7KH IROORZLQJ PDQVXVFULSW DSSHDUHG DV
3DFH 5 .HOOH\ DQG 5RQDOG %DUU\ ´)DVW &$5Vµ -RXUQDO RI 6WDWLVWLFDO &RPSXWDWLRQ DQG
6LPXODWLRQ 9ROXPH 1XPEHU S
:H JUDWHIXOO\ WKDQN WKH -RXUQDO RI 6WDWLVWLFDO &RPSXWDWLRQ DQG 6LPXODWLRQ DQG FRS\ULJKW RZQHU
23$ 2YHUVHDV 3XEOLVKHUV $VVRFLDWLRQ 19 IRU DOORZLQJ XV WR GLVWULEXWH WKLV HOHFWURQLF
YHUVLRQ RI WKH PDQXVFULSW
)$67 &$56
%\
5 .(//(@
7 −
7R QRUPDOL]H DQG PDNH & V\PPHWULFDO ZH SUH DQG SRVWPXOWLSO\ E\ 6
2
& = 6 & X +& X′ 6
7
7KLV RSHUDWLRQ ZLOO HQVXUH WKH ODUJHVW HLJHQYDOXH RI & HTXDOV 6HH 2UG IRU PRUH
GHWDLOV
& 7+( 0$;,080 /,.(/,+22' 6802)648$5(' (55256
7KH FRQYHQWLRQDO DSSURDFK WR FRPSXWLQJ &$5 HVWLPDWHV UHOLHV XSRQ VROYLQJ WKH &$5
QRUPDO HTXDWLRQV +RZHYHU WKH VROXWLRQ WR WKH HTXDWLRQV GHSHQGV XSRQ WKH
XQREVHUYDEOH SDUDPHWHU φ
; ′ , − φ& ; β = ; ′ , − φ& @′ @′ φ φ φ K φ @ /RRS RYHU 66(PO φ L φ φ φ K φ @ ZH FDQ SUHFRPSXWH DQ O E\ YHFWRU RI ORJ
O
GHWHUPLQDQWV 6HYHUDO DGYDQWDJHV DFFUXH WR FRPSXWLQJ WKHVH DV D JURXS )LUVW WKH RUGHULQJ
DOJRULWKPV IRU VSDUVH PDWULFHV DSSO\ WR DOO PDWULFHV ZLWK D JLYHQ SDWWHUQ RI VSDUVLW\ 7KLV
SDWWHUQ RI ]HURV DQG QRQ]HURV IRU OQ , − φ& ZRXOG DSSO\ IRU DOO SRVLWLYH φ +HQFH
FRPSXWLQJ WKH GHWHUPLQDQWV DV D JURXS DPRUWL]HV RYHU O LWHUDWLRQV WKH FRVW RI FRPSXWLQJ
WKH RSWLPDO RUGHULQJ 6HFRQG DV WKLV FRXOG SURYH D WLPH FRQVXPLQJ VWHS IRU VRPH
SUREOHPV FRPSXWLQJ WKHP DV D JURXS DOORZV WKH FRPSXWDWLRQV WR RFFXU GXULQJ QRQSHDN
XVDJH 7KLUG PRVW ILWWLQJ H[HUFLVHV LQYROYH H[SORUDWRU\ RU GLDJQRVWLF YDULDWLRQV LQ WKH
PRGHO ZKLFK ZRXOG DIIHFW WKH VSHFLILFDWLRQ RI ; DQG KHQFH = EXW QRW DIIHFW OQ , − φ& 7KXV
FKDQJHV LQ WKH VSHFLILFDWLRQ RI ; IXUWKHU DPRUWL]H WKH FRVW RI SUHFRPSXWLQJ OQ , − φ&
* &20387,1* 7+( &$5 /,.(/,+22'
*LYHQ WKH YDULRXV FRPSXWDWLRQV ZH FDQ GHILQH WZLFH WKH SURILOH ORJOLNHOLKRRG RYHU WKH
HQWLUH VLPXODWHG @ − Q OQ ##
66( φ @
*HQHUDWHG D FRPPRQ VHW RI 1 YHFWRUV RI HOHPHQWV HDFK XVLQJ WKH
0DWODE QRUPDO UDQGRP QXPEHU JHQHUDWRU :H SHUIRUP WKLV RSHUDWLRQ RQFH IRU
WKH HQWLUH VLPXODWLRQ 6FDOLQJ WKH FRPPRQ 1 HUURUV E\ σ JHQHUDWHV WKH 1
