riwhq ehhq irxqg lq wkhvh whvwv1 Lw lv
srvvleoh wkdw wkhvh vxuyh| gdwd pd| zhoo uhyhdo udwlrqdolw| diwhu doo1 Iru
ixuwkhu lpsolfdwlrqv ri sduwldo fxuuhqw lqirupdwlrq vhwv iru wkh whvwlqj ri
h!flhqf|/ vhh Plqirug dqg Shho +4<;7,1
753
Frqiurqwlqj Prghov zlwk Idfwv
DSSHQGL[ 47D5 FRPSRVLWH PRYLQJ0DYHUDJH
HUURU SURFHVVHV
Whvwlqj iru wkh h!flhqf| ri uhodwlrqvklsv lqyroylqj prylqj0dyhudjh huuruv
srvhv sureohpv1 Zh zlvk wr irup d prylqj0dyhudjh huuru surfhvv iurp
wkh frpsrvlwh huuru surfhvv
dyw.4 . eyw . fxw.4 . gxw
+4,
zkhuh yw.l dqg xw.l +l @ 4> 3, duh vhuldoo| xqfruuhodwhg udqgrp yduldeohv1
Zh ghqh wkh qhz prylqj0dyhudjh surfhvv
!w.4 . m!w @ dyw.4 . eyw . fxw.4 . gxw
+5,
zkhuh !w.l +l @ 4> 3, lv d vhuldoo| xqfruuhodwhg udqgrp yduldeoh1
Wkh phwkrg lv wr htxdwh wkh udwlr ri wkh yduldqfh wr wkh fryduldqfh
ri wkh huuru surfhvvhv rq wkh ohiw0 dqg uljkw0kdqg vlghv ri +5, +zkhuh wkh
yduldqfhv dqg fryduldqfhv ri wkh ohiw0 dqg uljkw0kdqg vlghv duh htxdo e|
ghqlwlrq d vlplodu phwkrg lv hpsor|hg iru kljkhu0rughu frpsrvlwh
surfhvvhv,1
Zh rewdlq wkdw
+4 . m 5 ,5!
+d5 . e5 ,5y . +f5 . g5 ,5x . 5+eg . df,fry+xy,
@
m5!
de5y . fg5x . +dg . ef,fry+xy,
+6,
zkhuh 5! > 5y > 5x duh wkh yduldqfhv ri !> y> x dqg fry+xy, lv wkh fryduldqfh
ehwzhhq x dqg y=
Htxdwlrq +6, lv d txdgudwlf htxdwlrq lq m +qrwh 5! fdqfhov,= Wkh
urrw ri wkh htxdwlrq zklfk kdv prgxoxv ohvv wkdq xqlw| lv fkrvhq vr
wkdw wkh surfhvv fdq eh vwdwlrqdu|1 Fohduo| wkh pdjqlwxgh ri m zloo
uh hfw uhodwlyh pdjqlwxghv ri yduldqfhv dqg fryduldqfhv1 Iru h{dpsoh
li d @ 4> e @ 4> f @ 4 dqg g @ 3> 5x @ 5y dqg fry+xy, @ 3 wkhq
3=8
=
m @ 6.8
5
Xvlqj wkh odj rshudwru lw lv dovr xvhixo wr qrwh wkdw zh fdq h{suhvv
!w.4 dv
!w.4 @
^dyw.4 . eyw . fxw.4 . gxw `
@
+4 . mO,
^dyw.4 . eyw . fxw.4 . gxw `i4 mO . m 5 O5 m 6 O6 . ===` +7,
Iurp +7, zh revhuyh wkdw !w.4 fdq eh h{suhvvhg dv dq lqqlwh vxp
ri wkh sdvw huuruv y dqg x1 Wklv lv lpsruwdqw lq vrph whvwv ri h!flhqf|
xqghu udwlrqdo h{shfwdwlrqv1 Vlqfh wkh huuruv lq wkh frpsrvlwh surfhvv
duh lq sulqflsoh dq| huuruv lq wkh hfrqrp| li d prylqj0dyhudjh huuru
surfhvv lv hvwlpdwhg mrlqwo| zlwk wkh rwkhu sdudphwhuv ri wkh prgho lw
Whvwlqj wkh Udwlrqdo H{shfwdwlrqv K|srwkhvlv
754
zloo lqgxfh fruuhodwlrqv ehwzhhq wkh uhjuhvvruv dqg wkh huuru whup zklfk
zloo uhvxow lq eldvhg dqg lqfrqvlvwhqw sdudphwhu hvwlpdwhv1 Iru lqvwdqfh
frqvlghu wkh wzr0shulrg dkhdg udwlrqdo h{shfwdwlrqv iruhfdvw huuru iru
wkh surfhvv/
|w @ |.
