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duh wkh frpsrqhqwv 4+d, dqg 4+e, lq Wdeoh 4/ zlwk wkh wzr frpsrqhqwv wrjhwkhu uhsuhvhqwlqj (cid:19)zkdw zh nqrz wkdw zh grq*w nqrz(cid:20) derxw wkh ixwxuh rxwfrph + 1 An(cid:5) Pdq| phdvxuhv h{lvw iru vxppdul}lqj surshuwlhv ri d iruhfdvw dqg lwv fruuhvsrqg0 lqj iruhfdvw huuru/ lqfoxglqj wkh eldv ri wkh iruhfdvw/ wkh yduldqfh ri wkh iruhfdvw huuru/ dqg wkh phdq vtxduh iruhfdvw huuru +PVIH,/ zklfk lv d frpprq vxppdu| phdvxuh ri d iruhfdvw huuru*v surshuwlhv1 Wkh PVIH lv gh(cid:31)qhg dv wkh phdq ri wkh vtxduhg iruhfdvw huuru/ h1j1/ HEe2 (cid:28) iru wkh iruhfdvw huuru e lq +7,1 Lq jhqhudo/ wkh zkroh An(cid:5) An(cid:5) glvwulexwlrq ri rxwfrphv lv ri lqwhuhvw zkhq frqvlghulqj iruhfdvw xqfhuwdlqw|1 Wkdw vdlg/ wkh sulpdu| phdvxuh ri iruhfdvw xqfhuwdlqw| lq hfrqrplfv lv wkh PVIH/ zklfk vlpsol(cid:31)hv wr wkh yduldqfh ri wkh iruhfdvw huuru zkhq wkh iruhfdvw lv xqeldvhg1 Wkh yduldqfh dqg PVIH fdswxuh lpsruwdqw dvshfwv ri iruhfdvw xqfhuwdlqw| dqg vr duh frqvlghuhg lq pxfk ri wkh glvfxvvlrq ehorz1 Vwloo/ wkh| kdyh vxevwdqwlyh olplwdwlrqv dv phdvxuhv ri iruhfdvw xqfhuwdlqw|> vhh Fohphqwv dqg Khqgu| +4<<6, iru d ghwdlohg dqdo|vlv1 Iru wkh vwdwlf prgho +4,/ wkh iruhfdvw huuru +7, kdv hdvlo| ghulydeoh surshuwlhv1 Lq idfw/ lwv glvwulexwlrq lv qrupdo= (cid:19) (cid:20) S e (cid:27) (cid:5) fcj2d(cid:129)n52 E A 52(cid:28)3(cid:129)o (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cMc +8, An(cid:5) An(cid:5) |’(cid:129) | dv iroorzv lpphgldwho| iurp qrupdolw| dqg lqghshqghqfh ri i(cid:4) j/ dqg khqfh wkh qru0 | (cid:9) pdolw| dqg lqghshqghqfh ri (cid:4) dqg K1 Ixuwkhu uhvxowv ghshqg rq wkh qdwxuh ri 5 1 An(cid:5) | Iru wkh vlpsoh fdvh zkhuh 5 lv dq lqwhufhsw +l1h1/ 5 ’ (cid:129),/ wkhq +8, vlpsol(cid:31)hv= | | (cid:19) (cid:20) e (cid:27) (cid:5) fcj2d(cid:129)nE(cid:129)*A(cid:28)o (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cM(cid:23) +9, An(cid:5) Ehfdxvh wkh iruhfdvw +(cid:9) lq +6, lv xqeldvhg iru wkh rxwfrph + / wkh yduldqfh ri An(cid:5) An(cid:5) e lv dovr wkh PVIH1 An(cid:5) Iljxuh 4d +wkh xsshu ohiw judsk lq Iljxuh 4, sorwv wkh PVIH iru K ’ f(cid:23)D dv d ixqf0 wlrq ri wkh iruhfdvw krul}rq (cid:5)/ zkhuh wkh xqlwv duh lq whupv ri j21 Iljxuh 4d frqwdlqv wzr fdofxodwlrqv ri wkh PVIH1 Wkh (cid:31)uvw lv wkh yduldqfh lq +9, iru d odujh hvwlpdwlrq vdpsoh +l1h1/ A ’ 4, dqg vr lv ghqrwhg wkh dv|pswrwlf PVIH +(cid:19)qr hvwlpdwlrq xqfhu0 wdlqw|(cid:20),1 Iurp +9,/ wkh dv|pswrwlf PVIH htxdov j2 iru doo iruhfdvw krul}rqv= khqfh lw lv d dw olqh dw xqlw| lq wkh judsk1 Wkh vhfrqg PVIH lv wkh yduldqfh lq +9, iru d vshfl(cid:31)f hvwlpdwlrq vdpsoh +khuh/ A ’ D, dqg vr lv ghqrwhg wkh (cid:31)qlwh vdpsoh PVIH +(cid:19)zlwk hvwlpdwlrq xqfhuwdlqw|(cid:20),15 Iurp +9,/ lw qxphulfdoo| htxdov (cid:129)(cid:23)2fj21 Lq vxppdu|/ iru dvwdwlf prgho zlwkrqo|dqlqwhufhsw/ wkh suhglfwdeoh xqfhuwdlqw| dv phdvxuhg e| wkh PVIH lv frqvwdqw dfurvv iruhfdvw krul}rqv1 Wkh PVIH ghfuhdvhv dvwkhhvwlpdwlrqvdpsohlqfuhdvhv/whqglqjwrdsrvlwlyhydoxh+wkhdv|pswrwlfPVIH, 5Zkloh d vdpsoh vl}h ri W @ 8 vhhpv yhu| vpdoo/ lw pd| qrw eh vr lq wklv frqwh{w/ zkhuh rqo| d vlqjoh sdudphwhu lv hvwlpdwhg1 Lq xquhvwulfwhg yhfwru dxwruhjuhvvlyh prghov dqg dxwruhjuhvvlyh glvwulexwhg odj prghov/ wkh qxpehu rihvwlpdwhg sdudphwhuv shu htxdwlrqlv frpprqo| 43((cid:24)63(ri wkh vdpsoh vl}h (cid:25) vlplodu wr wkh shufhqwdjh khuh1 ;

