Unit Root Log Periodogram Regression

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS

AT YALE UNIVERSITY

Box 2125, Yale Station

New Haven, Connecticut 06520

COWLES FOUNDATION DISCUSSION PAPER NO. 1244

Note: Cowles Foundation Discussion Papers are preliminary materials

circulated to stimulate discussion and critical comment.

Requests for single copies of a Paper will be filled by the

Cowles Foundation within the limits of the supply. References

in publications to Discussion Papers (other than mere

acknowledgment by a writer that he has access to such

unpublished material) should be cleared with the author to

protect the tentative character of these papers.

UNIT ROOT LOG PERIODOGRAM REGRESSION

Peter C. B. Phillips

December 1999

CORE Metadata, citation and similar papers at core.ac.uk

Xqlw Urrw Orj Shulrgrjudp Uhjuhvvlrq



Shwhu F1 E1 Skloolsv

Frzohv Irxqgdwlrq iru Uhvhdufk lq Hfrqrplfv

\doh Xqlyhuvlw|

4 Rfwrehu 4<<<

Devwudfw

Orj shulrgrjudp +OS, uhjuhvvlrq lv vkrzq wr eh frqvlvwhqw dqg wr kdyh d pl{hg qrupdo olplw glvwulexwlrq zkhq wkh phpru| sdudphwhu g @ 4= Jdxvvldq huuruv duh qrw uhtxluhg1 Whvwv ri g @ 4 edvhg rq OS uhjuhvvlrq duh frqvlvwhqw djdlqvw g ? 4 dowhuqdwlyhv exw lqfrqvlvwhqw djdlqvw g A 4 dowhuqdwlyhv1 D whvw edvhg rq d prglhg OS uhjuhvvlrq wkdw lv frqvlvwhqw lq erwk gluhfwlrqv lv surylghg1

MHO Fodvvlfdwlrq= F55

Nh| zrugv dqg skudvhv= Glvfuhwh Irxulhu wudqvirup/ iudfwlrqdo Eurzqldq prwlrq/ iudf0 wlrqdo lqwhjudwlrq/ orj shulrgrjudp uhjuhvvlrq/ orqj phpru| sdudphwhu/ qrqvwdwlrqdulw|/ vhplsdudphwulf hvwlpdwlrq dqg whvwlqj/ xqlw urrw1

4 Lqwurgxfwlrq

Iudfwlrqdo surfhvvhv duh jurzlqj lq srsxodulw| zlwk hpslulfdo uhvhdufkhuv lq hfrqrplfv dqg qdqfh1 Lq sduw/ wklv lv h{sodlqhg e| wkhlu fdsdflw| wr fdswxuh orqj udqjh fkdudfwhulvwlfv ri hfrqrplf gdwd wkdw hoxgh rwkhu prghov/ d ihdwxuh wkdw lv sduwlfxoduo| lpsruwdqw lq prg0 holqj wkh yrodwlolw| ri qdqfldo dvvhw uhwxuqv1 Lq sduw dovr/ prghov ri iudfwlrqdo lqwhjudwlrq duh dwwudfwlyh wr hpslulfdo uhvhdufkhuv ehfdxvh wkh| surylgh olehudwlrq iurp wkh fodvvlfdo gl0 fkrwrp| ri L+3, dqg L+4, wlph vhulhv dqg dssolfdwlrqv qrz fryhu d udqjh ri glhuhqw wlph vhulhv iurp dvvhw uhwxuqv dqg h{fkdqjh udwhv wr lqwhuhvw udwhv dqg lq dwlrq +vhh Edloolh/ 4<<8/ iru dq ryhuylhz,1 D qdwxudo jrdo lq pdq| ri wkhvh vwxglhv lv wr dvvhvv wkh h{whqw wr zklfk wkh vhulhv xqghu vwxg| pd| ghsduw iurp d vlpsoh xqlw urrw prgho1 Wr pdnh vxfk dq dvvhvvphqw/ dq lqihuhqwldo wkhru| wkdw lqfoxghv wkh xqlw urrw fdvh lv ghvludeoh/ hvshfldoo| rqh wkdw uhwdlqv jhqhudolw| zlwk uhjdug wr wkh vkruw phpru| frpsrqhqw ri wkh vhulhv1 Rqh ri wkh jrdov ri wkh suhvhqw sdshu lv wr surylghv vxfk d wkhru|1

W_{Wkdqnv jr wr wkh QVI iru uhvhdufk vxssruw xqghu Judqw Qr1 VEU <:0635<81 Frpsxwdwlrqv zhuh shuiruphg}

lq JDXVV dqg wkh sdshu zdv w|shg e| wkh dxwkru lq VZ5181

5 Rxu irfxv ri lqwhuhvw lv wkhuhiruh wkh hvwlpdwlrq ri wkh phpru| sdudphwhu cg* ri d iudf0 wlrqdoo| lqwhjudwhg surfhvv [|lq d prgho ri wkh irup

+4  O,_[|@ x|> +4,

zkhuh x| lv vwdwlrqdu| zlwk }hur phdq dqg frqwlqxrxv vshfwudo ghqvlw| i+, A 3= Lq vxfk

prghov/ d frpprqo| xvhg hvwlpdwru lq hpslulfdo zrun lv wkh orj shulrgrjudp +OS, hvwlpdwru +Jhzhnh dqg Sruwhu0Kxgdn/ 4<;6,g rewdlqhg iurp wkh iroorzlqj ohdvw vtxduhv uhjuhvvlrqe

orj +L%+r,, @ef g orje 4  hbr2. uhvlgxdo +5,

wdnhq ryhu ixqgdphqwdo iuhtxhqflhvqr@ 2Zr_? = v @ 4> ===> p

r iru vrph p ? q= Khuh e g @ 4₅ S₆ r'_S{rorj L%+r, 6 r'{2r > +6,

zkhuh L%+r, @ z%+r,z%+r,W lv wkh shulrgrjudp dqg z%+r, lv wkh glvfuhwh Irxulhu wudqv0

irup +giw,/ z%+r, @ I_2Z? S?|'[|h|br ri wkh wlph vhulhv [|= Wkh uhjuhvvlrq +5, lv prwlydwhg

e| wkh irup ri wkh orj vshfwuxp ri [| dqg kdv dsshdo ehfdxvh ri lwv qrqsdudphwulf wuhdwphqw

ri x| dqg wkh frqyhqlhqfh ri olqhdu ohdvw vtxduhv1 Xqghu Jdxvvldq dvvxpswlrqv dqg lq wkh

vwdwlrqdu| fdvh/ zkhuh g 5 +

2>2,> Urelqvrq +4<<8, ghyhorshg frqvlvwhqf| dqg dv|pswrwlf

qrupdolw| uhvxowv iru d yhuvlrq ri g zklfk wulpv rxw orz iuhtxhqflhv shulrgrjudp ruglqdwhve +l1h1 wdnhv v  o> iru vrph p A o A 4,/ dv vxjjhvwhg e| N qvfk +4<;9,1 Kxuylfk dqg Ehowudr +4<<6, kdyh ghyhorshg gdwd0gulyhq fulwhuld iru wkh vhohfwlrq ri p> dqg Kxuylfk/ Ghr/ dqg Eurgvn| +4<<;, h{whqg Urelqvrq*v +4<<8, uhvxowv wr lqfoxgh orz iuhtxhqflhv ruglqdwhv dqg qg dq rswlpdo fkrlfh ri wkh qxpehu ri shulrgrjudp ruglqdwhv lq wkh uhjuhvvlrq1 Wklv zrun surylghv d irxqgdwlrq ri dv|pswrwlf wkhru| ydolgdwlqj +5, iru xvh zlwk Jdxvvldq gdwd lq wkh vwdwlrqdu| fdvh1 Krzhyhu/ wkhuh lv suhvhqwo| qr wkhru| iru wkh xqlw urrw fdvh zkhuh g @ 4 lq +4,1