σ UDQGRP YDULDEOHV 7KLV SUDFWLFH UHIHUUHG WR DV ´FRUUHODWHG VDPSOLQJµ
5XELQVWHLQ JUHDWO\ UHGXFHV WKH YDULDQFH LQ 0RQWH &DUOR H[SHULPHQWV
:H VXEVHTXHQWO\ JHQHUDWHG WKH DXWRFRUUHODWHG GHSHQGHQW YDULDEOH < DFFRUGLQJ WR
< = ;β + σ , − φ& − X
ZKHUH X UHSUHVHQWV DQ Q E\ LWHU PDWUL[ RI 1 UDQGRP YDULDWHV ,Q DFWXDOLW\ ZH VROYHG WKH
FRUUHVSRQGLQJ HTXDWLRQ V\VWHP LQ IRU WKH FRHIILFLHQWV $6 UDWKHU WKDQ FRPSXWLQJ WKH
LQYHUVH DV WKLV JRHV PXFK IDVWHU
6HH *LOOH\ DQG 3DFH IRU D 0RQWH &DUOR VWXG\ RI WKH LQHTXDOLW\ UHVWULFWHG %D\HVLDQ HVWLPDWRU 6HH 3DFH DQG
*LOOH\ IRU PRUH LQIRUPDWLRQ RQ LQHTXDOLW\ UHVWULFWLRQV LQ UHJUHVVLRQ VHWWLQJV
, − φ& $6 = X
Q× Q Q× LWHU Q× LWHU
−
$6 = , − φ&
X
Q× LWHU Q× Q Q× LWHU
&RPSDUH WKLV WR WKH XVXDO &KROHVN\ GHFRPSRVLWLRQ DQG LQYHUVH IRUPXODWLRQ
, − φ& $, = ,
Q× Q Q× Q
Q× Q
−
$ = , − φ&
,
Q× Q Q× Q
$6 = $, X = , − φ& − X
Q× LWHU Q× Q Q× LWHU
Q× LWHU Q× Q
5HODWLYH WR UHTXLUHV VROYLQJ D ODUJHU V\VWHP Q E\ Q LQVWHDG RI Q E\ LWHU DQG
VXEVHTXHQWO\ PXOWLSO\LQJ DQ Q E\ Q PDWUL[ E\ D Q E\ LWHU PDWUL[ 2 QLWHU 6LQFH LWHU XVXDOO\
LV PXFK VPDOOHU WKDQ Q WKH LQYHUVH PHWKRG WDNHV VXEVWDQWLDOO\ ORQJHU IRU WKH VDPH UHVXOWV
6LQFH HIILFLHQW FRPSXWDWLRQ RI WKH GHWHUPLQDQW UHTXLUHG E\ PD[LPXP OLNHOLKRRG
HVWLPDWLRQ XVHV WKH &KROHVN\ GHFRPSRVLWLRQ RI , − φ& WKLV UHGXFHV WKH DGGLWLRQDO FRVW RI
VLPXODWLQJ WKH UDQGRP QXPEHUV +RZHYHU VROYLQJ WKH HTXDWLRQV E\ XQNQRZQV
IRU LWHU ULJKW KDQG VLGHV ZRXOG SURYH TXLWH GLIILFXOW ZLWKRXW UHVRUWLQJ WR VSDUVH PDWUL[
WHFKQLTXHV DV GHVFULEHG LQ & EHORZ
% 63$56,7< ,1 63$7,$/ 352%/(06
,I GLIIHUHQFLQJ DQ REVHUYDWLRQ ZLWK LWV QHDUE\ QHLJKERUV UHPRYHV PRVW RI WKH HIIHFWV RI
DXWRFRUUHODWLRQ WKH VSDWLDO ZHLJKWLQJ PDWUL[ & FDQ EH TXLWH VSDUVH )RU H[DPSOH LI DQ
REVHUYDWLRQ GLVSOD\V HUURU GHSHQGHQF\ RQO\ ZLWK LWV QHDUHVW P QHLJKERUV RQO\ P QRQ]HUR
HQWULHV H[LVW SHU URZ RI & 7KXV & ZLOO FRQWDLQ QP QRQ]HUR HOHPHQWV RXW RI Q SRVVLEOH
RQHV 7KLV SURGXFHV D PQ SURSRUWLRQ RI QRQ]HUR HOHPHQWV D SRSXODU PHDVXUH RI VSDUVLW\
)RU H[DPSOH ZLWK WKLV SUREOHP ZH XVHG IRXU QHLJKERUV IRU HDFK REVHUYDWLRQ LQ FRPSXWLQJ