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yduldeoh 5 lq prgho (cid:127) lv qrz + / exw wkh surshuwlhv ri wkh huuru whup (cid:4) dqg wkh | (cid:129) |3(cid:129) | qrwdwlrq duh rwkhuzlvh dv lq prgho (cid:127) +Vhfwlrq 614,1 Iru h{srvlwlrqdo frqyhqlhqfh/ lw (cid:129) lv dvvxphg wkdw mKm (cid:9) (cid:129) dqg wkdw + (cid:27) (cid:5)Efcj2*E(cid:129)(cid:3)K2(cid:28)(cid:28)16 f Wkh iruhfdvwv frqvlghuhg duh wkrvh edvhg rq ohdvw vtxduhv hvwlpdwlrq ri wkh frhi0 (cid:31)flhqw K lq prgho (cid:127) = 2 +(cid:9) ’ (cid:9) K+ | ’ A n(cid:129)c(cid:23)(cid:23)(cid:23)cA nMc +;, | |3(cid:129) (cid:9) zkhuh K lv wkh ohdvw vtxduhv hvwlpdwru ri wkh frh!flhqw K/ dqg lv dssur{lpdwho| glv0 wulexwhg dv (cid:7)EKcE(cid:129)(cid:3)K2(cid:28)*A(cid:28)1 Dv zlwk +5,/ htxdwlrq +;, pd| eh uh0lqgh{hg/ uhvxowlqj lq wkh iroorzlqj vhtxhqfh ri rqh0vwhs dkhdg iruhfdvwv= +(cid:9) ’ (cid:9) K+ (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cM(cid:23) +<, An(cid:5) An(cid:5)3(cid:129) Htxdwlrq +<, pd| eh zulwwhq pruh h{solflwo| dv= (cid:9) +(cid:9) ’ K+ An(cid:129) A (cid:9) +(cid:9) ’ K+ An2 An(cid:129) 1 1 1 (cid:9) +(cid:9) ’ K+ An(cid:5) An(cid:5)3(cid:129) 1 1 1 (cid:9) +(cid:9) ’ K+ AnM3(cid:129) AnM32 (cid:9) +(cid:9) ’ K+ (cid:23) +43, AnM AnM3(cid:129) Htxdwlrq +43, kljkoljkwv d frpsolfdwlrq zkhq iruhfdvwlqj iurp g|qdplf prghov= wkh rqh0shulrgdkhdgiruhfdvw+(cid:9) lvhdvlo|frqvwuxfwhgehfdxvh+ lvnqrzq/exwiruhfdvwv An(cid:129) A dworqjhu krul}rqv ghshqgxsrqixwxuh+xqnqrzq,rxwfrphv ri +1 Wkh xvxdo dssurdfk lvwruhsodfhwkrvhixwxuhrxwfrphve|wkhluiruhfdvwv1 Wkxv/wkhiruhfdvw+(cid:9) ehfrphv An2 (cid:9) (cid:9) K(cid:4)EK+ (cid:28) dqg/ pruh jhqhudoo|/ wkh (cid:5)0shulrg dkhdg iruhfdvw +(cid:9) lv= A An(cid:5) +(cid:9) ’ (cid:9) K(cid:5)+ (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cM(cid:23) +44, An(cid:5) A Wkh iruhfdvw huuru e lv wkh gl(cid:30)huhqfh ehwzhhq wkh rxwfrph + dqg lwv iruh0 An(cid:5) An(cid:5) fdvw +(cid:9) / dv lq wkh (cid:31)uvw olqh ri +7,/ exw wkh dqdo|wlfv gl(cid:30)hu iurp +7,1 Wr fdofxodwh wkh An(cid:5) iruhfdvwhuuru/ lwlvkhosixo(cid:31)uvwwrghulyh+ lqwhupvriwkhrevhuyhg+ EuLh (cid:16) (cid:9) (cid:5)(cid:28) An(cid:5) An(cid:16) dqg wkh xqrevhuyhg ixwxuh vkrfnv i(cid:4) c(cid:4) c(cid:23)(cid:23)(cid:23)c(cid:4) j1 E| uhshdwhg vxevwlwxwlrq An(cid:129) An2 An(cid:5) iurp +:,/ + lv= An(cid:5) + ’ K+ n(cid:4) An(cid:5) An(cid:5)3(cid:129) An(cid:5) ’ K(cid:4)EK+ n(cid:4) (cid:28)n(cid:4) An(cid:5)32 An(cid:5)3(cid:129) An(cid:5) 1 1 1 S ’ K(cid:5)+ n (cid:5)3(cid:129)K(cid:16)(cid:4) (cid:23) +45, A (cid:16)’f An(cid:5)3(cid:16) 6Wkhvh dvvxpswlrqv lpso| wkdw | lv lqwhjudwhg ri rughu }hur dqg hujrglf1 Vhh Edqhumhh/ Grodgr/ w Jdoeudlwk/ dqg Khqgu| +4<<6, iru ghwdlov1 43

Wkxv/ wkh iruhfdvw huuru e lv= An(cid:5) e ’ + (cid:3)+(cid:9) An(cid:5) An(cid:5) An(cid:5) ’ EK(cid:5)+ n S (cid:5)3(cid:129)K(cid:16)(cid:4) (cid:28)(cid:3) (cid:9) K(cid:5)+ A (cid:16)’f An(cid:5)3(cid:16) A ’ S (cid:5)3(cid:129)K(cid:16)(cid:4) nEK(cid:5) (cid:3) (cid:9) K(cid:5)(cid:28)+ (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cM(cid:23) +46, (cid:16)’f An(cid:5)3(cid:16) A Dv zlwk wkh vwdwlf prgho/ wkh g|qdplf prgho*v iruhfdvw huuru ghshqgv xsrq ixwxuh vkrfnv wr wkh hfrqrp| dqg xsrq hvwlpdwlqj udwkhu wkdq nqrzlqj wkh frh!flhqw K1 Wkh vwdwlf dqg g|qdplf iruhfdvw huuruv gr gl(cid:30)hu lq wkh irup ri wkhlu ghshqghqfh rq wkrvh wzr frpsrqhqwv/ dv frpsdulvrq ri +7, dqg +46, fodul(cid:31)hv1 Iluvw/ wkh iruhfdvw huuru ri wkh vwdwlf prgho ghshqgv rq wkh ixwxuh vkrfn lq