Wkh suhvhqw sdshu vwxglhv OS uhjuhvvlrq lq wklv xqlw urrw fdvh1 Wklv fdvh lv ri lqwhuhvw iru vhyhudo uhdvrqv1 Iluvw/ wkh xqlw urrw prgho lv dq lpsruwdqw vshfldo fdvh ri +4, zklfk kdv uhfhlyhg vxevwdqwldo dwwhqwlrq lq lwvhoi/ exw zklfk lv suhvhqwo| qrw fryhuhg e| wkh h{lvwlqj wkhru| ri vhplsdudphwulf hvwlpdwlrq ri g= Wkh fdvh lv sduwlfxoduo| lpsruwdqw lq hfrqrplf dssolfdwlrqv/ zkhuh/ dv lqglfdwhg hduolhu/ rqh lv riwhq frqfhuqhg derxw dvvhvvlqj wkh h{whqw ri wkh ghsduwxuh iurp d vlpsoh xqlw urrw prgho1 Wkh lqwhqvlyh lqwhuhvw lq whvwlqj wkh xqlw urrw k|srwkhvlv lq wkh hfrqrphwulf olwhudwxuh kdv irfxvvhg rq whvwv djdlqvw vwdwlrqdu| dowhuqdwlyhv zlwklq wkh frqwh{w ri dxwruhjuhvvlyh prghov/ udwkhu wkdq whvwv djdlqvw qrqvwdwlrqdu| iudfwlrqdo dowhuqdwlyhv1 Vhfrqg/ lw lv zhoo nqrzq wkdw wkh fruuhvsrqglqj vhplsdudphwulf xqlw urrw olplw wkhru| lv qrqvwdqgdug +Skloolsv/ 4<;:, dqg lw lv ri lqwhuhvw wr glvfryhu zkhwkhu wkhuh duh dq| xqxvxdo ihdwxuhv wr wkh olplw wkhru| lq wkh vhplsdudphwulf hvwlpdwlrq ri g zkhq g @ 4= Wklug/ lw kdv uhfhqwo| ehhq vkrzq e| Nlp dqg Skloolsv +4<<<d, wkdw g lv lqfrqvlvwhqw zkhq g A 4=e Vr g @ 4 wxuqv rxw wr eh wkh erxqgdu| fdvh iru frqvlvwhqw hvwlpdwlrq e| orj shulrgrjudp

6 uhjuhvvlrq1 Pruhryhu/ Nlp dqg Skloolsv vkrz wkdwg $e R 4 zkhq g A 4> dqg wklv dv|pswrwlf

eldv lq hvwlpdwlrq lv fhuwdlq wr eldv lqihuhqfh derxw ghsduwxuhv iurp xqlw urrw prgho li lw lv qrw fruuhfwhg1

Wkh zrun uhsruwhg lq wklv sdshu frpsohphqwv vrph rwkhu uhfhqw uhvhdufk rq OS uhjuhvvlrq dqg whvwlqj lq wkh qrqvwdwlrqdu| fdvh1 Kxuylfk dqg Ud| +4<<8, orrnhg dw wkh ehkdylru ri shul0 rgrjudp ruglqdwhv ri d iudfwlrqdoo| lqwhjudwhg surfhvv zlwk phpru| sdudphwhu g 5 ^3=8> 4=8, dqg irxqg hylghqfh ri eldv lq orj shulrgrjudp uhjuhvvlrq zkhq g A 4= Yhodvfr +4<<<d, vkrzhg frqvlvwhqf| ri dq OS hvwlpdwru wkdw wulpv rxw orz iuhtxhqf| ruglqdwhv/ xqghu Jdxvvldq dv0 vxpswlrqv dqg iru 

2 ? g ? 41 Nlp dqg Skloolsv +4<<<d, vkrzhg orj shulrgrjudp uhjuhvvlrq lv

frqvlvwhqw iru 

2 ? g ? 4 zlwkrxw uhtxlulqj Jdxvvldqlw| ru wulpplqj/ dv zhoo dv ghprqvwudwlqj

lqfrqvlvwhqf| iru g A 4= Vrph vlpxodwlrq uhvxowv fryhulqj qrqvwdwlrqdu| fdvhv zhuh uhsruwhg lq Kxuylfk dqg Ud| +4<<8, dqg Yhodvfr +4<<<e,/ erwk uhyhdolqj hylghqfh ri hvwlpdwlrq eldv zkhq g A 4= Ilqdoo|/ Urelqvrq +4<<7, dqg Wdqdnd +4<<<, vkrz krz wr xvh Odjudqjh pxowlsolhu dqg Zdog wkhru| iru whvwlqj ydoxhv ri g lq sdudphwulf prghov wkdw lqfoxgh erwk vwdwlrqdu| dqg qrq vwdwlrqdu| fdvhv/ exw h{foxgh zhdn qrqsdudphwulf ghshqghqfh> dqg Urelqvrq*v +4<<7, whvwv zhuh dssolhg e| Jlo0Dodôd dqg Urelqvrq +4<<:, wr wkh h{whqghg yhuvlrq ri wkh Qhovrq dqg Sorvvhu +4<;5, gdwd vhw1

Wkh dssurdfk lq wkh suhvhqw sdshu gudzv rq dq h{dfw uhsuhvhqwdwlrq ri wkh glvfuhwh Irxulhu wudqvirup +giw, lq wkh xqlw urrw fdvh1 Wklv uhsuhvhqwdwlrq zdv ghyhorshg e| wkh dxwkru lq rwkhu zrun +Skloolsv/ 4<<</ dqg Fruedh/ Rxoldulv dqg Skloolsv/ 4<<<, dqg wkh dvshfwv ri wkh jhqhudo wkhru| jlyhq lq Skloolsv +4<<<, wkdw zh qhhg khuh duh uhylhzhg eulh | lq Vhfwlrq 51 Vhfwlrq 6 jlyhv wkh olplw wkhru| iru g lq wkh xqlw urrw fdvh1 Jdxvvldqlw| lv qrw dvvxphge dqg wkh dssurdfk wr hvwdeolvklqj d fhqwudo olplw wkhruhp iru olqhdu frpelqdwlrqv ri qrqolqhdu ixqfwlrqv ri shulrgrjudp ruglqdwhv lv edvhg rq d qhz hpehgglqj dujxphqw wkdw uholhv rq d vwurqj dssur{lpdwlrq iru sduwldo vxpv ri olqhdu surfhvvhv/ zklfk lv jlyhq lq wkh Dsshqgl{ +Ohppd G, wrjhwkhu zlwk rwkhu whfkqlfdo uhvxowv wkdw duh qhhghg1 Wklv dssurdfk wr d FOW iru olqhdu frpelqdwlrqv ri glvfuhwh Irxulhu wudqvirupv lv xvhixo rxwvlgh wkh suhvhqw frqwh{w ri orj shulrgrjudp uhjuhvvlrq dqg lv ri vrph lqghshqghqw lqwhuhvw1 Vrph dgglwlrqdo wkhru| dqg lpsolfdwlrqv iru whvwlqj duh glvfxvvhg lq Vhfwlrq 71 Frqfoxglqj uhpdunv duh pdgh lq Vhfwlrq 8 dqg surriv duh jlyhq lq wkh Dsshqgl{ lq Vhfwlrq 91

5 Suholplqdulhv

Wklv vhfwlrq jlyhv h{solflw dvvxpswlrqv iru rxu ghyhorsphqw dqg eulh | uhylhzv vrph uhsuh0 vhqwdwlrqv ri wkh giw ri d iudfwlrqdoo| lqwhjudwhg wlph vhulhv jlyhq lq Skloolsv +4<<<,1 Zkloh wkhvh uhsuhvhqwdwlrqv duh ydolg lq erwk vwdwlrqdu| dqg qrqvwdwlrqdu| fdvhv/ wkh| zloo eh xvhg khuh rqo| lq wkh xqlw urrw fdvh/ zkhuh wkh| wdnh dq hvshfldoo| vlpsoh irup1

Wkh iudfwlrqdoo| lqwhjudwhg surfhvv [| lv ghqhg dv lq +4,/ zlwk x @ 3 iru doo m  3=

7 Dvvxpswlrq D Iru doo w A 3> x| kdv Zrog uhsuhvhqwdwlrq

x|@ F +O, %|@ " [ 'ff%|3> " [ 'fm mfm ? 4> F+4, 9@ 3> +7, zlwk %|@ llg3> 2 zlwk H+m%|m^, ? 4> iru vrph t A 7= Wkh t_{wk prphqw frqglwlrq rq wkh huuruv %}

| lv xvhixo lq wkh hpehgglqj dujxphqw xvhg

lq wkh surri ri rxu pdlq wkhruhp dqg lq ohppdv G dqg H ri Vhfwlrq 81 Wkh olqhdu surfhvv irupxodwlrq dqg vxppdelolw| frqglwlrq lq +7, fryhuv d zlgh fodvv ri vkruw phpru| surfhvvhv dqg/ dv lq Skloolsv +4<<<,/ hqdeohv xv wr xvh d ghfrpsrvlwlrq whfkqltxh wr ghyhors d frqyhqlhqw uhsuhvhqwdwlrq ri wkh giw ri d iudfwlrqdoo| lqwhjudwhg surfhvv1 Lq sduwlfxodu/ iurp wkhruhp 515 ri Skloolsv +4<<<, zh kdyh z+, @ z%+, G?  hb> g.s4 5q  h [bf+g,  h?b[hb?+g,  > +8, zkhuh h [b?+g, @Gh?b  h3bO> g[?@ ?3_[ R'f h gbRh3Rb[?3R= +9,

Zkhq x|@ 3 iru w  3/ dv lv dvvxphg deryh/ [|@ 3 iru w  3 dqg/ khqfh/[hbf+g, @ 3= Lq wklv

fdvh/ +8, ehfrphv z+, @ z%+, G?  hb_{> g}_sh?b 5qGh?b  h3b_{O> g}_[ ? @ z%+, G?  hb> gs4 5qh?b[hb?+g,= +:,