& X 7KLV UHVXOWHG LQ & KDYLQJ RQ DYHUDJH QHLJKERUV DQG D VSDUVLW\ RI
7KLV UHSUHVHQWV D YHU\ KLJK OHYHO RI VSDUVLW\ ZKLFK JURZV DV Q LQFUHDVHV
6SDUVLW\ UHVXOWV LQ D QXPEHU RI FRPSXWDWLRQDO JDLQV )LUVW LW GUDPDWLFDOO\ GHFUHDVHV WKH
VWRUDJH QHHGHG IRU & DQG , − φ& 8VLQJ WUDGLWLRQDO GHQVH WHFKQLTXHV & UHTXLUHV 0%
RI VWRUDJH GRXEOH SUHFLVLRQ 8VLQJ VSDUVH PDWUL[ WHFKQLTXHV & UHTXLUHV OHVV WKDQ .%
RI VWRUDJH 1DWXUDOO\ WKLV GLYHUJHQFH JURZV ZLWK Q
6HFRQG VSDUVLW\ JUHDWO\ DFFHOHUDWHV FRPSXWDWLRQV )RU H[DPSOH PXOWLSO\LQJ WKH Q E\ Q
PDWUL[ & ZLWK WKH Q E\ N PDWUL[ ; UHTXLUHV 2 NQ RSHUDWLRQV XVLQJ GHQVH PDWULFHV %DUULQJ
FRPSXWDWLRQDO ERRNNHHSLQJ WKH HTXLYDOHQW VSDUVH RSHUDWLRQ UHTXLUHV 2 NQP RSHUDWLRQV D
PXFK VPDOOHU QXPEHU 7KH UHDO EHQHILWV FRPH ZKHQ FRPSXWLQJ GHWHUPLQDQWV LQYHUVHV RU
VROYLQJ V\VWHPV RI HTXDWLRQV $OO RI WKHVH RSHUDWLRQV FDQ EXLOG XSRQ WKH &KROHVN\
GHFRPSRVLWLRQ RI D PDWUL[ DQG DOO UHTXLUH 2 Q RSHUDWLRQV ZKHQ XVLQJ GHQVH PDWULFHV
6SDUVLW\ KRZHYHU FDQ WRWDOO\ FKDQJH WKH RUGHU RI WKH QXPEHU RI RSHUDWLRQV UHTXLUHG LQ
WKHVH FRPSXWDWLRQV )RU H[DPSOH LI , − φ& KDG D EDQG VWUXFWXUH ZLWK KDOIEDQGZLGWK S
WKH &KROHVN\ GHFRPSRVLWLRQ RI , − φ& ZRXOG UHTXLUH 2 Q S + S RSHUDWLRQV *ROXE DQG
9DQ /RDQ S +HQFH IRU IL[HG EDQGZLGWKV WKH FRPSXWDWLRQV JURZ OLQHDUO\ ZLWK
Q WKH QXPEHU RI REVHUYDWLRQV
8QIRUWXQDWHO\ WKH H[LVWHQFH RI D SXUH EDQG VWUXFWXUH GRHV QRW DULVH YHU\ RIWHQ )LJXUH
D VKRZV WKH DFWXDO SORW RI WKH QRQ]HUR HOHPHQWV LQ , − φ& 7KH H[LVWHQFH RI VXFK
GLVSHUVHG RIIGLDJRQDOV FRXOG PDNH LW GLIILFXOW WR DFKLHYH FRPSXWDWLRQDO JDLQV
+RZHYHU RQH FDQ SHUPXWH WKH URZV RU FROXPQV RI , − φ& WR UHGXFH EDQGZLGWK RU WR
DFKLHYH RWKHU RSWLPL]DWLRQV $ YDULHW\ RI VXFK SHUPXWDWLRQV H[LVW VHH *HRUJH DQG /LX
IRU PRUH RQ WKH YDULRXV RUGHULQJV )RU H[DPSOH WKH UHYHUVH &XWKLOO0F.HH
DOJRULWKP DWWHPSWV WR UHRUGHU WKH URZV DQG FROXPQV RI WKH PDWUL[ WR FUHDWH D YDULDEOH EDQG
PDWUL[ DV VKRZQ LQ )LJXUH E )LJXUH E PDNHV WKH JDLQV RI H[SORLWLQJ VSDUVLW\ REYLRXV
/HVV REYLRXVO\ )LJXUH F VKRZV WKH SORW RI , − φ& SHUPXWHG XVLQJ WKH FROXPQ PLQLPXP