shulrg A n(cid:5) dorqh/ zkhuhdv wkh g|qdplf prgho*v iruhfdvw huuru frqwdlqv d zhljkwhg fxpxodwlrq ri vkrfnv iurp shulrg A n(cid:129) wkurxjk shulrg A n(cid:5)1 Vhfrqg/ wkh vwdwlf prgho*v iruhfdvw huuru ghshqgv (cid:9) rq wkh vlpsoh gl(cid:30)huhqfh ehwzhhq wkh frh!flhqw K dqg lwv hvwlpdwh K1 Wkh g|qdplf (cid:9) prgho*v iruhfdvw huuru ghshqgv xsrq wkh gl(cid:30)huhqfh ehwzhhq srzhuv ri K dqg K1 Iru (cid:9) wkh rqh0shulrg dkhdg iruhfdvw/ wkh g|qdplf prgho*v iruhfdvw huuru lv (cid:4) nEK(cid:3)K(cid:28)+ / An(cid:129) A (cid:9) sdudooholqj wkh vwdwlf prgho*v iruhfdvw huuru ri (cid:4) nEK(cid:3)K(cid:28)5 1 Dw rwkhu krul}rqv/ An(cid:129) An(cid:129) iruhfdvwv huuruv iurp wkh wzr w|shv ri prghov gl(cid:30)hu lq ghwdlo/ hyhq zkloh hdfk kdv wkh wzr frpsrqhqwv 4+d, dqg 4+e, iurp Wdeoh 41 Iurp +46,/ wkh surshuwlhv ri wkh g|qdplf prgho*v iruhfdvw huuruv ghshqg xsrq wkh glvwulexwlrq ri S (cid:5)3(cid:129)K(cid:16)(cid:4) dqg EK(cid:5) (cid:3) (cid:9) K(cid:5)(cid:28)/ dqg xsrq wkh lqlwldo frqglwlrq iru (cid:16)’f S An(cid:5)3(cid:16) iruhfdvwlqj/ l1h1/ + 1 Zkloh (cid:5)3(cid:129)K(cid:16)(cid:4) lv ylvxdoo| wkh prvw frpsoh{ ri wkh wkuhh A (cid:16)’f An(cid:5)3(cid:16) whupv/ lwv glvwulexwlrq lv wkh prvw vwudljkwiruzdug/ dv wkh ixwxuh vkrfnv i(cid:4) j duh S An(cid:5)3(cid:16) qrupdoo| glvwulexwhg dqg vhuldoo| lqghshqghqw1 Wkh phdq ri (cid:5)3(cid:129)K(cid:16)(cid:4) lv }hur/ (cid:16)’f An(cid:5)3(cid:16) dqg lwv yduldqfh lv wkh vxp ri wkh yduldqfhv ri lwv frpsrqhqwv= S (cid:15)@hE (cid:5)3(cid:129)K(cid:16)(cid:4) (cid:28) ’ (cid:15)@hE(cid:4) (cid:28)n(cid:15)@hEK(cid:4) (cid:28)n(cid:4)(cid:4)(cid:4)n(cid:15)@hEK(cid:5)3(cid:129)(cid:4) (cid:28) (cid:16)’f An(cid:5)3(cid:16) An(cid:5) An(cid:5)3(cid:129) An(cid:129) ’ j2nK2j2 n(cid:4)(cid:4)(cid:4)nK2E(cid:5)3(cid:129)(cid:28)j2 # $ (cid:129)(cid:3)K2(cid:5) ’ j2 (cid:23) +47, (cid:129)(cid:3)K2 Wkh h{dfw glvwulexwlrq ri wkh iruhfdvw huuruv lv frpsolfdwhg e| wkh qrqolqhdulw| (cid:9) K(cid:5) dqg (cid:9) e| wkh ghshqghqfh ri K rq + 1 Krzhyhu/ xqghu rqh vwdqgdug vhw ri dssur{lpdwlrqv/ A EK(cid:5)(cid:3) (cid:9) K(cid:5)(cid:28) lv glvwulexwhg dv (cid:5)EfcdE(cid:5)K(cid:5)3(cid:129)(cid:28)2E(cid:129)(cid:3)K2(cid:28)*Ao(cid:28)/ wkh yduldeoh + kdv lwv xqfrqgl0 A wlrqdo glvwulexwlrq(cid:5)Efcdj2*E(cid:129)(cid:3)K2(cid:28)o(cid:28)/ dqgwkhwhupEK(cid:5)(cid:3) (cid:9) K(cid:5)(cid:28) dqg+ duh lqghshqghqwo| A glvwulexwhg1 Xvlqj wkhvh dssur{lpdwlrqv/ wkh iruhfdvw huuru iurp wkh g|qdplf prgho kdv d phdq ri +dssur{lpdwho|, }hur dqg d yduldqfh +dssur{lpdwho|, htxdo wr= # $ (cid:129)(cid:3)K2(cid:5) j2E(cid:5)K(cid:5)3(cid:129)(cid:28)2 j2 n c +48, (cid:129)(cid:3)K2 A zklfk lv dovr lwv +dssur{lpdwh, (cid:31)qlwh vdpsoh PVIH1 Wkh (cid:31)uvw whup lq +48, lv wkh dv|pswrwlf PVIH lq +47,1 Wkh vhfrqg whup fdswxuhv wkh frpsolfdwhg ghshqghqfh ri 44

EK(cid:5) (cid:3) (cid:9) K(cid:5)(cid:28)+ rq irxu idfwruv= wkh iruhfdvw krul}rq (cid:5)/ wkh dxwruhjuhvvlyh frh!flhqw K/ A wkh hvwlpdwlrq vdpsoh vl}h A/ dqg wkh htxdwlrq huuru yduldqfh j21 Iljxuh 4e sorwv wkh dv|pswrwlf PVIH lq +47, dv d ixqfwlrq ri wkh iruhfdvw krul0 }rq (cid:5)1 Iurp +47,/ wkh dv|pswrwlf PVIH htxdov j2 iru (cid:5) ’ (cid:129)/ j2 nK2j2 iru (cid:5) ’ 2/ j2nK2j2nKej2 iru (cid:5) ’ (cid:12)/ dqg vr rq/ whqglqj wr j2*E(cid:129)(cid:3)K2(cid:28) iru odujh (cid:5)1 Lq Iljxuh 4e/ K ’ f(cid:23)D/ vr wkhvh ydoxhv duh j2/ (cid:129)(cid:23)2Dj2/ (cid:129)(cid:23)(cid:12)(cid:129)j2/ dqg vr rq/ whqglqj wr (cid:129)(cid:23)(cid:12)(cid:12)j21 Wkh +dv|pswrwlf, suhglfwdeoh xqfhuwdlqw| lqfuhdvhv dv wkh iruhfdvw krul}rq lqfuhdvhv> dqg wkdw suhglfwdeoh xqfhuwdlqw| lv erxqghg ryhu doo iruhfdvw krul}rqv1 Iljxuh 4e dovr sorwv wkh +dssur{lpdwh, (cid:31)qlwh vdpsoh PVIHlq +48,1 Wklv PVIHlv odujhuwkdq+rudwohdvwdvodujhdv,wkhdv|pswrwlfPVIH=hvwlpdwlrqxqfhuwdlqw|dggv wr wkh lqkhuhqw xqfhuwdlqw| iurp wkh ixwxuh vkrfnv/ dv iru wkh vwdwlf prgho1 Wkh (cid:31)qlwh vdpsoh PVIH lv dovr erxqghg ryhu doo iruhfdvw krul}rqv1 Dv Iljxuh 4e ghprqvwudwhv/ wkh (cid:31)qlwh vdpsoh PVIH qhhg qrw lqfuhdvh prqrwrqlfdoo| lq wkh iruhfdvw krul}rq= dv (cid:5) lqfuhdvhv/ wkh (cid:31)qlwh vdpsoh PVIH pd| ghfuhdvh dv zhoo dv lqfuhdvh1 Wkh h{sodqdwlrq iru wklv qrq0prqrwrqlflw| wxuqv rq wkh whup j2E(cid:5)K(cid:5)3(cid:129)(cid:28)2*A lq +48,1 Wkdw whup kdv d frpsolfdwhg ghshqghqfh rq wkh iruhfdvw krul}rq (cid:5)/ exw lq dq| fdvh lv dozd|v srvlwlyh iru qrq}hur K dqg whqgv wr }hur dv (cid:5) ehfrphv odujh1 Iru hfrqrplf dqg srolf| dqdo|vlv/ qrq0prqrwrqlflw| lq wkh PVIH fdq eh zruu|lqj= wkh iruhfdvw krul}rqv ri juhdwhvw hfrqrplf ru srolf| lqwhuhvw pd| kdyh wkh juhdwhvw +suhglfwdeoh, xqfhuwdlqw|1 Wr vxppdul}h/ iru d g|qdplf (cid:31)uvw0rughu dxwruhjuhvvlyh prgho/ wkh suhglfwdeoh xqfhuwdlqw| dv phdvxuhg e| wkh PVIH whqgv wr lqfuhdvh lq wkh iruhfdvw krul}rq/ dowkrxjk hvwlpdwlrq xqfhuwdlqw| pd| lqgxfh qrq0prqrwrqlflw|1 Dv zlwk wkh PVIH iurpwkhvwdwlfprgho/ wkhg|qdplf prgho*vPVIHghfuhdvhvdvwkhhvwlpdwlrqvdpsoh lqfuhdvhv/ whqglqj wr d srvlwlyh ydoxh +wkh dv|pswrwlf PVIH, iru d odujh hvwlpdwlrq vdpsoh1 Wkh PVIHv iurp g|qdplf dqg vwdwlf prghov gl(cid:30)hu prvw qrwdeo| wkurxjk wkhlu ghshqghqfh +ru odfn wkhuhri, rq wkh iruhfdvw krul}rq1 Wkh iroorzlqj vxevhfwlrq/ dorqj zlwk Vhfwlrq 7/ h{dplqhv wklv lvvxh lq juhdwhu ghwdlo1 616 D Frpsdulvrq Wklv vxevhfwlrq frpsduhv wkh suhglfwlyh xqfhuwdlqw| iurp wkh vwdwlf dqg g|qdplf prghov lq wkh wzr suhylrxv vxevhfwlrqv/ irfxvlqj rq wkh urohv ri frh!flhqw hvwlpdwlrq dqg wkh iruhfdvw krul}rq1 Wr kljkoljkw wkh odwwhu/ wzr hpslulfdo prghov ri X1N1 qhw qdwlrqdo lqfrph duh h{dplqhg1 Iljxuhv4ddqg4esorwwhgwkhPVIHviruwkhvwdwlfdqgg|qdplfprghov/hydoxdwhg dw d vshfl(cid:31)f ydoxh ri wkh uhjuhvvlrq sdudphwhu K= K ’ f(cid:23)D1 Iljxuhv 4f dqg 4g sorw wkh frpsdudeoh judskv iru d gl(cid:30)huhqw ydoxh ri K= K ’ f(cid:23).1 Iru wkh vwdwlf prgho +Iljxuh 4f,/ erwk wkh dv|pswrwlf dqg (cid:31)qlwh vdpsoh PVIHv uhpdlq xqfkdqjhg1 Lq idfw/ wkh PVIHv iru wkh vwdwlf prgho duh lqyduldqw wr K/ dv +8, dqg +9, lpso|1 Iru wkh g|qdplf prgho +Iljxuh 4g,/ erwk wkh dv|pswrwlf dqg (cid:31)qlwh vdpsoh PVIHv kdyh dowhuhg1 Hdfk lqfuhdvhv pruh udslgo| dw vpdoo iruhfdvw krul}rqv> dqg wkh| whqg wr d 45

odujhu PVIH iru odujh (cid:5)= wr (cid:129)(cid:23)bSj2/ udwkhu wkdq wr (cid:129)(cid:23)(cid:12)(cid:12)j21 Iru pxowl0shulrg dkhdg iruhfdvwv dw d jlyhq krul}rq (cid:5)/ erwk wkh dv|pswrwlf dqg (cid:31)qlwh vdpsoh PVIHv lqfuhdvh lq wkh frh!flhqw K/ dv iroorzv iurp wkh vxp lq wkh vhfrqg olqh ri +47, dqg iurp wkh vhfrqg whup lq +48,1 Wkxv/ wkh vwdwlf qdwxuh ri wkh vwdwlf prgho ryhu wkh hvwlpdwlrq +lq0vdpsoh, shulrg fduulhv wkurxjk wr wkh +suhglfwdeoh, surshuwlhv ri lwv iruhfdvwv dqg iruhfdvw huuruv1 Wlph ghshqghqfh fkdudfwhul}hv wkh qdwxuh ri wkh g|qdplf prgho/ erwk lq vdpsoh dqg ryhu wkh iruhfdvw shulrg1 Wkhvh uhvxowv vkrz wkdw fkdudfwhulvwlfv ri iruhfdvwv duh prgho ghshqghqw1 Vrphzkdw vxusulvlqjo|/ fkdudfwhulvwlfv ri iruhfdvwv duh uhodwlyho| gdwd lqghshqghqw1 Wr