Htxdwlrq +:, vkrzv wkdw wkh h{dfw uhodwlrq ehwzhhq z%+, dqg z+, lqyroyhv d fruuhfwlrq

whup wkdw ghshqgv rq [hb?+g,= Zkhq g @ 4> zh kdyh wkh vlpsolfdwlrq [hb?+4, @ hb[?>

dqg +8, ehfrphv z+, @  4  hb_z %+, . h b s 5q  h?b_[ ? [f  > +;, ru vlpso| z+, @  4  hb_z %+, . h b s 5qh?b[?>

zkhq [f @ 3> d uhvxow jlyhq hduolhu lq Fruedh/ Rxoldulv dqg Skloolsv +4<<<,1 Uhzulwlqj wklv

odvw htxdwlrq dv z%+, @ _{4  h}z+,_b h E?nb 4  hb s[_5q? +<, @ _{4  h}4 _bz+,  hE?nbs[? 5q  > +43,

8 lw lv lpphgldwho| dssduhqw wkdw erwk frpsrqhqwv zloo lq xhqfh wkh dv|pswrwlf ehkdylru ri wkh gdwd giw z%+, zkhq g @ 4=

Wkh uhsuhvhqwdwlrq +8, vkrzv wkdw wkh giw ri d iudfwlrqdoo| lqwhjudwhg surfhvv frpsulvhv wzr glvwlqfw frpsrqhqwv1 Wkh uvw ri wkhvh lv wkh giw ri wkh lqqrydwlrqv x|> vfdohg e| wkh

wudqvihu ixqfwlrq ri wkh glhuhqflqj owhu/ G?



hb_{> g}_{= Wkh vhfrqg lqyroyhv d zhljkwhg vlqx0}

vrlgdo vxp/ [hb?+g,/ ri wkh revhuydwlrqv [|= Zkhq g @ 4/ erwk frpsrqhqwv vlpsoli|1 Wkh

wudqvihu ixqfwlrq lv wkhq vlpso| G?



hb_{> 4} _{@ 4  h}b _{dqg wkh vlqxvrlgdo vxp +9, ehfrphv}

h

[b?+4, @ hb[?> zklfk ghshqgv rqo| rq wkh qdo vdpsoh revhuydwlrq1

Vlqfh wkh olplw ehkdylru ri wkhvh wzr frpsrqhqwv lv udwkhu hdvlo| rewdlqhg lq wkh xqlw urrw fdvh/ zh fdq h{shfw wr ghyhors giw dv|pswrwlfv iru lqwhjudwhg surfhvvhv e| dqdo|}lqj wkh wzr whupv lq +<,/ udwkhu wkdq e| dwwhpswlqj wr zrun gluhfwo| zlwk wkh giw ri [|lwvhoi1 Vrph

uhvxowv dorqj wkhvh olqhv duh suhvhqwhg lq Skloolsv +4<<<, dqg Fruedh/ Rxoldulv dqg Skloolsv +4<<<,1 Pruhryhu/ wkh uhsuhvhqwdwlrqv iroorz e| dojheudlf vlpsolfdwlrq dqg vr wkhvh/ dqg uhvxowv rewdlqhg iurp wkhp/ wxuq rxw qrw wr ghshqg xsrq glvwulexwlrqdo vshflfdwlrqv olnh Jdxvvldqlw|1

Dw wkh ixqgdphqwdo iuhtxhqflhv r@ 2Zr_? > v @ 4> ===> q> +43, ehfrphv

z%+r, @ _{4  h}4_br  z+r,  hbrs[? 5q  = +44, Wkh frpsrqhqw +5q,3*2_{[? lq +44, lv lqghshqghqw ri iuhtxhqf| dqg lq xhqfhv wkh dv|ps0}

wrwlf ehkdylru ri z%+r, iru doo ydoxhv ri v1 Wklv surshuw| phdqv wkdw z%+r, lv vsdwldoo|

fruuhodwhg dfurvv doo wkh ixqgdphqwdo iuhtxhqflhv1 Lq hhfw/ wkhuh lv ohdndjh dfurvv doo wkh ixqgdphqwdo iuhtxhqflhv iurp wkh }hur iuhtxhqf| +v @ 3,/ l1h1 iurp z+f, @ +5q,3*2[?=

D ixuwkhu frpsolfdwlqj idfwru lq orj shulrgrjudp uhjuhvvlrq lv wkdw rqh qhhgv wr zrun zlwk d orjdulwkplf ixqfwlrq ri wkh shulrgrjudp ruglqdwhv1 Lq hhfw/ wkh prgho xqghuo|lqj wkh hpslulfdo uhjuhvvlrq +5, lqyroyhv wkh orjdulwkp ri wkh vtxduhg prgxoxv ri +44,/ l1h1/

orj mz%+r,m2 @ 5 orj4  hbr. orjz+r,  hbrs[?

5q 

2= +45,

Wkh vhfrqg whup lq +45, lqyroyhv wkh giw ri wkh vwdwlrqdu| huuruv/ z+r, > frxsohg zlwk wkh

ohdndjh idfwru z+f, @ +5q,3*2[?iurp v @ 3= Wkh dv|pswrwlf dqdo|vlv ri orj shulrgrjudp

uhjuhvvlrq uhtxluhv wkdw zh wuhdw erwk wkh qrqolqhdulw| lq +45, dqg wkh vsdwldo fruuhodwlrq dulvlqj iurp wklv ohdndjh1

6 Orj0Shulrgrjudp Uhjuhvvlrq lq wkh Xqlw Urrw Fdvh

Wkh dssurdfk zh dgrsw lq uhvroylqj wkh whfkqlfdo gl!fxowlhv mxvw phqwlrqhg lv wr h{sdqg wkh suredelolw| vsdfh lq zklfk wkh surfhvvhv duh ghqhg lq vxfk d zd| wkdw wkh gdwd fdq eh uhsuhvhqwhg doprvw vxuho| dqg xs wr d qhjoljleoh huuru lq whupv ri d Eurzqldq prwlrq wkdw

9 lv ghqhg rq wkh vdph vsdfh1 Wkh dujxphqw lqyrnhv dq doprvw vxuh lqyduldqfh sulqflsoh dqg hpehgglqj1

Pruh vshflfdoo|/ ghqh wkh sduwldo vxp surfhvv V& @S&'x iru n  4> dqg Vf@ 3> iru

n @ 31 Vlqfh x kdv qlwh prphqwv ri rughu t A 5s A 7> zh fdq h{sdqg wkh suredelolw| vsdfh

dv qhfhvvdu| wr vhw xs d sduwldo vxp surfhvv wkdw lv glvwulexwlrqdoo| htxlydohqw wr V& dqg d

Eurzqldq prwlrq E+, zlwk yduldqfh 5i+3, rq wkh vdph vsdfh iru zklfk

vxs

f$&$?mV&E+n,m @ r@r+q



R,> +46,

jlylqj d xqlirup dssur{lpdwlrq wr V& ryhu 3  n  q lq whupv ri wkh Eurzqldq prwlrq

E= Lq +46,/ s vdwlvhv t A 5s A 7> vr wkdw s A 5= Zkhq t dqg s duh odujh/ wkh huuru rughu ri pdjqlwxgh lq wklv dssur{lpdwlrq ehfrphv vpdoo dqg lw lv erxqghg/ ru R@r+4,> zkhq

V& lv Jdxvvldq1 Zkhq wkh frpsrqhqwv x lq V& duh lqghshqghqw/ doprvw vxuh lqyduldqfh

sulqflsohv ru vwurqj dssur{lpdwlrqv ri wkh w|sh +46, kdyh ehhq suryhg e| pdq| dxwkruv xvlqj d ydulhw| ri whfkqltxhv/ d srsxodu uhfhqw dssurdfk ehlqj wkh Kxqjduldq frqvwuxfwlrq/ h1j1 Vkrudfn dqg Zhooqhu +4<;9, dqg Fvùujø/ P1 dqg O1 Kruyäwk +4<<6,1 D vwurqj dssur{lpdwlrq uhvxow mxvwli|lqj wkh uhsuhvhqwdwlrq +46, lq wkh fdvh zkhuh x| lv d olqhdu surfhvvhv vdwlvi|lqj

Dvvxpswlrq D lv jlyhq lq ohppd G lq wkh Dsshqgl{1 Vhwwlqj V?& @ q3*2 &'x> zh fdq

zulwh wklv dssur{lpdwlrq lq wkh irup vxs f$&$?  V?& E+n_q,@ r@r # 4 q23R $ = +47,