"
[
l xwl .
l@3
"
[
l ywl
+8,
l@3
Zh kdyh wkdw
|w @ Hw5 |w . 3 xw . 4 xw4 . 3 yw . 4 yw4
+9,
Dowkrxjk wkh huuru whup lv d prylqj0dyhudjh surfhvv/ lq d whvw ri
h!flhqf| lw zrxog eh lqdssursuldwh wr hvwlpdwh mrlqwo| wkh sdudphwhuv
ri wkh prgho dqg wkh prylqj0dyhudjh surfhvv
|w @ 3 . 4 Hw5 |w . !w . m!w4
=
+:,
=
dqg whvw wkdw 3 @ 3> 4 @ 4= Dv vkrzq deryh wkh huuru whup fdq eh zulw0
whq dv dq lqqlwh vxppdwlrq ri suhylrxv huuruv1 Wklv lv dovr d surshuw|
ri wkh iruhfdvw vr wkdw wkhuh zloo eh fruuhodwlrq ehwzhhq wkh huuru dqg
wkh h{sodqdwru| yduldeoh1 Dq dssursuldwh surfhgxuh lv wr hvwlpdwh +:,
e| ohdvw vtxduhv dqg hpsor| vwdqgdug huuruv zklfk duh prglhg wr doorz
iru wkh vhuldo fruuhodwlrq lq wkh huuru whup +vhh h1j1 Kdqvhq dqg Krgulfn/
4<;3,1
Lw lv dovr xvhixo wr qrwh wkdw dq DUPD iruhfdvw ri d yduldeoh/ hyhq
dvvxplqj wkh xqghuo|lqj prgho lv olqhdu/ lv ohvv h!flhqw wkdq d udwlrqdo
h{shfwdwlrqv iruhfdvw li wkh uhgxfhg irup ri d yduldeoh lqfoxghv frp0
srvlwh prylqj0dyhudjh huuruv1 Vxssrvh iru looxvwudwlrq d yduldeoh/ |w / lv
jhqhudwhg e| wkh surfhvv
|w @ xw xw4 . yw
+;,
Wkh udwlrqdo rqh0shulrg dkhdg iruhfdvw/ Hw4 |w @ xw4 = Ohw wkh
prylqj0dyhudjh surfhvv iru +;, eh jlyhq e| |w @ !w m!w4 +m A 3, vr
d
|w @ m!w4 = Wkh dvvrfldwhg
wkdw wkh rqh0shulrg DUPD iruhfdvw lv Hw4
iruhfdvw huuruv duh
|w Hw4 |w @ xw . yw
iru wkh udwlrqdo h{shfwdwlrq dqg iru wkh DUPD iruhfdvw
xw xw4 . yw
d
|w Hw4
|w @ !w @
+4 mO,
+<,
+43,
Dvvxplqj iru vlpsolflw| wkdw fry+xy, lv }hur wkh yduldqfh ri wkh ud0
wlrqdo h{shfwdwlrqv iruhfdvw lv
5x . 5y
+44,
755
Frqiurqwlqj Prghov zlwk Idfwv
Wkh DUPD iruhfdvw huuru fdq eh zulwwhq dv
d
|w @ xw +4 m,xw4 m+4 m,xw5 m 5 +4 m,xw6 . = = =
|w Hw4
. yw . myw4 . m 5 yw5 . = = =
+45,
4
@ 4 . mO . m 5 O5 . m 6 O6 . = = = ,
+Uhfdoo wkdw +4mO,
Dowkrxjk wkh DUPD iruhfdvw huuru dsshduv vhuldoo| fruuhodwhg wklv lv
lq idfw qrw wkh fdvh dv vxevwlwxwlrq iru m lq whupv ri wkh yduldqfhv ri x
dqg y/ wkrxjk phvv|/ zloo ghprqvwudwh1
Wkh yduldqfh ri wkh DUPD iruhfdvw huuru lv jlyhq e|
5x . 5x +4 m,5 ^4 . m 5 . m 7 . m 9 . = = = ` .
5y i4 . m 5 . m 7 . m 9 . = = = j +46,
zklfk zh fdq vlpsoli| dv
5x .
+4 m,5 5x
5y
.
+4 m 5 ,
+4 m 5 ,
+47,
Ehfdxvh m lv ohvv wkdq rqh wkh yduldqfh ri wkh DUPD iruhfdvw lv
juhdwhu wkdq wkh udwlrqdo h{shfwdwlrqv iruhfdvw huuru1 Hvvhqwldoo| lqiru0
pdwlrq lv orvw lq iruhfdvwlqj wkh frpsrvlwh huuru surfhvv1 Lq dgglwlrq wkh
lqqrydwlrq iurp wkh DUPD surfhvv/ !w / wkrxjk vhuldoo| xqfruuhodwhg/
zloo qrw qhfhvvdulo| eh ruwkrjrqdo wr yduldeohv wkdw duh fruuhodwhg zlwk
sdvw x dqg y lqqrydwlrqv/ gxh wr wkh lpsolflw ghshqghqfh ri wkh DUPD
lqqrydwlrq rq wkhvh yduldeohv1