looxvwudwh wkhvh uhvxowv/ wkh uhpdlqghu ri wklv vxevhfwlrq frqvlghuv wzr prghov (cid:25) rqh vwdwlf dqg wkh rwkhu g|qdplf (cid:25) wkdw jhqhudol}h ru h{sdqg xsrq wkh prghov lq Vhfwlrqv614dqg6151 Wkh(cid:31)uvwlvduhjuhvvlrqprghozlwkdqlqwhufhswdqgolqhduwuhqg/ dqg wkh vhfrqg lv d udqgrp zdon prgho zlwk guliw1 Erwk kdyh sod|hg lpsruwdqw urohv lq vwdwlvwlfdo dqg hfrqrplf prghoolqj> vhh Er{ dqg Mhqnlqv +4<:3,/ Khqgu|/ Sdjdq/ dqgVdujdq +4<;7,/ Khqgu|+4<<8/ Fkdswhu :,/ dqgFohphqwv dqgKhqgu|+5333, lqwhu dold1 Iru hdvh ri h{srvlwlrq/ xqfhuwdlqw| iurp frh!flhqw hvwlpdwlrq lv ljqruhg1 Wkh (cid:31)uvw prgho lv d vwdwlf prgho zlwk erwk dq lqwhufhsw dqg d olqhdu wuhqg= (cid:127)W G + ’ K nK |n(cid:4) (cid:4) (cid:27) (cid:5)WEfcj2(cid:28) | ’ (cid:129)c(cid:23)(cid:23)(cid:23)cAcA n(cid:129)c(cid:23)(cid:23)(cid:23)cA nMc +49, (cid:129) | (cid:129) 2 | | zkhuh K lv wkh lqwhufhsw dqg K lv wkh frh!flhqw rq wkh wuhqg |1 Wkh iruhfdvwv duh= (cid:129) 2 +(cid:9) ’ K nK (cid:4)EA n(cid:5)(cid:28) (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cMc +4:, An(cid:5) (cid:129) 2 vr wkh iruhfdvw huuruv duh= e ’ (cid:4) (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cMc +4;, An(cid:5) An(cid:5) dqg wkh PVIH lv= (cid:15)@hEe (cid:28) ’ j2 (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cM(cid:23) +4<, An(cid:5) Dv lq wkh vlpsohu vwdwlf prgho zlwk rqo| dq lqwhufhsw/ wkh +dv|pswrwlf, PVIH grhv qrw ghshqg xsrq wkh iruhfdvw krul}rq (cid:5)1 Wkh PVIH lq +4<, lv wkh (cid:19)suhglfwhg(cid:20) PVIH/ lq wkdw j2 lv wkh ydoxh ri wkh PVIH/ dvvxplqj wkdw prgho (cid:127)W lq +49, lv wkh GJS1 Zlwk vrph glvwulexwlrqdo dvvxpswlrqv (cid:129) derxw(cid:4)/wkhsuhglfwhgPVIHfdqsurylghwkhedvlvirufdofxodwlqjfrq(cid:31)ghqfhlqwhuydov iru wkh iruhfdvwv lq +4:,/ dv lv frpprq lq pdq| hfrqrphwulfv vriwzduh sdfndjhv1 Lq jhqhudo/ wkh suhglfwhg PVIH gl(cid:30)huv iurp wkh dfwxdo PVIH/ dv zkhq wkh dvvxphg prgho lv qrw wkh GJS> vhh Fohphqwv dqg Khqgu| +5333,1 Wrlooxvwudwh wkh surshuwlhv ri wkh suhglfwhgPVIHhpslulfdoo|/ frqvlghu prghoolqj uhdo qhw qdwlrqdo lqfrph lq wkh Xqlwhg Nlqjgrp ryhu 4<:3(cid:24)4<<6 dqg iruhfdvwlqj lw wkurxjk 53431 Wkh gdwd duh iurp Hulfvvrq/ Khqgu|/ dqg Suhvwzlfk +4<<;,/ dqg wkh 46

10.2 Trend model 10.2 Random walk model Actual value Actual value Fitted Fitted Forecast Forecast 10 10 9.8 9.8 9.6 9.6 9.4 9.4 9.2 9.2 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 Iljxuh5= Dfwxdo/(cid:31)wwhg/dqgiruhfdvwydoxhviurpwkhwuhqgdqgudqgrpzdonprghovri dqqxdouhdoqhwqdwlrqdolqfrphiruwkhXqlwhgNlqjgrp+lqorjv,/zlwk<8(frq(cid:31)ghqfh lqwhuydov iru wkh iruhfdvwv1 prgho lv +49,/ zkhuh + lv wkh orjdulwkp ri uhdo qhw qdwlrqdo lqfrph1 Iljxuh 5d vkrzv wkh uhvxowv iurp hvwlpdwlqj wklv prgho dqg iruhfdvwlqj iurp lw1 Wkh ohiw kdoi ri Ilj0 xuh 5d sorwv dfwxdo lqfrph dqg wkh (cid:31)wwhg ydoxhv ri lqfrph iurp wkh hvwlpdwhg prgho ryhu 4<:3(cid:24)4<<61 Wkh uljkw kdoi ri Iljxuh 5d sorwv wkh prgho*v iruhfdvwv wkurxjk 53431 Wkh yhuwlfdo eduv durxqg wkrvh iruhfdvwv uhsuhvhqw wkh dqwlflsdwhg bDI frq(cid:31)ghqfh lqwhuydov iru lqfrph/ zklfk duh urxjko| (cid:9)2j(cid:9) +qdpho|/ soxv0ru0plqxv wzlfh wkh hvwl0 pdwhg htxdwlrq vwdqgdug huuru,1 Zkloh wklv prgho iru lqfrph grhv lqfoxgh d wuhqg/ wkdw wuhqg lv ghwhuplqlvwlf/ lpso|lqj wkdw lwv ixwxuh ydoxhv duh nqrzq/ dv zhoo dv lwv fxuuhqw dqg sdvw ydoxhv1 Vr/ lq hvvhqfh/ wklv prgho lv vwdwlf/ dqg wkh dqwlflsdwhg iruhfdvw xqfhuwdlqw| lv frqvwdqw dfurvv gl(cid:30)huhqw iruhfdvw krul}rqv1 Wkh vhfrqg prgho lv d udqgrp zdon zlwk dq lqwhufhsw= (cid:127)W G + ’ @n+ n(cid:4) (cid:4) (cid:27) (cid:5)WEfcj2(cid:28) | ’ (cid:129)c(cid:23)(cid:23)(cid:23)cAcA n(cid:129)c(cid:23)(cid:23)(cid:23)cA nMc +53, 2 | |3(cid:129) | | zkhuh @ lv wkh lqwhufhsw/ dqg wkh frh!flhqw rq wkh odjjhg ghshqghqw yduldeoh lv frqvwudlqhg wr htxdo xqlw|1 47