Dv vkrzq lq +69, dqg +6:, lq wkh Dsshqgl{/ wkh dssur{lpdwlrq +47, ohdgv wr wkh vlpxowdqhrxv uhsuhvhqwdwlrq ri wkh giw z+r, @ s4 5 ] _ f h 2Zro_{gE +u, . R} @r _p q  . r@r # 4 q23R $ @ r. R@r _p q  . r@r # 4 q23R $ > +48,

zlwk wkh huuru pdjqlwxgh kroglqj xqlirupo| ryhu v @ 4> ===> p> dqg wkh uhsuhvhqwdwlrq [? s 5q @ 4 s 5E +4, . r@r # 4 q23R $ @  . .r@r # 4 q23R $ = +49,

Wkh yduldeohv +r,6r' dqg  wkdw dsshdu lq +48 dqg +49, duh Jdxvvldq dqg lqghshqghqw1

Wkh vlpsoh irup ri +48, dqg +49, hqdeohv xv wr ghyhors d olplw wkhru| iru iuhtxhqf| dyhudjhv ri wkh vhfrqg whup lq +45,1 Orj shulrgrjudp dv|pswrwlfv iroorz gluhfwo|1 Wkh rxwfrph lv wkh iroorzlqj uhvxow/ zklfk jlyhv wkh dv|pswrwlf glvwulexwlrq rig zkhq g @ 4=e 614 Wkhruhp Ohw [| iroorz +4, zlwk g @ 4 dqg x| vdwlvi| +7,= Li

p2

q .

s p

: wkhq _s pg  ge $_PQ  3>4₇2+Z ,] " f Q  3>4₇2+Z ,sgi +Z , gZ +4;, Khuh/ Z lv "2  zlwk sgi +Z , @ ^52+2,`3h3`*2Z3  2 dqg 2_{+Z, @ h}3` [" 'f Z m$ q # +4 . m,2. #<sub>+4 . m,</sub>r<sub></sub> 5 7<sub>h</sub>3`[" 'f Z m$ # +4 . m, 6 8 2 >

zkhuh #+}, @ _{+},@+}, lv wkh svl ixqfwlrq/ wkh orjdulwkplf ghulydwlyh ri wkh jdppd ixqfwlrq}

> dqg #+}, lv wkh wuljdppd ixqfwlrq/ lwv uvw ghulydwlyh1

Frqglwlrq +4:, rq wkh iuhtxhqf| edqg ir> 4 ? v ? pj uhvwulfwv wkh udqjh ri wkh hhfwlyh

vdpsoh vl}h p +wkh qxpehu ri shulrgrjudp ruglqdwhv, lq wkh orj shulrgrjudp uhjuhvvlrq +5,1 Wkh frqglwlrq krogv zkhq p @ R+q23B, iru dq|  A 3/ zkhq s A 9=

Wkh yduldqfh ri wkh olplw glvwulexwlrq +4;, lv 427 wlphv 4 52+ 2, ] _" f 3 E Ch3` [" 'f Z m$ q # +4 . m,2. #+4 . m,r 5 7<sub>h</sub>3`[" 'f Z m$ # +4 . m, 6 8 24 F Dh3`*2Z32gZ @ s4 5 " [ 'f q # +4 . m,2. #<sub>+4 . m,</sub>r m$ ] <sub>"</sub> f h 3 2`Z32gZ s4 5 " [ 'f " [ &'f # +4 . m, # +4 . n, m$n$ ] _" f h 3D 2`Zn&32gZ @ s4 5 " [ 'f q # +4 . m,2. #_{+4 . m,}r m$ m . 2    2 _n 2 s4 5 " [ 'f " [ &'f # +4 . m, # +4 . n, m$n$ m . n .  2   D 2 _n&n 2 @ ₉2 .s4 6 " [ 'f k # +4 . m,2S_&'  &2 l   2    2  _ m$ s4 8 " [ 'f " [ &'f # +4 . m, # +4 . n, m$n$ ₄ 5  n& ₅ 8 _n& +4<, vlqfh 4 s 5 " [ 'f # +4 . m,2 m$ m . 2    2 _n 2 @ 4 s 6 " [ 'f # +4 . m,2 2    2  _ m$ > #+4 . m, @ ₉2 [ &' 4 n2> +53,

;

Iljxuh 4= Ghqvlwlhv ri orj shulrgrjudp hvwlpdwru= q @ 533> p @ ^qfSD_`=

dqg # +4 . m, @ + F .S_&'  & m  4 F m @ 3 F @ 3=8::548 +Hxohu*v frqvwdqw,= +54,

zkhuh +53, dqg +54, duh vwdqgdug 0 h1j1 Judgvkwh|q dqg U|}kln +4<98/ ;1698 ) ;1699,1 Wkh vxppdwlrq lq wkh wklug whup ri +4<, fdq eh zulwwhq dv

" [ 'f " [ &'f # +4 . m, # +4 . n, m$n$ ₄ 5  n& ₅ 8 _n& @ [" 'f # +4 . m, 2    2 D _ m$ " [ &'f # +4 . n, n$ ₄ 5. m  & ₅ 8 _& = Khqfh/ wkh yduldqfh ri wkh olplw glvwulexwlrq rispg  ge lv

2 _ @  2 57 .7s46 " [ 'f k # +4 . m,2S_{&' &}2 l   2    2  _ m$  4 7s8 " [ 'f # +4 . m, 2    2 D _ m$ " [ &'f # +4 . n, n$ ₄ 5 . m  & ₅ 8 _& =

< Qxphulfdo hydoxdwlrq jlyhv 2

_ @ 3=6<7;> zklfk lv voljkwo| vpdoohu wkdq 2@57 @ 3=7445> wkh

olplwlqj yduldqfh ri spg  ge  lq wkh vwdwlrqdu| fdvh1 Wkxv/ wkh olplw glvwulexwlrq ri wkh orj shulrgrjudp hvwlpdwru lq wkh xqlw urrw fdvh/ dowkrxjk pl{hg qrupdo/ kdv ohvv glvshuvlrq wkdq lq wkh vwdwlrqdu| fdvh1 Wklv lv frquphg lq qlwh vdpsoh vlpxodwlrqv1 Iljxuh 4 vkrzv wkh vdpsolqj glvwulexwlrqv ri wkh orj shulrgrjudp hvwlpdwru +edvhg rq 43/333 uhsolfdwlrqv, iru vhyhudo ydoxhv ri g zkhq q @ 533 dqg p @qfSD_{1 Wkh uhgxfwlrq lq wkh yduldqfh zkhq}

g @ 4 lq frpsdulvrq zlwk g @ 3=7 lv dssduhqw/ dv lv wkh grzqzdug eldv wrzdugv xqlw| zkhq g A 4= Wkh odwwhu lv v|vwhpdwlfdoo| h{soruhg lq Nlp dqg Skloolsv +4<<<d,1

7 Whvwlqj d Xqlw Urrw djdlqvw Iudfwlrqdo Dowhuqdwlyhv

Wkhruhp 614 vkrzv wkdw wkh frqyhqwlrqdo OS uhjuhvvlrq wkhru|/ yl}1 wkh olplw glvwulexwlrq Q3>Z2

2e



iru spg  ge  wkdw lv nqrzq wr dsso| lq wkh vwdwlrqdu| fdvh +Urelqvrq/ 4<<8/ Kxuylfk hw do/ 4<<;,/ lv lqydolg dw g @ 4= Qhyhuwkhohvv/ wkh olplw wkhru| lq wkhruhp 614 frxog hdvlo| eh xvhg iru whvwlqj g @ 4 vlpso| e| dgmxvwlqj fulwlfdo ydoxhv wr frqirup zlwk wkh fruuhfw pl{hg qrupdo olplw glvwulexwlrq jlyhq lq +4;,1 Vxfk fulwlfdo ydoxhv frxog uhdglo| eh frpsxwhg e| qxphulfdo phwkrgv1 Wkh uhvxowlqj whvw zrxog lqyroyh wkh frpsxwdwlrq rispg  4e dqg fdoleudwlrq djdlqvw wkh fulwlfdo ydoxhv iurp wkh olplw glvwulexwlrq +4;,1 Vlqfh lw lv nqrzq +Nlp dqg Skloolsv/ 4<<<d, wkdw OS uhjuhvvlrq lv frqvlvwhqw iru ₂ ? g ? 4/ vxfk d whvw zrxog eh frqvlvwhqw djdlqvw vxfk dowhuqdwlyhv1 Krzhyhu/ lw lv dovr nqrzq +Nlp dqg Skloolsv/ 4<<<d, wkdw OS uhjuhvvlrq lv lqfrqvlvwhqw zkhq g A 4> dqg lqghhg g $e R 4 iru doo g A 4= Wkxv/ wkh