GHJUHH DOJRULWKP ZKLOH )LJXUH G VKRZV WKH SORW RI , − φ& SHUPXWHG XVLQJ WKH V\PPHWULF
PLQLPXP GHJUHH DOJRULWKP
7DEOH , VKRZV WKH WLPLQJV DVVRFLDWHG ZLWK FRPSXWLQJ WKH ORJGHWHUPLQDQWV XVLQJ WKH
RULJLQDO UDQGRP UHYHUVH &XWKLOO0F.HH FROXPQ PLQLPXP GHJUHH RUGHULQJV DQG
V\PPHWULF FROXPQ PLQLPXP GHJUHH RUGHULQJV IRU , − φ& $V 7DEOH , PDNHV FOHDU WKH
RUGHULQJ RI WKH URZV DQG FROXPQV PDWWHUV ZLWK WKH V\PPHWULF PLQLPXP GHJUHH RUGHULQJ
UHGXFLQJ H[HFXWLRQ WLPHV E\ RYHU WKH RULJLQDO RUGHULQJ ,QWULJXLQJO\ WKH PRUH
LQWXLWLYH UHYHUVH &XWKLOO0F.HH RUGHULQJ DFWXDOO\ SHUIRUPHG ZRUVH WKDQ WKH RULJLQDO
RUGHULQJ EXW PXFK EHWWHU WKDQ WKH UDQGRP RUGHULQJ $V D ZRUVW FDVH VFHQDULR WKH UDQGRP
RUGHULQJ SURGXFHG FRPSXWDWLRQDO WLPHV ZRUVH WKDQ WKH RSWLPDO RUGHULQJ E\ D IDFWRU RI
$OO FRPSXWDWLRQV XVHG WKH 0DWODE ODQJXDJH UXQQLQJ RQ D 0K] 3HQWLXP FRPSXWHU
7R SODFH WKHVH UHVXOWV LQ SHUVSHFWLYH /L WRRN WKH HLJHQYDOXH URXWH WR FRPSXWLQJ
GHWHUPLQDQWVÂÂ /L XVHG DQ ,%0 56 0RGHO DQG D &0 SDUDOOHO SURFHVVLQJ
7KH XVH RI FROXPQ PLQLPXP GHJUHH RUGHULQJ GHVWUR\V V\PPHWU\ +HQFH WKH GHWHUPLQDQW FRPSXWDWLRQV XVH
/8 GHFRPSRVLWLRQ
ÂÂ 7KH VSDUVH PDWUL[ WHFKQRORJ\ PDNHV LW PXFK IDVWHU WR FRPSXWH WKH QHFHVVDU\ GHWHUPLQDQWV WKDQ WR ILQG WKH
GHWHUPLQDQW YLD HLJHQYDOXHV WKH VWDQGDUG SUDFWLFH LQ WKLV DUHD 2UG :KLOH VSDUVH HLJHQYDOXH URXWLQHV
H[LVW WKHVH VWLOO UHTXLUH PRUH WLPH XVXDOO\ WKDQ WKH GLUHFW FRPSXWDWLRQ RI D VHW RI GHWHUPLQDQWV
VXSHUFRPSXWHU 7KH &0 KDG SURFHVVRUV HDFK ZLWK 0% RI ORFDO PHPRU\ DQG IRXU
YHFWRU XQLWV )RU D E\ VSDWLDO ZHLJKW PDWUL[ WKH 56 UHTXLUHG
VHFRQGV ZKLOH WKH &0 UHTXLUHG VHFRQGV ,Q FRQWUDVW LW WRRN RQO\ VHFRQGV WR
FRPSXWH GLIIHUHQW GHWHUPLQDQWV RI D E\ PDWUL[ RQ D 0K] 3HQWLXP
FRPSXWHU XVLQJ 0DWODE +HQFH WKH VSDUVH WHFKQRORJ\ HPSOR\HG KHUH DOORZHG D SHUVRQDO
FRPSXWHU RQ D ODUJHU SUREOHP WR H[FHHG WKH SHUIRUPDQFH RI D VXSHUFRPSXWHU RQ D VPDOOHU
SUREOHP
)LQDOO\ HYHQ VXSHUFRPSXWHUV KDYH PHPRU\ OLPLWDWLRQV 7KH XVH RI VSDUVH PDWUL[
WHFKQRORJ\ KDV DOORZHG XV WR KDQGOH SUREOHPV ZLWK REVHUYDWLRQV 3DFH DQG %DUU\
