Asymptotic Approximations in the Near-Integrated Model with a Non-Zero Initial Condition

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9815

PERRON, Pierre

VODOUNOU, Cosme

Asymptotic Approximations in the Near-Integrated

Model with a Non-Zero Initial Condition

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Département de sciences économiques

Université de Montréal

Faculté des arts et des sciences C.P. 6128, succursale Centre-Ville Montréal (Québec) H3C 3J7 Canada http://www.sceco.umontreal.ca SCECO-information@UMontreal.CA Téléphone : (514) 343-6539 Télécopieur : (514) 343-7221

Ce cahier a également été publié par le Centre interuniversitaire de recherche en économie quantitative (CIREQ) sous le numéro 3297.

This working paper was also published by the Center for Interuniversity Research in Quantitative Economics (CIREQ), under number 3297.

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CAHIER 9815

ASYMPTOTIC APPROXIMATIONS IN THE NEAR-INTEGRATED MODEL WITH A

NON-ZERO INITIAL CONDITION

Pierre PERRON1 and Cosme VODOUNOU2

1

Centre de recherche et développement en économique (C.R.D.E.) and Département de sciences économiques, Université de Montréal, and Department of Economics, Boston University

2

Institut National de la Statistique et de l'Analyse Économique, Bénin

December 1997

_________________

This paper is drawn from chapter 1 of Vodounou's Ph.D. Dissertation at the Université de Montréal. Perron acknowledges financial support from the Social Sciences and Humanities Research Council (SSHRC) of Canada, the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the Fonds pour la Formation de chercheurs et l'aide à la recherche (FCAR) of Québec. The authors also thank Jon Knight for useful conversations and for providing them with the draft notes referred to as Knight and Satchell (1990).

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RÉSUMÉ

Ce papier considère plusieurs approximations asymptotiques dans le modèle autorégressif de premier ordre presque intégré avec une condition initiale non nulle. En premier, nous élargissons le travail de Knight et Satchell (1993) qui ont considéré le cas d'une marche aléatoire avec une condition initiale nulle. Ensuite, nous dérivons l'expansion de la fonction génératrice de moments conjointe qui est pertinente dans ce cadre théorique plus général. Nous considérons aussi, comme approximation alternative, l'expansion stochastique de Phillips (1987c) et l'approximation en temps continu de Perron (1991). Nous assurons comment ces méthodes alternatives donnent ou non une approximation adéquate pour la distribution en échantillon fini de l'estimateur de moindres carrés dans un modèle AR(1). Les résultats montrent que, quand la condition initiale est non nulle, l'approximation en temps continu de Perron (1991) fonctionne très bien, alors que les autres offrent seulement des améliorations lorsque la condition initiale est nulle.

Mots clés : expansion d'Edgeworth, asymptotiques en temps continu, expansion stochastique, fonction de répartition, modèle autorégressif

ABSTRACT

This paper considers various asymptotic approximations in the near-integrated first-order autoregressive model with a non-zero initial condition. We first extend the work of Knight and Satchell (1993), who considered the random walk case with a zero initial condition, to derive the expansion of the relevant joint moment generating function in this more general framework. We also consider, as alternative approximations, the stochastic expansion of Phillips (1987c) and the continuous time approximation of Perron (1991). We assess how these alternative methods provide or not an adequate approximation to the finite-sample distribution of the least-squares estimator in a first-order autoregressive model. The results show that, when the initial condition is non-zero, Perron's (1991) continuous time approximation performs very well while the others only offer improvements when the initial condition is zero.

Key words : Edgeworth expansion, continuous-time asymptotics, stochastic expansion, distribution function, autoregressive model

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(13)

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(14)

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(16)

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(17)

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(18)

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(EA ' 2-e o o ooA

c

zkhuh o lv wkh frpsoh{ frqmxjdwh ri o1 Diwhu vrph ghyhorsphqwv zh fdq zulwh (EA ' momA _{EULt EAw n f t? EAw c}

zkhuh

f ' E mom ULt w * Emom t? w Zh qrz frqvlghu wkh h{sdqvlrq ri f dqg w Zh kdyh= f ' A_B o_AQ n _Ao 2oQo B A !Q* ' oQ B n o oQA n oQo B A ' _B oQn_Bo_A o_Qn B n EA! _c dqg w ' |@?! B A 2oo2 B2_A *2 * n o_A n _Ao2₂ ' _AB oo2 B2_A o_A _Ao2₂ ' _AB o B2_AEB2n o2 n EA! Khqfh/ ULtEAw ' ULtB _BAo EB2_{n o} 2 ' ULt B n _BAo EB2_{n o} 2 t? Bc dqg t?EA w ' t?B _BAo EB2_{n o} 2 ' t? B _BAo EB2_{n o} 2 ULt B