whvw zrxog eh lqfrqvlvwhqw djdlqvw dowhuqdwlyhv g A 4=

Wkh wkhru| lq Vhfwlrqv 5 dqg 6 jlyhv xv d vlpsoh zd| ri frqvwuxfwlqj d ydolg whvw1 Zh qhhg dq hvwlpdwru ri g wkdw lv frqvlvwhqw iru erwk g  4 dqg g A 4 wr dffrpsolvk wklv1 Frqvlvwhqw hvwlpdwlrq ri g durxqg xqlw| lv srvvleoh e| xvlqj d qhz hvwlpdwru fdoohg prglhg orj shulrgrjudp uhjuhvvlrq vxjjhvwhg lq Skloolsv +4<<<,1 Khuh/ wkh hvwlpdwru fdq eh rewdlqhg txlwh vlpso| iurp +<,/ zklfk dw wkh ixqgdwphqwdo iuhtxhqflhv ir = v @ 4> ===> pj lv

z%+r, @ _{4  h}z+_br,_r  h br 4  hbr [? s 5q= +55, Zh frqvwuxfw wkh prglhg giw y%+r, @ z%+r, . h br 4  hbr [? s 5q> +56,

dqg dvvrfldwhg shulrgrjudp ruglqdwhv L+r, @ y%+r, y%+r,W= Revhuyh wkdw y%+r, dqg

L+r, duh erwk revhuydeoh ixqfwlrqv ri wkh gdwd1 Zlwk wklv vlpsoh gdwd wudqvirupdwlrq zh

kdyh wkh iroorzlqj h{dfw prgho

orj L+r, @ f  5 orj4  hbr. orjL_i+r, +3, >

43 zlwk f @ orj i+3, = Prglhg OS uhjuhvvlrq lv vlpso| d ohdvw vtxduhv uhjuhvvlrq ri orj L+r,

rq orj4  hbr_{= Vhw dr @ orj}_{4  h}br_{ dqg {r @ dr d> zlwk d @ p}3S6 r'dr= Wkhq/ wkh prglhg OS hvwlpdwru lv vlpso| h g @ 4 5 S₆ r'_S{rorj L+r, 6 r'{2r = +57,

Wkh olplw glvwulexwlrq ri wklv hvwlpdwru zkhq g @ 4 lv jlyhq lq wkh iroorzlqj uhvxow1 714 Wkhruhp Ohw [| iroorz +4, zlwk g @ 4 dqg x| vdwlvi| +7,= Li

p2 q . s p q23R $ 3> wkhq s pg  gh $_Q # 3>₅₇2 $ = +58,

Wkxv/g kdv wkh vdph olplw glvwulexwlrq dw g @ 4 dv wkh zhoo nqrzq olplw glvwulexwlrq ri wkhh OS hvwlpdwru lq wkh vwdwlrqdu| fdvh +Urelqvrq/ 4<<8/ dqg Kxuylfk hw do1/ 4<<;,1 Lq idfw/ Nlp dqg Skloolsv +4<<<e, vkrz wkdw wkh olplw wkhru| +58, iru g dssolhv iru ydoxhv ri g ryhu wkhh udqjh 3 ? g ? 5= Lq sduwlfxodu/ g lv frqvlvwhqw iru ydoxhv ri g durxqg xqlw| dqg qrw htxdoh wr xqlw|1 Wkxv/ d vhplsdudphwulf whvw iru d xqlw urrw djdlqvw iudfwlrqdo dowhuqdwlyhv fdq eh edvhg vlpso| rq wkh vwdwlvwlf

]_@

s

pg  4h  @s57 >

zlwk fulwlfdo ydoxhv rewdlqhg iurp wkh vwdqgdug qrupdo glvwulexwlrq1 Wklv whvw lv frqvlvwhqw djdlqvw erwk g ? 4 dqg g A 4 iudfwlrqdo dowhuqdwlyhv1

8 Frqfoxglqj Uhpdunv

Orj shulrgrjudp uhjuhvvlrq lv vkrzq wr eh frqvlvwhqw dqg wr kdyh d pl{hg qrupdo olplw glvwulexwlrq zkhq g @ 4= Wklv fdvh wxuqv rxw wr ghqh wkh erxqgdu| iru frqvlvwhqw hvwlpdwlrq ri g e| orj shulrgrjudp uhjuhvvlrq1 Iru odujhu ydoxhv ri g> lw lv nqrzq wkdwg $e R 4 +Nlp dqg

Skloolsv/ 4<<<d,1 Wkxv/ OS uhjuhvvlrq hqfrxqwhuv qrqvwdqgdug olplw wkhru| dw wkh xqlw urrw erxqgdu|/ mxvw olnh wkh vhuldo fruuhodwlrq frh!flhqw/ exw ri d glhuhqw irup1

Lq ylhz ri wkh lqfrqvlvwhqf| ri OS uhjuhvvlrq zkhq wkh phpru| sdudphwhu g A 4> wkh OS hvwlpdwrug lv xqvxlwdeoh iru whvwlqj d xqlw urrw djdlqvw wzr0vlghg iudfwlrqdo dowhuqdwlyhv1e Iruwxqdwho|/ wkhuh lv d vlpsoh dowhuqdwlyh surfhgxuh edvhg rq prglhg OS uhjuhvvlrq zklfk surgxfhv frqvlvwhqw hvwlpdwhv ri g iru g A 4 dqg iru g  41 Whvwv iru d xqlw urrw edvhg rq wkh prglhg OS hvwlpdwru duh hdv| wr frqvwuxfw/ lqyroyh rqo| wkh vwdqgdug qrupdo glvwulexwlrq dqg dsso| xqghu qrqsdudphwulf zhdn ghshqghqfh iru wkh vkruw phpru| frpsrqhqw1 Wkh| wkhuhiruh vhhp vxlwdeoh iru jhqhudo lpsohphqwdwlrq lq hpslulfdo uhvhdufk1 Dv lq wkh fdvh ri

44 frqyhqwlrqdo xqlw urrw whvwv/ zkhq wkhuh duh ghwhuplqlvwlf wuhqgv lq wkh prgho/ lw lv ghvludeoh wr h{wudfw wkh wuhqgv xqghu wkh qxoo dv lq Vfkplgw dqg Skloolsv +4<<5, ehiruh lpsohphqwlqj wkh ]_whvw jlyhq khuh1 Dgglwlrqdoo|/ lvvxhv uhodwlqj wr qglqj dq crswlpdo* fkrlfh ri wkh qxpehu

ri ruglqdwhv p lq wkh frqvwuxfwlrq ri wkh hvwlpdwrug/ dv zhoo dv ghyhorslqj gdwd0ghwhuplqhgh phwkrgv ri fkrrvlqj p duh vwloo wr eh frqvlghuhg1

9 Dsshqgl{

Wkh uvw wzr ohppdv uhsruw prphqw h{suhvvlrqv zklfk fdq eh rewdlqhg vwudljkwiruzdugo| iurp nqrzq uhvxowv1 Wkh| duh jlyhq khuh iru frqyhqlhqfh1 Ohppd F jlyhv wkh olplw glvwulewlrq ri d vdpsoh dyhudjh ri qrqolqhdu ixqfwlrqv ri fruuhodwhg Jdxvvldq yduldwhv/ zklfk lv xvhixo lq surylqj wkh pdlq wkhruhp1 Ohppd G jlyhv d vwurqj dssur{lpdwlrq uhvxow iru sduwldo vxpv ri olqhdu surfhvvhv vdwlvi|lqj Dvvxpswlrq D zlwk t A 7= D zhdnhu dssur{lpdwlrq uhvxow wkdw krogv zkhq t A 5 lv jlyhq lq Ohppd H1

914 Ohppd D Li Uh +d, > Uh +, A 3> wkhq ] _" f +orj }, h 3@5_}D3_{g} @}  +, dD ^# +,  orj +d,` > dqg <sub>] "</sub> f +orj }, 2<sub>h</sub>3@5<sub>}</sub>D3<sub>g} @</sub>  +, dD q ^# +,  orj +d,`2. #_+,r_>

zkhuh # +, lv wkh svl ixqfwlrq/ wkh orjdulwkplf ghulydwlyh ri wkh jdppd ixqfwlrq  +, > dqg #+, > wkh uvw ghulydwlyh ri # +,/ lv wkh wuljdppd ixqfwlrq1

915 Surri Vwduw e| glhuhqwldwlqj zlwk uhvshfw wr  wkh htxdwlrq  +,

dD @

] _"

f h

3@5_}D3_g}>

jlylqj wkh uvw uhvxow _+,

dD   +, orj d_dD @  +,_dD ^# +,  orj +d,` @

] _"

f h

3@5_}D3_{orj +}, g}>}

zkhuh # +, lv wkh orjdulwkplf ghulydwlyh ri wkh jdppd ixqfwlrq1 Vxevhtxhqw glhuhqwldwlrq |lhogv wkh vhfrqg uhvxow1  +, dD q ^# +,  orj +d,`2. #+,r@] " f h 3@5_}D3_{+orj },}2_g}=