IRUWKFRPLQJ $ GHQVH VSDWLDO ZHLJKW PDWUL[ ZRXOG KDYH UHTXLUHG JLJDE\WHV GRXEOH
SUHFLVLRQ ZKLFK ZRXOG KDYH IXUWKHU H[DFHUEDWHG WKH RYHUDOO FRPSXWDWLRQDO GLIILFXOWLHV
& 0217( &$5/2 (;3(5,0(17 7,0,1*6
6LPXODWLQJ D &$5 QHHG QRW WDNH YHU\ ORQJ 7R VKRZ WKLV ZH PHDVXUHG WLPHV LQ WKH
GLIIHUHQW FRPSXWDWLRQDO VWDJHV IRU VLPXODWLQJ UHJUHVVLRQV IRU D SDUWLFXODU FDVH φ ,W
WRRN VHFRQGV IURP WKH WLPH WKH PDWUL[ & DQG WKH ORJGHWHUPLQDQWV ZHUH ORDGHG WR
JHQHUDWH ; DQG RWKHU PDWULFHV QHHGHG DQG WR SHUIRUP WKH GHFRPSRVLWLRQV ,W WRRN
DGGLWLRQDO VHFRQGV WR JHQHUDWH WKH VSDWLDOO\ DXWRFRUUHODWHG UHDOL]DWLRQV RI < 7KLV
LQYROYHG FRPSXWLQJ D &KROHVN\ GHFRPSRVLWLRQ RI , − φ& DQG VROYLQJ D E\
V\VWHP WLPHV 6SDUVLW\ SURYLGHV DQ LQFUHGLEOH ERRVW WR WKHVH FRPSXWDWLRQV 7KH
FRPSXWDWLRQ RI WKH PD[LPXP OLNHOLKRRG HVWLPDWHV WRRN RQO\ VHFRQGV ORQJHU
+HQFH WKH WRWDO WLPH IRU FRPSXWLQJ ZDV VHFRQGV FRQGLWLRQDO XSRQ WKH
SUHFRPSXWDWLRQ RI WKH GHWHUPLQDQWV DQG &
' 0217( &$5/2 (;3(5,0(17 5(68/76
7KH VLPXODWLRQ UHVXOWV LQ 7DEOH ,, PDWFK VRPH RI WKRVH UHSRUWHG LQ WKH OLWHUDWXUH XVLQJ
UHJXODU ODWWLFHV )LUVW WKH PD[LPXP OLNHOLKRRG HVWLPDWRU VOLJKWO\ XQGHUHVWLPDWHV WKH WUXH
GLIIHUHQFLQJ SDUDPHWHU φ +RZHYHU ERWK HVWLPDWRUV VKRZ YHU\ VPDOO DPRXQWV RI ELDV
a a
HVSHFLDOO\ ZKHQ H[DPLQLQJ WKH PHGLDQ YDOXHV RI φPO φHJOV 6HFRQG WKH YDULDQFH RI ERWK
HVWLPDWRUV GHFUHDVHV ZLWK ULVLQJ φ 7KH LQHTXDOLW\ UHVWULFWLRQV GR QRW VHHP WR DIIHFW WKH
UHVXOWV PXFK H[FHSW IRU ZKHQ φ LV ,Q WKLV FDVH WKH LQHTXDOLW\ DLGV ERWK HVWLPDWRUV
6HH +DLQLQJ S DQG &UHVVLH S IRU D UHYLHZ RI VWXGLHV ILQGLQJ WKHVH HIIHFWV
( 352),/( /,.(/,+22' 3/276
$V DQ H[DPSOH RI WKH XVH RI VLPXODWLRQ IRU LQIHUHQFH )LJXUHV D E F DQG G VKRZ
a
WKH SURILOH OLNHOLKRRG SORWV IRU φ DFURVV LWHUDWLRQV IRU φ 7KH DVWHULVN
PO
RQ HDFK SURILOH OLNHOLKRRG FXUYH GHQRWHV WKH PD[LPXP SRLQW 7KH YHUWLFDO OLQH VHJPHQW
IURP WKH ORZHVW WR WKH KLJKHVW FXUYH VKRZV WKH ORFDWLRQ RI WKH WUXH YDOXH RI φ 7KH ODUJH GRW
GHQRWHV WKH PHDQ ZKLOH WKH V\PERO × GHQRWHV WKH PHGLDQ IRU DOO WKH SURILOH OLNHOLKRRGV