Vlqfh mom ' EEmom ULt w2_{n Emom t? w}2*2_{/ zh dovr kdyh=}

mom ' no_A n _Ao2₂2n_AB2₂ 2o_B2_Ao2 *2 ' n 2o_A n o2n 2o2n B2 A2 *2 ' n o A n 2o2n B2 2A2 47

(19)

Lw iroorzv wkdw momA _{' i T} % A *? # n o A n 2o2n B2 2A2 $& ' i T % A # o A n 2o2 n B2 2A2 o2 2A2 $& ' i TEo i T # 2o2 n B2 o2 2A $ ' i TEo % n2o2n B2 o2 2A &

Lw uhpdlqv wr frqvlghu wkh h{suhvvlrq ULtEA w n f t?EA w Xvlqj wkh uhvxowv deryh= ULtEAw n f t?EAw ' ULt B n o

BAEB2n o2 t? B B on _Bo₂2_A o2 n B2 t? B o BAEB2n o2 ULt B H{sdqglqj wklv odvw h{suhvvlrq ohdgv wr=

ULtEAw n f t?EAw ' _B2_A2 n EA!2c

zlwk ' ULt B ot? B_B c 2 ' o2Eo2 n B2 B2oEB2n o2 _{t? B} B o2EB2n o2 ULt B Zh qdoo| jhw (EA ' i TEo % n2o2n B2 o2 2A & 2 B2_A ' i TEo 2AB2 22 B2E2o2n B2 o2 c dqg zh ghgxfh wkdw= (EA !*2 _'!*2 i TEo*2 + n eA B2 22 B2E2o2n B2 o2 , H{sdqglqj wklv h{suhvvlrq/ zh kdyh= (EA!*2 _{' i TEo} *2 # ULt B ot? B_B $!*2 +D17, ; ? = n Eo2 n B2E2o2 oB2 eAB2_{ULt B o} t? B_B t? B B Eo2 n B2E2o2n B2 eAB2_{ULt B o} t? B_B ULt B < @ > 48

(20)

Zh qrz frqvlghu wkh frpsrqhqw ri wkh prphqw0jhqhudwlqj ixqfwlrq lqyroylqj +f ' V zklfk lv= [ ' i T # V2 A V2 Ak V2_k2 2 $ i T#_A n k2 V₂2(EA _(EA $ Vxevwlwxwlqj k dqg k2 _{e| wkh dssur{lpdwlrqv +D15, dqg +D16,/zh kdyh=}

i T # V2 A V2 Ak V2_k2 2 $ ' i TEV₂2 + V2_Ao , c dqg i T#_A n k2 V₂2(EA _(EA $ ' i T # V2 2 $ + nV2E n 2o_2A , c vxssrvlqj wkdw

(EA *(EA ' n *Ac zklfk lv wr eh vkrzq ehorz1 Khqfh/zh kdyh=

[ ' n V_2A2 n EA!2

Wr ghwhuplqh /zh uvw h{sdqg (EA dv iroorzv=

(EA ' mom) !EULt EEA w n f t? EEA w

' mom) !dULt EA w n w t? EA w n f Et? EAw w ULt EAwo ' mom) !iE wf ULtEA w n Ew n f t?EA wj

Wkh h{sdqvlrqv ri E wf dqg Ew n f wr rughu EA!_{duh/uhvshfwlyho|=}

wf ' n o_Ac dqg w n f ' o_B n B_A Be_o 2 o2 n B2 Khqfh/zh kdyh=

(EA ' mom) !iEULt B ot? B_B

n B2_AEoB2Eo2n B2 o2Eo2n B2 n Bet? B_B n _B2_AULt Bjc zlwk ' oB2n o2Eo2n B2 Zh ghgxfh wkdw (EA (EA ' n o2 n B2 AULt B ot? *_* t? B B ' n Ac 49