45 916 Ohppd E Horj"2 D+,  @ h3B*2[" 'f  B 2 _ m$ # _ 5 . m  . orj 5> Yduorj"2 D+,  @ h3B*2[" 'f  B 2 _ m$ + #₅ . m2. # 5 . m ,  5 9 7h3B*2 " [ 'f  B 2 _ m$ # _ 5 . m 6 : 8 2 = 917 Surri Wkh ghqvlw| ri "2 D+, lv sgi +}, @ h3B*235*2 D 2  5D2 " [ 'f _ 7 _{ }}D 23n m$D 2   @ h3B*235*2_}D 23 D 2  5D2 fI _ 5>7}  > zkhuhfI+d> }, @S"'f-E@5= Vr Horj"2 D+,  @ h3B*2 D 2  5D2 " [ 'f  B e _ m$D 2   ] _" f h 35*2_}D 23norj +}, g} @ h3B*2 D₂5D2 " [ 'f  B e _ m$D 2   D₂ . m   2 D 2n  # 5 . m  . orj 5 @ h3B*2[" 'f  B 2 _ m$  #₅ . m. orj 5 @ h3B*2[" 'f  B 2 _ m$ # _ 5 . m  . orj 5= Qh{w Horj"2 D+, ₂ @ h3B*2 D 2  5D2 " [ 'f  B e _ m$D 2   ] _" f h 35*2_}D 23n+orj },2g} @ h3B*2 D₂5D2 " [ 'f  B e _ m$D₂_ D₂ . m   2 D 2n + # 5 . m  . orj +5,2. # 5 . m , @ h3B*2[" 'f  B 2 _ m$ + #₅ . m. orj +5,2. #₅ . m,>

46 dqg wkhq Yduorj"2 D+,  @ Horj"2 D+, ₂ Horj"2 D+, ₂ @ h3B*2[" 'f  B 2 _ m$ + #₅ . m. orj +5,2. #₅ . m,  5 9 7h3B*2 " [ 'f  B 2 _ m$ # _ 5 . m  . orj +5, 6 : 8 2 @ h3B*2[" 'f  B 2 _ m$ + #₅ . m2. #₅ . m,  5 9 7h3B*2[" 'f  B 2 _ m$ # _ 5 . m 6 : 8 2 > jlylqj wkh wzr vwdwhg uhvxowv1

918 Ohppd F Ohw r@ I_2ZUfh2ZrogE +u, dqg  @ I_2ZUfgE +u, > zkhuh E lv Eurzqldq

prwlrq zlwk yduldqfh 5i+3,= Wkhq/ li {r @ dr  d zkhuh dr @ orj4  hbr dqg d @

p3S6 r'dr> 4 s p 6 [ ' {rorj mr m2 $_PQ  3> 2_+Z,_] " f Q  3> 2_+Z,_{sgi +Z , gZ>} zkhuh 2+Z, @ h3` [" 'f Z m$ q # +4 . m,2. #+4 . m,r 5 7<sub>h</sub>3`[" 'f Z m$ # +4 . m, 6 8 2 > dqg Z lv fkl0vtxduhg zlwk rqh ghjuhh ri iuhhgrp dqg sgi +Z , @ ^52+₂,`3h3`*2Z32=

919 Surri Vhw _r@ s4 5 ] _ f h 2Zro_{gE +u, @}t_i +3, ] _ f h 2Zro_{gZ +u, @}t_i +3,]r> vd|/ dqg  @ti+3,\

zkhuh Z lv vwdqgdug Eurzqldq prwlrq/ dqg i]rj6_  llg QS+3> 4, dqg lv lqghshqghqw ri \>

zklfk lv Q+3> 4,= Lw lv frqyhqlhqw wr zulwh ]r @ r . 2rl= Wkh frpsrqhqwv r> 2r lq wklv

ghfrpsrvlwlrq duh lqghshqghqw dqg hdfk lv Q3> 2

 = Wkhq orj m_r m2 @ orj +i+3,, . orj

k

+_r \ ,2. 2 2r

47 @ orj4₅i+3,  . orjk5q+r \ ,2. 22r rl @ orj4₅i+3,  . orj ^Jrt` > vd|1 +59, Frqglwlrqdo rq \> r \ lv Q  \> 2  > dqg vr/ frqglwlrqdo rq \> Jrt @ +r \ , 2_{. }2 2r 4@5 @ 5 q +r \ ,2. 22r r  "22+, =

Wkxv/ frqglwlrqdo rq \ / wkh idplo| iJrtj6_ duh lqghshqghqw dqg lghqwlfdoo| glvwulexwhg qrq0

fhqwudo fkl0vtxduhg yduldwhv zlwk wzr ghjuhhv ri iuhhgrp dqg qrqfhqwudolw| sdudphwhu  zkhuh  @ # \ 4@s5 $₂ @ 5\2₌

Lw iroorzv iurp Ohppd E dqg +54, wkdw H +orj Jrtm\ , @ H  orj"2 2+,  m\@ h3B*2[" 'f  B 2 _ m$ # +4 . m, . orj 5 @ h3B*2[" '  B 2 _ m$ 3 C[ &' 4 n F 4 D_{ h}3B*2_{F . orj 5} @ h3B*2[" '  B 2 _ m$  [ &' 4 n Fh3B*2 F . orj 5 @ t> vd|1 Ixuwkhu/

Ydu +orj Jrtm\ , @ Ydu

 orj"2 2+,  m\ @ h3B*2[" 'f  B 2 _ m$ q # +4 . m,2. #+4 . m,r  5 9 7h3B*2[" 'f  B 2 _ m$ # +4 . m, 6 : 8 2 @ 2t> vd|1

Wkxv/ frqglwlrqdo rq \> orj Jrt lv llg t> 2t= Lw iroorzv wkdw/ frqglwlrqdo rq \>

4 s_p[6 ' {rorj Jrt @ s4_p 6 [ ' {r+orj Jrt  t,

48 vdwlvhv wkh Olqghehuj0Ihoohu fhqwudo olplw wkhruhp +f1i1 Urelqvrq/ 4<<8/ s1 43:3, dqg zh kdyh 4 s p 6 [ '{rorj Jrt   _t $ Q_ 3 C_3>2 t 3 C _olp 6<" 4 p 6 [ '{ 2 r 4 D 4 D _{@ Q} 3 C_3>2 t 3 C _olp 6<" 4 p 6 [ '{ 2 r 4 D 4 D @ Q3> 2t  > vlqfh olp 6<" 4 p 6 [ ' {2 r @ ] _ f +orj {, 2_{g{ }]  f +orj {, g{ 2 @ 4= Xqfrqglwlrqdoo|/ zh wkhuhiruh kdyh

4 s p 6 [ '{rorj Jrt _ $ PQ3> 2t  @ ] " 3"Q  3> 2t  sgi +\ , g\ @ ] " f Q  3> 2+Z ,sgi +Z, gZ> +5:, zkhuh 2+Z, @ h3` [" 'f Z m$ q # +4 . m,2. #+4 . m,r  5 7<sub>h</sub>3`[" 'f Z m$ # +4 . m, 6 8 2 > dqg Z @ \2 _{@ "}2 = Lw iroorzv iurp +6;,/ +59, dqg +5:, wkdw 4 s p 6 [ ' {rorj mr m2 @ s4_p 6 [ ' {rorj Jrt $_ ] _" f Q  3> 2_{+Z ,}_{sgi +Z, gZ>} dv vwdwhg1

91: Ohppd G Ohw V& @ S&'x iru n  4> dqg Vf @ 3> iru n @ 3> zkhuh x vdwlvhv

Dvvxpswlrq D exw zlwk H m%|m^? 4 iru vrph t A 5s A 7= Wkhq/ wkh suredelolw| vsdfh rq zklfk

wkh x dqg V&duh ghqhg fdq eh h{sdqghg lq vxfk d zd| wkdw wkhuh lv d surfhvv glvwulexwlrqdoo|

htxlydohqw wr V& dqg d Eurzqldq prwlrq E+, zlwk yduldqfh 5i+3, rq wkh qhz vsdfh iru

zklfk vxs f$&$?  _sV& qE+qn,  @ r@r+ 4 q23R,= +5;,

49 91; Surri Lq ylhz ri Dvvxpswlrq D/ zh pd| xvh wkh EQ ghfrpsrvlwlrq +vhh Skloolsv dqg Vror/ 4<<5, wr zulwh F +O, @ F +4, .F +O, +O  4,h zkhuhF +O, @h S" 'fhfO zlwk hf @S"r'nfr dqgS"'fmhfm ? 4= Wkhq/ x|@ F +4, %|.h%|3h%|> zlwkh%|@F +O, %h |> dqg V|@ F +4, | [ ' % .h%fh%|@ V#|.h%fh%|> zkhuh V#| @ S|' dqg  @ F +4, %= Qh{w/ vlqfh  lv llg zlwk phdq }hur dqg qlwh

prphqwv ri rughu t A 5s A 7/ zh pd| xvh d vwurqj dssur{lpdwlrq wr wkh sduwldo vxp surfhvv V#|ri 1 Lq sduwlfxodu/ e| d uhvxow ri Nrpoöv/ Pdmru dqg Wxvqäg| +vhh Fvùujr dqg Kruydwk/