)LJXUHV D DQG G φ LOOXVWUDWH KRZ WKH LQIRUPDWLRQ PDWUL[ DSSURDFK FRXOG UXQ
LQWR SUREOHPV 1RWH LQ )LJXUH F φ WKH SURILOH OLNHOLKRRG LV DV\PPHWULF )LJXUH E
VKRZV DQ LQWHUPHGLDU\ FDVH φ
7KH VLPXODWHG SURILOH OLNHOLKRRGV GLVWLOO JUHDW DPRXQWV RI LQIRUPDWLRQ DERXW WKH
PD[LPD RI WKH OLNHOLKRRGV DQG WKHLU JOREDO EHKDYLRU DFURVV SRVVLEOH UHDOL]DWLRQV VLPXODWLRQ
WULDOV 2QH FDQ LPPHGLDWHO\ REWDLQ D IHHO IRU WKH UREXVWQHVV RI LQIHUHQFH IRU WKH LPSRUWDQW
VSDWLDO GLIIHUHQFLQJ SDUDPHWHU φ
&21&/86,21
,Q D VSDWLDO FRQWH[W PD[LPXP OLNHOLKRRG WHFKQLTXHV KDYH D UHSXWDWLRQ RI EHLQJ VORZ WR
FRPSXWH SURQH WR HUURU WKURXJK ORFDO RSWLPD DQG RIIHULQJ RQO\ DV\PSWRWLF LQIHUHQFH 7KH
DSSURDFK DGRSWHG KHUH WR FRPSXWLQJ FRQGLWLRQDO DXWRUHJUHVVLRQV DWWDFNV HDFK RI WKHVH
SUREOHPV )LUVW WKH XVH RI VSDUVH PDWULFHV DQG UHVWUXFWXULQJ WKH VXPRIVTXDUHG HUURUV
IXQFWLRQ JUHDWO\ DFFHOHUDWHV FRPSXWDWLRQV 6HFRQG WKH HYDOXDWLRQ RI WKH OLNHOLKRRG RYHU D
JULG RI YDOXHV RI WKH LPSRUWDQW VSDWLDO GLIIHUHQFLQJ SDUDPHWHU JUHDWO\ HQKDQFHV UREXVWQHVV
WR ORFDO RSWLPD 7KLUG WKH TXLFN VLPXODWLRQ RI WKH FRQGLWLRQDO DXWRUHJUHVVLRQ FRXSOHG ZLWK
WKH SURILOH OLNHOLKRRG LQIRUPDWLRQ SURYLGHV D ZHDOWK RI LQIRUPDWLRQ IRU FRQGXFWLQJ
LQIHUHQFH LQ DOO VDPSOH VL]HV
$V DQ LOOXVWUDWLRQ RI WKH HIILFDF\ RI WKHVH WHFKQLTXHV ZH FRPSXWHG D VLPXODWHG
FRQGLWLRQDOO\ DXWRUHJUHVVLYH YHFWRUV RI < RI REVHUYDWLRQV E GHWHUPLQDQWV RI D
E\ PDWUL[ DQG F FRPSXWHG HVWLPDWHV ZLWK WKH SURILOH OLNHOLKRRG YDOXHV
7KLV WRRN VHFRQGV VHFRQGV IRU WKH GHWHUPLQDQWV DQG IRU WKH HVWLPDWHV ,Q
FRQWUDVW /L RQ D VPDOOHU SUREOHP WRRN ORQJHU RQ D VXSHUFRPSXWHU WR FDOFXODWH WKH
GHWHUPLQDQWV WKDQ ZH DFFRPSOLVKHG RQ D SHUVRQDO FRPSXWHU
+RSHIXOO\ DGGLWLRQV OLNH WKHVH WR WKH VSDWLDO VWDWLVWLFV WRROER[ ZLOO HQDEOH XVHUV WR KHOS
FRSH ZLWK WKH LQFUHDVLQJ IORZ RI VSDWLDO LQIRUPDWLRQ
$&.12:/('*(0(176
:H ERWK ZRXOG OLNH WR WKDQN WKH 8QLYHUVLW\ RI $ODVND IRU LWV JHQHURXV UHVHDUFK VXSSRUW
,Q DGGLWLRQ 3DFH ZRXOG OLNH WR DFNQRZOHGJH VXSSRUW IURP WKH &HQWHU IRU 5HDO (VWDWH DQG
8UEDQ (FRQRPLF 6WXGLHV 8QLYHUVLW\ RI &RQQHFWLFXW )LQDOO\ ZH ZRXOG OLNH WKH WKDQN %LOO