(21)

zlwk ' o2n B2 ULt B ot? *_* t? B B Ilqdoo|/ C8 Ec ' ! +_n V2 o2 n B2 2A ULt B ot? *_* t? B B " D_(-A!Q*

Frpelqlqj wklv uhvxow zlwk +D17, dqg uhduudqjlqj |lhogv wkh uhvxow vwdwhg lq wkh Wkhruhp1 Ghulydwlrq ri +45,1 Iurp Shuurq +4<<9,/ wkh mrlqw prphqw jhqhudwlqj ixqfwlrq ri E~ESc c gESc fdq eh h{suhvvhg dv=

C8 Ec ' i T # eA V A e S 2S $ ScEc c +D18, zkhuh ScEc ' [SEc i T + V 2A E n S b k i T E n S b [_SEc l,c zlwk

[SEc ' i2b i TEES n * dEb n n S i TEb n Eb S i TEbojQ* c

dqg

b ' 2 2S S

Wr rewdlq wkh dowhuqdwlyh uhsuhvhqwdwlrq/ zh vwduw zlwk wkh iroorzlqj lghqwlwlhv= i TEb ' SEb rEb dqg i TEb ' SEb n rEb zklfk ghqh wkh k|shuerolfixqfwlrqv SEb hw rEb1 Zh fdq zulwh=

Eb n n S i TEb n Eb S i TEb ' 2bSEb 2oQrEbc

zlwk oQ ' S n dqg zh ghgxfh wkdw= Eb n n S i TEb n Eb S i TEb 2b ' SEb oQ rEb b Xvlqj B ' b/ zh kdyh B ' b ' 2 2S S dqg

SEb oQrEb_b ' ULt B oQt? B_B

Lw lv wkhq hdv| wr vkrz wkdw [SEc fdq eh zulwwhq dv= [SEc ' i TEES n *2 ULt B oQ B t? B _!Q* / 4:

(22)

dqg zh ghgxfh wkdw 5c-c ' i TEoQ*2 ULt B o_BQ t? B!Q* i T + V_2AEoQ b % _{ULt B}i TEb_o_Q * t? * &, Wkh h{suhvvlrq +45, lv rewdlqhg frpelqlqj wklv odvw uhvxow zlwk +D18,1

(23)

Uhihuhqfhv

^4` Ehujvwurp/ D1U1 +4<;7,= Frqwlqxrxv Wlph Vwrkdvwlf Prghov dqg Lvvxhv ri Djjuhjdwlrq ryhu Wlph/ lq ]1 Juloolfkhv dqgP1G1 Lqwuloljdwru +hgv1,/ Kdqgerrn ri Hfrqrphwulfv/ yro1 5/ Qruw0Kroodqg/ 4479045431

^5` Erenrvnl/ P1 M1 +4<;6,= K|srwkhvlv Whvwlqj lq Qrqvwdwlrqdu| Wlph Vhulhv/ Xqsxe0 olvkhgSk1 G1 Glvvhuwdwlrq/ Ghsduwphqw ri Vwdwlvwlfv/ Xqlyhuvlw| ri Zlvfrqvlq1

^6` Fdydqdjk/ F1 +4<;9,= Urrwv Orfdo wr Xqlw|/ pdqxvfulsw/ Kduydug Xqlyhuvlw|1

^7` Fkdq/ Q1 K1 dqgF1 ]1 Zhl +4<;:,= Dv|pswrwlf Lqihuhqfh iru Qhduo| Qrqvwdwlrqdu| DU+4, Surfhvvhv/ Wkh Dqqdov ri Vwdwlvwlfv 48/ 4383043961

^8` Glfnh|/ G1 D1 dqgIxoohu/ Z1 D1 +4<:<,= Glvwulexwlrq ri wkh Hvwlpdwruv iru Dxwruhjuhv0 vlyh Wlph Vhulhv zlwk d Xqlw Urrw/ Mrxuqdo ri wkh Dphulfdq Vwdwlvwlfdo Dvvrfldwlrq :7/ 75:07641