Iroorzlqj +44,/ +46,/ dqg wkh (cid:31)uvw wzr olqhv ri +47,/ wkh (cid:5)0vwhs dkhdg iruhfdvwv duh= +(cid:9) ’ @(cid:5)n+ (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cMc +54, An(cid:5) A wkh iruhfdvw huuruv duh= S e ’ (cid:5)3(cid:129)(cid:4) (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cMc +55, An(cid:5) (cid:16)’f An(cid:5)3(cid:16) dqg wkh PVIH lv= (cid:15)@hEe (cid:28) ’ (cid:5)j2 (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cM(cid:23) +56, An(cid:5) Wkh suhglfwhgPVIH lqfuhdvhv lq wkh iruhfdvw krul}rq (cid:5)/ dqglq idfw lqfuhdvhv olqhduo| lq (cid:5)/ dqg zlwkrxw erxqg1 Wkh suhglfwhg PVIH lq +56, dvvxphv wkdw prgho (cid:127)W lq 2 +53, lv wkh GJS1 Iljxuh5esorwvwkhdfwxdo/(cid:31)wwhg/dqgiruhfdvwydoxhviurpwklvudqgrpzdonprgho ri lqfrph/ xvlqj wkh vdph vdpsoh shulrgv iru hvwlpdwlrq dqg iruhfdvwlqj dv zlwk wkh vwdwlf prgho lq Iljxuh 5d1 Wkh frq(cid:31)ghqfh lqwhuydov iru wkh udqgrp zdon iruhfdvwv lq Iljxuh 5e lqfuhdvh yhu| vxevwdqwldoo| dv wkh iruhfdvw krul}rq lwvhoi lqfuhdvhv/ frqwudvw0 lqj zlwk frq(cid:31)ghqfh lqwhuydov ri (cid:31){hg zlgwk lq Iljxuh 5d1 Iljxuhv 5d dqg 5e sruwud| wzr yhu| gl(cid:30)huhqw sdwwhuqv iru wkh dqwlflsdwhg +ru suhglfwhg, iruhfdvw xqfhuwdlqw|/ dqg wkhlu frpsdulvrq looxvwudwhv krz prgho fkrlfh fdq d(cid:30)hfw wkrvh sdwwhuqv1 H{dfwo| wkh vdph vhulhv lv ehlqj prghoohg dqg iruhfdvw lq Iljxuhv 5d dqg 5e= rqo| wkh prghov wkhpvhoyhv gl(cid:30)hu1 Pruh jhqhudoo|/ vwdwlf prghov riwhqlpso|suhglfwhg iruhfdvw xqfhuwdlqw|wkdw lv wlph lqyduldqw ru qhduo|vr/ zkhuhdv g|qdplf prghov jhqhudoo|lpso| wlph0ghshqghqw suhglfwhgiruhfdvw xqfhuwdlqw|/ riwhq lqfuhdvlqj lq wkh iruhfdvw krul}rq1 Prghov (cid:127) dqg (cid:127) suhvhqw vwdwlf dqg g|qdplf uhodwlrqvklsv dv eodfn dqg zklwh/ (cid:129) 2 exw lq sudfwlfh d zkroh vshfwuxp ri prghov h{lvwv zlwk erwk vwdwlf dqg g|qdplf dvshfwv1 Wkh vlpsohvw h{dpsoh lv= (cid:127) G + ’ K + nK 5 n(cid:4) c +57, (cid:12) | (cid:129) |3(cid:129) 2 | | zkhuh K dqg K duh wkh frh!flhqwv ri odjjhg + dqg fxuuhqw 51 Prgho (cid:127) lqfoxghv (cid:129) 2 (cid:12) prghov (cid:127) dqg (cid:127) dv vshfldo fdvhv zlwk K ’ f dqg K ’ f uhvshfwlyho|1 Zkhq K (cid:129) 2 (cid:129) 2 (cid:129) dqg K duh erwk qrq}hur/ prgho (cid:127) kdv erwk g|qdplf dqg vwdwlf ihdwxuhv1 Dqdo|wlfdo 2 (cid:12) h{dplqdwlrq ri prgho (cid:127) *v suhglfwdeoh iruhfdvw xqfhuwdlqw| lv ihdvleoh exw uhyhdov (cid:12) uhodwlyho| olwwoh eh|rqg zkdw kdv ehhq vhhq iurp prghov (cid:127) dqg (cid:127) vhsdudwho|1 (cid:129) 2 Wklv vhfwlrq kdv frqvlghuhg prghov dqg iruhfdvwv ri lqglylgxdo yduldeohv rqo|/ dqg qrw ri vhwv ri yduldeohv1 \hw/ hfrqrplfv w|slfdoo| lv derxw uhodwlrqvklsv ehwzhhq ydul0 deohv/ zkhuh wkrvh yduldeohv pd| lqwhudfw zlwk hdfk rwkhu/ hlwkhu frqwhpsrudqhrxvo|/ ru dw d odj/ ru erwk1 Wkh qh{w vhfwlrq wkxv wxuqv wr v|vwhp iruhfdvwv dqg wkhlu dvvrfl0 dwhg suhglfwdeoh xqfhuwdlqw|1 Lq wkh odqjxdjh ri Wdeoh 5/ wkh lvvxhv duh wkh glphqvlrq dqg frpsohwhqhvv ri wkh uhodwlrqvkls ri lqwhuhvw dqg ri wkh iruhfdvw1 48