+XJKHV IRU KDYLQJ GLVFXVVHG WKH SDSHU DW WKH $PHULFDQ 5HDO (VWDWH 6RFLHW\ PHHWLQJV
%LEOLRJUDSK\
%DUU\ 5RQDOG 3 DQG 5 .HOOH\ 3DFH ´.ULJLQJ ZLWK /DUJH 'DWD 6HWV 8VLQJ 6SDUVH
0DWUL[ 7HFKQLTXHVµ &RPPXQLFDWLRQV LQ 6WDWLVWLFV &RPSXWDWLRQ DQG 6LPXODWLRQ
%DUU\ 5RQDOG 3 DQG -D\ 0 9HU +RHI ´%ODFNER[ .ULJLQJ ³ .ULJLQJ ZLWKRXW
6SHFLI\LQJ 9DULRJUDP 0RGHOVµ -RXUQDO RI $JULFXOWXUDO %LRORJLFDO DQG (QYLURQPHQWDO
6WDWLVWLFV
%HVDJ -XOLDQ ( ´6SDWLDO ,QWHUDFWLRQ DQG WKH 6WDWLVWLFDO $QDO\VLV RI /DWWLFH 6\VWHPVµ
-RXUQDO RI WKH 5R\DO 6WDWLVWLFDO 6RFLHW\ %
%HVDJ -XOLDQ ( ´6WDWLVWLFDO $QDO\VLV RI 1RQODWWLFH 'DWDµ 7KH 6WDWLVWLFLDQ
&UHVVLH 1RHO $& 6WDWLVWLFV IRU 6SDWLDO 'DWD 5HYLVHG HG 1HZ
*HRUJH $ODQ DQG -RVHSK /LX &RPSXWHU 6ROXWLRQ RI /DUJH 6SDUVH 3RVLWLYH 'HILQLWH
6\VWHPV (QJOHZRRG &OLIIV 3UHQWLFH+DOO
*LOOH\ 2: DQG 5 .HOOH\ 3DFH ´8VLQJ ,QHTXDOLW\ 5HVWULFWLRQV WR 7DPH
0XOWLFROOLQHDULW\ LQ +HGRQLF 3ULFLQJ 0RGHOVµ 5HYLHZ RI (FRQRPLFV DQG 6WDWLVWLFV
*ROXE * + DQG & ) 9DQ /RDQ 0DWUL[ &RPSXWDWLRQV VHFRQG HGLWLRQ -RKQ
+RSNLQV
+DLQLQJ 5REHUW 6SDWLDO 'DWD $QDO\VLV LQ WKH 6RFLDO DQG (QYLURQPHQWDO 6FLHQFHV
&DPEULGJH 8QLYHUVLW\ 3UHVV
+R 6KX
0XOWLIDFWRU $QDO\VLV RI 9DULDQFH DQG &RYDULDQFHµ -RXUQDO RI 6WDWLVWLFDO &RPSXWDWLRQ DQG
6LPXODWLRQ
/L %LQ ´,PSOHPHQWLQJ 6SDWLDO 6WDWLVWLFV RQ 3DUDOOHO &RPSXWHUVµ LQ $UOLQJKDXV 6
HG 3UDFWLFDO +DQGERRN RI 6SDWLDO 6WDWLVWLFV &5& 3UHVV %RFD 5DWRQ
0DUGLD .9 DQG $- :DWNLQV ´2Q 0XOWLPRGDOLW\ RI WKH /LNHOLKRRG LQ WKH 6SDWLDO
/LQHDU 0RGHOµ %LRPHWULND
0HHNHU :4 DQG /$ (VFREDU ´7HDFKLQJ DERXW $SSUR[LPDWH &RQILGHQFH
5HJLRQV %DVHG RQ 0D[LPXP /LNHOLKRRG (VWLPDWHVµ 7KH $PHULFDQ 6WDWLVWLFLDQ
2UG -. (VWLPDWLRQ 0HWKRGV IRU 0RGHOV RI 6SDWLDO ,QWHUDFWLRQ -RXUQDO RI WKH
$PHULFDQ 6WDWLVWLFDO $VVRFLDWLRQ
3DFH 5 .HOOH\ DQG 5RQDOG %DUU\ ´6SDUVH 6SDWLDO $XWRUHJUHVVLRQVµ 6WDWLVWLFV DQG
3UREDELOLW\ /HWWHUV
3DFH 5 .HOOH\ DQG 2: *LOOH\ ´,PSURYLQJ 3UHGLFWLRQ DQG $VVHVVLQJ
6SHFLILFDWLRQ 4XDOLW\ LQ 1RQ/LQHDU 6WDWLVWLFDO 9DOXDWLRQ 0RGHOVµ -RXUQDO RI %XVLQHVV
DQG (FRQRPLFV 6WDWLVWLFV