^9` Hydqv/ J1E1 dqgQ1H1 Vdylq +4<;4d,= Wkh Fdofxodwlrq ri wkh Ohdvw Vtxduhv Hvwlpdwru ri wkh Sdudphwhu lq d Udqgrp Zdon Prgho/ Wkh Dqqdov ri Vwdwlvwlfv </ 44470444;1 ^:` Hydqv/ J1E1 dqgQ1H1 Vdylq +4<;4e,= Whvwlqj iru Xqlw Urrwv= 4/ Hfrqrphwulfd 7</

:860::;1

^;` Ixoohu/ Z1D1 +4<:9,= Lqwurgxfwlrq wr Vwdwlvwlfdo Wlph Vhulhv/ Qhz \run= Mrkq Zloh|1 ^<` Jxuodqg/ M1 +4<7;,= Lqyhuvlrq Irupxodh iru wkh Glvwulexwlrq ri Udwlrv/ Dqqdov ri

Pdwkhpdwlfdo Vwdwlvwlfv 4</ 55;056:1

^43` Klvdpdwvx K1 dqgN1 Pdhndzd +4<<7,= Wkh Glvwulexwlrq ri wkh Gxuelq 0 Zdwvrq Vwdwlvwlf lq LqwhjudwhgdqgQhdu0LqwhjudwhgPrghov/ Mrxuqdo ri Hfrqrphwulfv 94/ 69:0 6;51

^44` Nqljkw/ M1O1 dqgV1H1 Vdwfkhoo +4<<3,= Hgjhzruwk H{sdqvlrq ri wkh Ohdvw Vtxduhv Hvwlpdwru lq wkh Qhdu LqwhjudwhgPrgho/ Xqsxeolvkhggudiw qrwhv1

^45` Nqljkw/ M1O1 dqgV1H1 Vdwfkhoo +4<<6,= Dv|pswrwlf H{sdqvlrqv iru Udqgrp Zdon zlwk Qrupdo Huuruv/ Hfrqrphwulf Wkhru| </ 69606:91

^46` Pdqq/ K1E1 dqgD1 Zdog+4<76,= Rq wkh Vwdwlvwlfdo Wuhdwphqw ri Olqhdu Vwrfkdvwlf Glhuhqfh Htxdwlrqv/ Hfrqrphwulfd 44/ 4:605531

^47` Shuurq/ S1 +4<;<,= Wkh Fdofxodwlrq ri wkh Olplwlqj Glvwulexwlrq ri wkh Ohdvw Vtxduhv Hvwlpdwru lq d Qhdu0LqwhjudwhgPrgho/ Hfrqrphwulf Wkhru| 8/ 57405881

(24)

^48` Shuurq/ S1 +4<<4d,= D Frqwlqxrxv Wlph Dssur{lpdwlrq wr wkh Xqvwdeoh Iluvw0rughu Dxwruhjuhvvlyh Surfhvv= Wkh Fdvh Zlwkrxw dq Lqwhufhsw/ Hfrqrphwulfd 8</ 54405691 ^49` Shuurq/ S1 +4<<4e,= D Frqwlqxrxv Wlph Dssur{lpdwlrq wr wkh Vwdwlrqdu| Iluvw0rughu

Dxwruhjuhvvlyh Prgho/ Hfrqrphwulf Wkhru| :/ 56905851

^4:` Shuurq/ S1 +4<<9,= Wkh Dghtxdf| ri Dv|pswrwlf Dssur{lpdwlrqv lq Wkh Qhdu0 Lqwhjudwhg Dxwruhjuhvvlyh Prgho zlwk Ghshqghqw Huuruv/ Mrxuqdo ri Hfrqrphwulfv :3/ 64:06831

^4;` Skloolsv/ S1F1E1 +4<::,= Dssur{lpdwlrqv wr Vrph Ilqlwh Vdpsoh Glvwulexwlrqv Dvvrfl0 dwhg zlwk d Iluvw Rughu Vwrfkdvwlf Glhuhqfh Htxdwlrq/ Hfrqrphwulfd 78/ 79607;81 ^4<` Skloolsv/ S1F1E1 +4<;:d,= Wlph Vhulhv Uhjuhvvlrq zllwk Xqlw Urrwv/ Hfrqrphwulfd 88/

5::06351

^53` Skloolsv/ S1F1E1 +4<;:e,= Wrzdugv d Xqlhg Dv|pswrwlf Wkhru| iru Dxwruhjuhvvlrq/ Elrphwulnd :7/ 868087:1