4<<6/ s14;, zh fdq h{sdqg wkh suredelolw| vsdfh dv qhfhvvdu| wr vhw xs d sduwldo vxp surfhvv wkdw lv glvwulexwlrqdoo| htxlydohqw wr V#& dqg d Eurzqldq prwlrq E+, zlwk yduldqfh 5i+3,

rq wkh vdph vsdfh iru zklfk

vxs

f$&$?mV#&E+n,m @ r@r+q



^,> +5<,

jlylqj d xqlirup dssur{lpdwlrq wr V#& ryhu 3  n  q lq whupv ri wkh Eurzqldq prwlrq E=

Qh{w/ vlqfh

mV&E+n,m  mV#&E+n,m . mh%fh%&m

zh kdyh vxs f$&$?  _sV& qE+nq,   vxs f$&$?  V_s#& qE+qn,  . 5 vxs f$&$? m_sh%&m q= +63, Qrz vxs f$&$? m_sh%&m q @ r@r # 4 q23R $ +64, krogv li pd{ f$&$? mh%&m qR @ r@r+4, = +65, Exw S % pd{ f$&$? mh%&m qR A  & @ Spd{ f$&$? mh%&m^ q^R A  ^ @ S % 4 q^R ? [ &' mh%&m^4 k mh%&m^ A q ^ R^lA ^ & ? H S_? &'mh%&m^4 k mh%&m^A q ^ R^l qR^^ @ H  mh%&m^4 k mh%&m^ A q ^ R^l q^R3^ >

4: e| vwdwlrqdulw| rih%&= Lw iroorzv wkdw

" [ ?'S % pd{ f$&$? mh%&m qR A  & ? [" ?' H +mh%&m^, q^R3 ? 4

vlqfh t A 5s= Uhvxow +65, wkhq iroorzv e| wkh Eruho Fdqwhool ohppd1 Zh ghgxfh iurp +5<,/ +63, dqg +64,1wkdw vxs f$&$?  _sV& qE+qn,  @ r@r # 4 q23R $ > jlylqj wkh vwdwhg uhvxow1

91< Uhpdun Lw lv dssduhqw iurp wkh deryh surri wkdw li doo zh qhhg lv vxs f$&$?  _sV& qE+nq,  @ rR # 4 q23R $ > +66, wkhq lq sodfh ri +65, wkh iroorzlqj lv vx!flhqw pd{ f$&$? mh%&m qR @ rR+4, = +67, Dssduhqwo|/ +67, krogv li S % pd{ f$&$? mh%&m qR A  & @ Spd{ f$&$? mh%&mR q A R  @ S % 4 q ? [ &' mh%&mR4 ^mh%&mR A qR` A R & ? H + S_?

&'mh%&mR4 ^mh%&mR A qR`,

q^R^

@ H +mh%&mR4 ^m_%hR&mRA qR`, $ 3>

zklfk zloo eh vr zkhq H +mh%&mR, ? 4= E| Plqnrzvnl*v lqhtxdolw|/ zh kdyh

H +mh%&mR, @ H 3 C  " [ 'f h f%&3    R4 D_? 3 C[" 'f+m h fmRH m%&3mR,  R 4 D R @ 3 C[" 'fm h fm 4 D R H m%&mR ? 3 C[" 'fm mfm 4 D R H m%&mR=

Wkxv/ +66, krogv li wkh prphqw frqglwlrq lq Dvvxpswlrq D lv vlpso| H +m%&mR, ? 4 iru dq|

s A 5= Vwdwhg irupdoo|/ wkh uhvxow lv jlyhq dv Ohppd H ehorz1 Dnrqrp +4<<6/ wkhruhp 6, jdyh d vlplodu uhvxow/ xvlqj d glhuhqw phwkrg ri surri1

4; 9143 Ohppd H Ohw V& @ S&'x iru n  4> dqg Vf @ 3> iru n @ 3> zkhuh x vdwlvhv

Dvvxpswlrq D exw zlwk H m%|mR ? 4 iru vrph s A 5= Wkhq/ wkh suredelolw| vsdfh rq zklfk wkh

x dqg V& duh ghqhg fdq eh h{sdqghg lq vxfk d zd| wkdw wkhuh lv d surfhvv glvwulexwlrqdoo|

htxlydohqw wr V& dqg d Eurzqldq prwlrq E+, zlwk yduldqfh 5i+3, rq wkh qhz vsdfh iru

zklfk vxs f$&$?  _sV& qE+ n q,  @ rR+ 4 q23R,=

9144 Surri ri Wkhruhp 614 Vhw dr@ orj4  hbrdqg {r@ drd> zkhuh d @ p3S6r'dr=

Wkhq/ 5g @e S₆ r'{_Srorj mz%+r,m2 6 r'{2r > dqg/ iurp +45,/

orj mz%+r,m2 @ 5 orj4  hbr. orjz+r,  hbrs[?

5q  2 vr wkdw 5spg  4e @  I₆S6_r'{rorj_{z+r, }I 2Z?hbr[?  2  6S6r'{2r = +68,

Qh{w/ zh surfhhg wr qg d pruh frqyhqlhqw uhsuhvhqwdwlrq iru z+r, dqg [?@sq lq +68,1

Xvlqj sduwldo vxppdwlrq/ zh zulwh iru v @ 4> ===> p z+r, @ s4 5q ? [ |'h 2Zr| ? x_|@  # 4 s 5q ? [ |'h 2Zr| ? V_| $  s4 5q ? [ |'V|3  h2Zr|?  @ s4 5qV? 4 s 5q ? [ |' V|3h2ZrE|3?  h2Zr?  4  @ s4 5 V? s q 4 s 5 ? [ |' V_s|3 q h 2ZrE|3 ?  h2Zr?  4  = Lq ylhz ri wkh hpehgglqj +47, V_s|3 q @ E _{w  4} q  . r@r # 4 q23R $ >

zkhuh wkh huuru pdjqlwxgh krogv xqlirupo| lq w @ 4> ==> q= Wkhq/ zh kdyh z+r, @ s4 5 % E +4, . r@r # 4 q23R $&  s4 5 ? [ |' % Ew  4_q . r@r # 4 q23R $& h2ZrE|3? h2Zr?  4 @ s4 5E +4,  4 s 5 ? [ |' % Ew  4_q . r@r # 4 q23R $& h2ZrE|3? h2Zr?  4. r_@r # 4 q23R $ @ s4 5E +4,  5vl s 5 ] _ f % E +u, . r@r # 4 q23R $& h2Zro_gu_{4 . R}p q  . r@r # 4 q23R $

4< @ s4 5E +4,  5vl s 5 ] _ f E +u, h 2Zro_{gu . R} @r _p q  . r@r # 4 q23R $ @ s4 5 ] _ f h 2Zro_{gE +u, . R} @r _p q  . r@r # 4 q23R $ > +69,

zkhuh wkh huuru pdjqlwxgh krogv xqlirupo| lq v  p= Zkhq v @ 3> zh kdyh lq wkh vdph zd| [? s 5q @ 4 s 5E +4, . r@r # 4 q23R $ = +6:, Qrz xvh wkh qrwdwlrq r @ s4₅ ] _ f h 2Zro_{gE +u, >  @ s}4 5E +4, =

Wkh yduldwhv irj6r' duh lqghshqghqw frpsoh{ Jdxvvldq QS+3> i+3,, dqg duh lqghshqghqw

ri > zklfk lv uhdo Jdxvvldq Q+3> i+3,,= Zulwlqj z+r, @ r. R@r _p q  . r@r # 4 q23R $ > dqg [? s 5q @  . r@r # 4 q23R $ >

dqg xvlqj wkh idfw wkdwS6r'm{rm @ R +p, +f1i1 Urelqvrq/ 4<<8/ s1 439:,/ zh kdyh

4 s p 6 [ r'{rorj  z+r, s4 5qhbr[?  2 @ s4 p 6 [ r'{rorj   r  . R@r _p q  . r@r # 4 q23R $   2 @ s4 p 6 [ r' {r 5 7_{orj mr m}2_{. orj} 4 . R@r _p q  . r@r # 4 q23R $   26 8 @ s4 p 6 [ r'{rorj mr m 2_{. R} # 4 s p 6 [ r'm{rm $ % R@r _p q  . r@r # 4 q23R $& @ s4 p 6 [ r' {rorj mr m2. Rsp 5 7_R@rp q  . r@r # 4 q23R $₂6 8 @ s4_p[6 r' {rorj mr m2. R@r # p2 q $ . r@r # _s p q23R $ @ s4 p 6 [ r' {rorj mr m2. r@r+4, +6;, zkhq p2 q . s p q23R $ 3= +6<,