7 Suhglfwdeoh Xqfhuwdlqw| iru V|vwhpv Lq Vhfwlrq 614 deryh/ ixwxuh ydoxhv ri wkh h{rjhqrxv yduldeoh 5 zhuh dvvxphg wr eh nqrzq zkhq iruhfdvwlqj +1 Wklv dvvxpswlrq lv vhqvleoh iru ghwhuplqlvwlf yduldeohv vxfk dv dq lqwhufhsw ru d olqhdu wuhqg1 Krzhyhu/ iru prvw hfrqrplf yduldeohv/ wkh yduldeohv wkhpvhoyhv zrxog qhhg wr eh iruhfdvw zkhq iruhfdvwlqj +1 Grlqj vr ohdgv gluhfwo| wr iruhfdvwlqj iurp v|vwhpv ri htxdwlrqv1 Wklv vhfwlrq (cid:31)uvw frqvlghuv d vlp0 soh exw klvwrulfdoo| lpsruwdqw wzr0yduldeoh v|vwhp +Vhfwlrq 714, dqg wkhq wxuqv wr d yhfwru dxwruhjuhvvlyh prgho +ru YDU/ Vhfwlrq 715,/ zklfk lv wkh edvlv iru pxfk hpslulfdo iruhfdvwlqj lq hfrqrplfv1 Dv lq Vhfwlrq 6/ wkh irfxv lv rq wkh lqkhuhqw suh0 glfwdeoh xqfhuwdlqw|1 Wkh suhglfwdeoh xqfhuwdlqw| iurp hvwlpdwlrq lv fdofxodeoh/ erwk dqdo|wlfdoo| dqg qxphulfdoo|/ exw lw surylghv ohvv gluhfw lqvljkw gxh wr wkh lqfuhdvhg frpsoh{lw| ri wkh prghov1 714 D Vlpsoh G|qdplf V|vwhp Wkh vwdwlf prgho lq +4, h{suhvvhv + lq whupv ri 5 dqg d glvwxuedqfh (cid:4) 1 Wklv vxevhf0 | | | wlrq dxjphqwv wkdw prgho zlwk dq htxdwlrq iru 5 lwvhoi/ vr wkdw iruhfdvwv ri + fdq eh | frqvwuxfwhg iurp iruhfdvwv ri 5 zkhq wkh ixwxuh ydoxhv ri 5 duh xqnqrzq1 Pdq| srvvleoh prghov h{lvw iru 5 1 Rqh sodxvleoh ghwhuplqdqw ri 5 lv + = wkdw | | |3(cid:129) lv/ + Judqjhu0fdxvhv 51 Vhh Judqjhu +4<9<, iru wkh ghyhorsphqw ri wkh frqfhsw ri Judqjhu fdxvdolw|/ dqg Khqgu| dqg Pl}rq +4<<<, iru uhdvrqv zk| Judqjhu fdxvdolw| lv vr suhydohqw lq hfrqrplf gdwd1 Wkxv/ lw lv ri lqwhuhvw wr frqvlghu prgho (cid:127) = e + % & #% & % &$ + ’ K5 n(cid:4) (cid:4) f j2 f (cid:127) G | | | | (cid:27) (cid:5)W c c +58, e 5 ’ S+ n(cid:15) (cid:15) f f /2 | |3(cid:129) | | zkhuh S lv wkh frh!flhqw rq odjjhg + lq wkh htxdwlrq iru 5> dqg (cid:15) lv wkh glvwxuedqfh | lq wkdw htxdwlrq1 Wkh wzr glvwxuedqfhv E(cid:4) G (cid:15) (cid:28) duh dvvxphg wr eh qrupdoo| dqg | | lqghshqghqwo| glvwulexwhg/ kdylqj phdq }hur dqg +uhvshfwlyho|, yduldqfhv j2 dqg /21 Wkh fryduldqfh ehwzhhq (cid:4) dqg (cid:15) lv }hur= wkdw lv/ wkh htxdwlrq iru + lv d frqglwlrqdo | | prgho ri + jlyhq5 / dqgK lv wkh frh!flhqw lpsolhg e| wkdw frqglwlrqlqj> vhh Hulfvvrq | | +4<<5d, iru dq lqwurgxfwru| h{srvlwlrq rq frqglwlrqlqj dqg +uhodwhgo|, h{rjhqhlw|1 Wkhvh dvvxpswlrqv derxw qrupdolw| dqg frqglwlrqlqj duh frqyhqlhqw/ exw duh qrw fhqwudo wr wkh surshuwlhv ri wkh iruhfdvwv1 Lq hfrqrplfv/ +58, lv nqrzq dv d frezhe prgho/ zklfk pd| fkdudfwhul}h d pdunhw zlwk odjv lq wkh surgxfwlrq surfhvv/ dv ri djulfxowxudo frpprglwlhv1 Lq wkh frezhe prgho/+ dqg5 duhlqwhusuhwhgdvwkhorjvrisulfhdqgtxdqwlw|/uhvshfwlyho|1 Ghqrwlqj wkrvh orjv dv R dqg ^/ +58, ehfrphv= + % & #% & % &$ R ’ K^ n(cid:4) (cid:4) f j2 f (cid:127)W G | | | | (cid:27) (cid:5)W c (cid:23) +59, e ^ ’ SR n(cid:15) (cid:15) f f /2 | |3(cid:129) | | 49

Wkh frezhe prgho +59, kdv wkh iroorzlqj lqwhusuhwdwlrq1 Wkh htxdwlrq iru R lv | ghulyhg iurp d ghpdqg htxdwlrq= wkh sulfh R fohduv wkh pdunhw iru d jlyhq txdqwlw| | ^ vxssolhg1 Wkh ydoxh (cid:129)*K lv wkh sulfh hodvwlflw| ri ghpdqg1 Wkh htxdwlrq iru ^ lv d | | vxsso| htxdwlrq/ fdswxulqj +iru lqvwdqfh, krz pxfk iduphuv ghflgh wr surgxfh wklv |hdu +^ ,/ ghshqglqj xsrq wkh sulfh wkdw wkh| zhuh deoh wr rewdlqlqwkh suhylrxv |hdu | +R ,1 Wkh ydoxh S lv wkh sulfh hodvwlflw| ri vxsso|1 Vhh Wlqehujhq +4<64, dqg Vxlwv |3(cid:129) +4<88, iru slyrwdo frqwulexwlrqv rq wkh frezhe prgho/ dqg Khqghuvrq dqg Txdqgw +4<:4/ ss1 475(cid:24)478, iru dq h{srvlwlrq1 Iruhfdvwlqj iurp wkh frezhe prgho jhqhudol}hv rq wkh phfkdqlvp iru iruhfdvwlqj iurp wkh (cid:31)uvw0rughu dxwruhjuhvvlyh prgho lq Vhfwlrq 615/ vr zh uhyhuw wr wkh E+ G 5(cid:28) qrwdwlrq lq +58,1 Ljqrulqj hvwlpdwlrq/ wkh iruhfdvwv ghulyh iurp +58, zlwkrxw wkh glvwxuedqfhv/ zkrvh ixwxuh ydoxhv duh e| gh(cid:31)qlwlrq xqnqrzq= + +(cid:9) ’ K5 An(cid:5) An(cid:5) (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cM(cid:23) +5:, 5(cid:9) ’ S+ An(cid:5) An(cid:5)3(cid:129) Htxdwlrq +5:, pd| eh zulwwhq pruh h{solflwo| dv= % & % & +(cid:9) K5 An(cid:129) ’ An(cid:129) 5(cid:9) S+ An(cid:129) A % & % & +(cid:9) K5 An2 ’ An2 5(cid:9) S+ An2 An(cid:129) 1 1 1 % & % & +(cid:9) K5 An(cid:5) ’ An(cid:5) 5(cid:9) S+ An(cid:5) An(cid:5)3(cid:129) 1 1 1 % & % & +(cid:9) K5 AnM3(cid:129) ’ AnM3(cid:129) 5(cid:9) S+ AnM3(cid:129) AnM32 % & % & +(cid:9) K5 AnM ’ AnM (cid:23) +5;, 5(cid:9) S+ AnM AnM3(cid:129) Irupdoo|/ +5;, sdudoohov +43,/ zlwk wkh iruphu gholqhdwlqj sdluv ri iruhfdvwv dw d jlyhq krul}rq (cid:5)/ udwkhu wkdq d vlqjoh iruhfdvw dw hdfk krul}rq1 Doo yduldeohv h{fhsw + A rq wkh uljkw0kdqg vlgh ri +5;, duh gdwhg lq wkh ixwxuh dqg khqfh xqnqrzq/ vr dq lpsohphqwdeoh iruhfdvwlqj dojrulwkp lv vwloo uhtxluhg1 Frqvlghu jhqhudwlqj wkh iruhfdvwv iru + (cid:31)uvw1 Vxevwlwxwlqj wkh vhfrqg htxdwlrq ri +5:, lqwr wkh (cid:31)uvw rewdlqv= +(cid:9) ’ KS+ (cid:5) ’ (cid:129)c(cid:23)(cid:23)(cid:23)cMc +5<, An(cid:5) An(cid:5)3(cid:129) zklfk lv lghqwlfdo wr +<,/ h{fhsw wkdw wkh dxwruhjuhvvlyh frh!flhqw lv KS/ udwkhu wkdq K lwvhoi1 Doo M iruhfdvwv iru + fdq eh jhqhudwhg iurp +5<,/ hpsor|lqj wkh dssurdfk lq 4:

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$ billion 100 100 static model, one quarter ahead static model, multiple quarters ahead Actual value 0 0 Forecast 95% confidence interval -100 -100 -200 -200 1985 1986 1987 1988 1985 1986 1987 1988 100 100 dynamic model, one quarter ahead dynamic model, multiple quarters ahead 0 0 -100 -100 -200 -200 1985 1986 1987 1988 1985 1986 1987 1988 Iljxuh 6= Dfwxdo dqg iruhfdvw ydoxhv iurp wzr prghov ri wkh uhdo X1V1 wudgh edodqfh/ erwk rqh txduwhu dqg pxowlsoh txduwhuv dkhdg/ zlwk <8( frq(cid:31)ghqfh lqwhuydov iru wkh iruhfdvwv1 ghwhuplqlqj wkh hpslulfdo uroh ri Lwhpv 5+d,(cid:24)5+f, lq wkh iruhfdvw huuru wd{rqrp| ri Wdeoh 41 Vshfl(cid:31)fdoo|/ Lwhpv 5+d,(cid:24)5+f, uh hfwglvfuhsdqflhv ehwzhhqwkh prgho xvhgiru iruhfdvwlqj dqg wkh dfwxdo ehkdylru ri wkh hfrqrp|/ zlwk Lwhp 5+d, w|slfdoo| ehlqj sulpdulo| uhvsrqvleoh iru iruhfdvw idloxuh lq hfrqrphwulf prghov1 Irupdo vwdwlvwlfdo dqdo|vlv ri prgho0edvhg iruhfdvwv lq oljkw ri suhglfwdeoh iruhfdvw xqfhuwdlqw| (cid:25) wkurxjk whvwv ri sdudphwhu frqvwdqf| (cid:25) kdv ehhq fhqwudo wr hydox0 dwlqj dqg lpsurylqj hpslulfdo hfrqrplf prghov1 Fkrz +4<93, surylghv wkh lqlwldo ghyhorsphqw ri wkhvh hydoxdwlrq whfkqltxhv/ exloglqj rq Ilvkhu +4<55,/ zlwk Kdqvhq +4<<5d/ 4<<5e, dqg Dqguhzv +4<<6, frqvwuxfwlqj frqvwdqf| whvwv zlwk dq xqnqrzq euhdnsrlqw1 Jrogihog +4<:6/ 4<:9,/ Mxgg dqg Vfdgglqj +4<;5,/ dqg Eded/ Khqgu|/ dqg Vwduu +4<<5, h{dplqh frqvwdqf| whvwv lq wkhlu fhqwudo uroh lq prghoolqj wkh gh0 pdqg iru prqh|1 Dw d pruh jhqhudo ohyho/ frqvwdqf| whvwv duh nh| wr hydoxdwlqj wkh Oxfdv +4<:9, fulwltxh/ zlwk frqvwdqf| whvwv surylglqj wkh sulpdu| hpslulfdo edvlv iru uhixwlqj wkh Oxfdv fulwltxh> vhh Khqgu| +4<;;,/ Hqjoh dqg Khqgu| +4<<6,/ Hulfvvrq dqg Lurqv +4<<8,/ dqg Khqgu| +5333,1 Suhglfwdeoh iruhfdvw xqfhuwdlqw| lv dovr fhqwudo wr fdofxodwlqj iruhfdvw hqfrpsdvvlqj whvw vwdwlvwlfv> vhh Fkrqj dqg Khqgu| +4<;9,/ 54

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