5LSOH\ %ULDQ ' 6SDWLDO 6WDWLVWLFV 1HZ
5LSOH\ %ULDQ ' 6WDWLVWLFDO ,QIHUHQFH IRU 6SDWLDO 3URFHVVHV &DPEULGJH &DPEULGJH
8QLYHUVLW\ 3UHVV
5XELQVWHLQ 59 6LPXODWLRQ DQG WKH 0RQWH &DUOR 0HWKRG 1HZ
6RQH $NLR DQG 'DQLHO *ULIILWK ´7UDGH2IIV $VVRFLDWHG ZLWK 1RUPDOL]LQJ &RQVWDQW
&RPSXWDWLRQDO 6LPSOLILFDWLRQV IRU (VWLPDWLQJ 6SDWLDO 6WDWLVWLFDO 0RGHOVµ -RXUQDO RI
6WDWLVWLFDO &RPSXWDWLRQ DQG 6LPXODWLRQ
:DUQHV -- DQG %ULDQ 5LSOH\ ´3UREOHPV ZLWK /LNHOLKRRG (VWLPDWLRQ RI
&RYDULDQFH )XQFWLRQV RI 6SDWLDO *DXVVLDQ 3URFHVVHVµ %LRPHWULND
=LPPHUPDQ 'DOH ´&RPSXWDWLRQDOO\ ([SORLWDEOH 6WUXFWXUH RI &RYDULDQFH 0DWULFHV
DQG *HQHUDOL]HG &RYDULDQFH 0DWULFHV LQ 6SDWLDO 0RGHOVµ -RXUQDO RI 6WDWLVWLFDO
&RPSXWDWLRQ DQG 6LPXODWLRQ
7DEOH , ³ 'HWHUPLQDQW &RPSXWDWLRQ 7LPHV
2UGHULQJ 7LPH SHU 'HWHUPLQDQWV
5DQGRP
2ULJLQDO
5HYHUVH &XWKLOO0F.HH
&ROXPQ 0LQLPXP 'HJUHH
6\PPHWULF 0LQLPXP 'HJUHH
7DEOH ,, ³ 0/ &$5 6LPXODWLRQ 6WDWLVWLFV
φ a a a a
PHDQ φ
PO
PHGLDQ φ
PO
σ φ
PO
UDQJH φ
PO
Figure 1a − Original Ordering
0
500
1000
1500
2000
2500
0 500 1000 1500 2000 2500
nz = 17659
Figure 1b − Reverse Cuthill−McKee Ordering
0
500
1000
1500
2000
2500
0 500 1000 1500 2000 2500
nz = 17659
Figure 1c − Column Minimum Degree Ordering
0
500
1000
1500
2000
2500
0 500 1000 1500 2000 2500
nz = 17659
Figure 1d − Symmetric Minimum Degree Ordering
0
500
1000
1500
2000
2500
0 500 1000 1500 2000 2500
nz = 17659
Figure 2a − Profile Likelihood for φ of 0.05
3800
3600
3400
Scaled Log−likelihood
3200
3000
2800
2600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
φ
Figure 2b − Profile Likelihood for φ of 0.25
3800
3700
3600
3500
Scaled Log−likelihood
3400
3300
3200
3100
3000
2900
2800
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
φ
Figure 2c − Profile Likelihood for φ of 0.50
3700
3600
3500
3400
Scaled Log−likelihood
3300
3200
3100
3000
2900
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
φ
Figure 2d − Profile Likelihood for φ of 0.95
3400
3200
3000
2800
Scaled Log−likelihood
2600
2400
2200
2000
1800
1600
1400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
φ