53 Wkhq/ iurp +6;, dqg Ohppd F zh ghgxfh wkdw 4 s p 6 [ r'{rorj  z+r, s4 5qhbr[?  2 $_ ] _" f Q  3> 2+Z,sgi +Z , gZ= +73, Ilqdoo|/ xvlqj +73, dqg p3S6 r'{2r $ 4 lq +68,/ zh rewdlq s pg  4e $_PQ  3>4₇2_{+Z ,}_] " f Q  3>4₇2_{+Z ,}_{sgi +Z , gZ>}

jlylqj wkh uhtxluhg uhvxow1

9145 Surri ri Wkhruhp 714 Iurp +57, zh kdyh h g @ 4₅ S₆ r'_S{rorj L+r, 6 r'{2r > dqg vr 5spg  4h @  I 6 S₆ r'{rorj mz+r,m2  6S6r'{2r = +74,

Surfhhglqj dv lq wkh surri ri wkhruhp 614/ zh kdyh 4 s p 6 [ r'{rorj mz+r,m 2 _{@ s}4 p 6 [ r'{rorj mrm 2_{. r} @r+4, > +75,

xqghu +6<,/ dqg wkhq/ mxvw dv lq wkh surri ri Ohppd F exw zlwk \ @ 3> zh jhw 4 s_p[6 r' {rorj mrm2 $_Q # 3>₉2 $ = +76,

Qrwh wkdw wklv fdvh iroorzv gluhfwo| iurp Ohppd F zkhq Z @ 3 d=v1 dqg wkhq wkh olplw glvwulexwlrq lv qrupdo zlwk yduldqfh

2_{+3, @ #+4, @} 2

9 =

iurp +53, ru Judgvkwh|q dqg U|}kln +4<98/ ;1699,1 Frpelqlqj +75, dqg +76, zlwk p3S6

r'{2r $ 4 lq +74,/ zh rewdlq 5spg  4h $_Q # 3>2 9 $ > jlylqj wkh vwdwhg uhvxow1

54

Uhihuhqfhv

^4` Dnrqrp/ M1 +4<<6,1 Frpsruwhphqw dv|pswrwltxh gx whpsv g*rffxsdwlrq gx surfhvvxv ghv vrpphv sduwlhoohv1 Dqqdov ri wkh Lqvwlwxwh ri Khqul Srlqfduì/ 5</ 8:;41

^5` Fruedh/ G1/ V1 Rxoldulv dqg S1 F1 E1 Skloolsv +4<<<,1 Edqg Vshfwudo Uhjuhvvlrq zlwk Wuhqglqj Gdwd/ \doh Xqlyhuvlw|/ plphrjudskhg1

^6` Fvùujr/ P1 Dqg O1 Kruyäwk +4<<6,1 Zhljkwhg Dssur{lpdwlrqv lq Suredelolw| dqg Vwdwlv0 wlfv1 Qhz \run= Zloh|1

^7` Jhzhnh/ M1 dqg V1 Sruwhu0Kxgdn +4<;6,1 Wkh Hvwlpdwlrq dqg Dssolfdwlrq ri Orqj Php0 ru| Wlph Vhulhv Prghov/ Mrxuqdo ri Wlph Vhulhv Dqdo|vlv 7/ 554056:1

^8` Jlo0Dodôd/ O1 D1 dqg S1 P1 Urelqvrq +4<<:,1 Whvwlqj ri Xqlw Urrw dqg Rwkhu Qrq0 vwdwlrqdu| K|srwkhvlv lq Pdfurhfrqrplf Wlph Vhulhv/ Mrxuqdo ri Hfrqrphwulfv ;3/ 574059;1

^9` Judgvkwh|q L1 V1 dqg L1 P1 U|}kln +4<98,1 Wdeohv ri Lqwhjudov/ Vhulhv dqg Surgxfwv1 Qhz \run= Dfdghplf Suhvv1

^:` Kxuylfk/ F1 P1 dqg E1 N1 Ud| +4<<8,1 Hvwlpdwlrq ri wkh Phpru| Sdudphwhu iru Qrq0 vwdwlrqdu| ru Qrqlqyhuwleoh Iudfwlrqdoo| Lqwhjudwhg Surfhvvhv/ Mrxuqdo ri Wlph Vhulhv Dqdo|vlv 49/ 4:0751

^;` Kxuylfk/ F1 P1 dqg N1 L1 Ehowudr +4<<6,1 Dxwrpdwlf Vhplsdudphwulf Hvwlpdwlrq ri wkh Phpru| Sdudphwhu ri d Orqj0Phpru| Wlph Vhulhv/ Mrxuqdo ri Wlph Vhulhv Dqdo|vlv 48/ 5;806351

^<` Kxuylfk/ F1 P1/ U1 Ghr dqg M1 Eurgvn| +4<<;,1 Wkh Phdq Vtxduhg Huuru ri Jhzhnh dqg Sruwhu Kxgdn*v Hvwlpdwru ri wkh Phpru| Sdudphwhu ri d Orqj Phpru| Wlph Vhulhv/ Mrxuqdo ri Wlph Vhulhv Dqdo|vlv 4</ 4<0791

^43` Nlp/ F1 V1 dqg S1 F1 E1 Skloolsv +4<<<d,1 Orj Shulrgrjudp Uhjuhvvlrq= wkh Qrqvwd0 wlrqdu| Fdvh1 \doh Xqlyhuvlw|/ plphrjudskhg1

^44` Nlp/ F1 V1 dqg S1 F1 E1 Skloolsv +4<<<e,1 Prglhg Orj Shulrgrjudp Uhjuhvvlrq1 \doh Xqlyhuvlw|/ plphrjudskhg1

^45` N qvfk/ K1 U1 +4<;9,1 Glvfulplqdwlrq ehwzhhq orqj udqjh ghshqghqfh dqg prqrwrqlf wuhqgv1 Mrxuqdo ri Dssolhg Suredelolw|/ 56/ 435843631

^46` N qvfk/ K1 U1 +4<;:,1 Vwdwlvwlfdo Dvshfwv ri Vhoi0Vlplodu Surfhvvhv/ Surfhhglqjv ri wkh Iluvw Zruog Frqjuhvv ri wkh Ehuqrxool Vrflhw| Yro1 4/ 9:0:7/ YQX Vflhqfh Suhvv1 ^47` Skloolsv/ S1 F1 E1 +4<<<,1 Glvfuhwh Irxulhu Wudqvirupv ri Iudfwlrqdo Surfhvvhv/ \doh

Xqlyhuvlw|/ plphrjudskhg1

^48` Skloolsv/ S1 F1 E1 dqg Y1 Vror +4<<5,1 Dv|pswrwlfv iru olqhdu surfhvvhv/ Dqqdov ri Vwdwlvwlfv 53/ <:443341

55 ^49` Urelqvrq/ S1 P1 +4<<7,1 H!flhqw whvwv ri qrqvwdwlrqdu| k|srwkhvhv/ Mrxuqdo ri wkh

Dphulfdq Vwdwlvwlfdo Dvvrfldwlrq/ Yro1 ;</ 47530476:1

^4:` Urelqvrq/ S1 P1 +4<<8,1 Orj Shulrgrjudp Uhjuhvvlrq ri Wlph Vhulhv zlwk Orqj Phpru| Ghshqghqfh/ Wkh Dqqdov ri Vwdwlvwlfv 56/ 437;043:51

^4;` Vfkplgw/ S1 dqg S1 F1 E1 Skloolsv +4<<5,1 OP whvwv iru d xqlw urrw lq wkh suhvhqfh ri ghwhuplqlvwlf wuhqgv/ R{irug Exoohwlq ri Hfrqrplfv dqg Vwdwlvwlfv 87/ 58:5;:1

^4<` Vkrudfn/ J1 U1 Dqg M1 D1 Zhooqhu +4<;9,1 Hpslulfdo Surfhvvhv zlwk Dssolfdwlrqv wr Vwdwlvwlfv1 Qhz \run= Zloh|1

^53` Wdqdnd/ N +4<<<,/ Wkh Qrqvwdwlrqdu| Iudfwlrqdo Xqlw Urrw/ Hfrqrphwulf Wkhru| 48/ 87<08;51

^54` Yhodvfr/ F1 +4<<<d,1 Qrq0Vwdwlrqdu| Orj0Shulrgrjudp Uhjuhvvlrq/ Mrxuqdo ri Hfrqr0 phwulfv <4/ 65806:41

^55` Yhodvfr/ F1 +4<<<e,1 Jdxvvldq Vhplsdudphwulf Hvwlpdwlrq ri Qrq0Vwdwlrqdu| Wlph Vh0 ulhv/ Mrxuqdo ri Wlph Vhulhv Dqdo|vlv 53/ ;:045:1