^54` Skloolsv/ S1F1E1 +4<;:f,= Dv|pswrwlf H{sdqvlrqv lq Qrqvwdwlrqdu| Yhfwru Dxwruhjuhv0 vlrqv/ Hfrqrphwulf Wkhru| 6/ 7809;1

^55` Skloolsv/ S1F1E1 dqg S1 Shuurq +4<;;,= Whvwlqj iru d Xqlw Urrw lq Wlph Vhulhv Uhjuhv0 vlrq/ Elrphwulnd :8/ 66806791

^56` Uxelq/ K1 +4<83,= Frqvlvwhqf| ri Pd{lpxp Olnholkrrg Hvwlpdwhv lq Wkh H{sorvlyh Fdvh/ lq W1 F1 Nrrspdqv +hg1,/ Vwdwlvwlfdo Lqihuhqfh lq G|qdplf Hfrqrplf Prghov/ Qhz \run= Mrkq Zloh|1

^57` Vdwfkhoo/ V1H1 +4<;7,= Dssur{lpdwlrq wr wkh Ilqlwh Vdpsoh Glvwulexwlrq Iru Qrqvwdeoh Iluvw Rughu Vwrfkdvwlf Glhuhqfh Htxdwlrqv/ Hfrqrphwulfd 85/ 45:4045;<1

^58` Yrgrxqrx/ F1 +4<<:d,= Hvvdlv hq ìfrqrphwulh wkìrultxh hw hq ìfrqrpìwulh gh od qdqfh dssoltxìv dx{ prgëohv g|qdpltxhv/ Xqsxeolvkhg SkG Glvvhuwdwlrq/ Gìsduwhphqw gh Vflhqfhv Ìfrqrpltxhv/ Xqlyhuvlwì gh Prqwuìdo1

^59` Yrgrxqrx/ F1 +4<<:e,= Dgìtxdwlrq ghv dssur{lpdwlrqv g*Hgjhzruwk gdqv oh prgëoh DU+4, vdeoh dyhf ydohxu lqlwldoh {h/ pdqxvfulsw/ Xqlyhuvlwì gh Prqwuìdo1

^5:` Zklwh/ M1V1+4<8;,= Wkh Olplwlqj Glvwulexwlrq ri wkh Vhuldo Fruuhodwlrq Frh!flhqw lq wkh H{sorvlyh Fdvh/ Dqqdov ri Pdwkhpdwlfdo Vwdwlvwlfv 5</ 44;;044<:1

(25)

Wdeoh 4= Fulwlfdo ydoxhv ri wkh h{dfw glvwulexwlrq ri W +A , 4( 518( 8( 43( <3( <8( <:18( <<( f@08 @ 3=3 W@ 43 0<1458 0:198; 0916;4 071<97 51663 51<3: 617:8 71587 W@ 58 0451;58 043155; 0;1696 0916<9 516<8 61333 6176; 7135; W@ 83 04719:4 04419;4 0<1775 0:1375 51764 61334 61788 61<:5 W@433 0481:8; 045173; 0<1<46 0:1677 51777 61366 617:; 61<<5 @ 3=8 W@ 43 0:1:83 091888 081775 071558 51398 51889 51<95 61769 W@ 58 043186< 0;1:44 0:1456 081754 51498 51:55 61476 618:; W@ 83 0451394 0<1863 0:1;73 081<68 51546 51::5 61539 61979 W@433 0461638 04317;: 0;1647 091599 51545 51::: 614<6 61947 @ 4=3 W@ 43 081975 071:89 061<74 061383 419<6 513:< 516<8 51:46 W@ 58 0:173: 091386 081335 061;:; 41;48 51593 51943 51<63 W@ 83 0;17;; 091<63 081949 07155; 41;95 51668 519:: 6136; W@433 0;1<9< 0:14<6 081<89 071797 41;9< 51674 51:46 61389 @ 5=3 W@ 43 0615;5 051:3; 051578 041:5: 4149: 41797 419<7 41<6< W@ 58 07137; 06163; 051:3; 0513<< 415:; 418<7 41;9< 51485 W@ 83 071795 0619;6 061364 05164: 41656 41989 41<76 51565 W@433 0719<9 061;64 06144: 0516;8 41666 419:8 41<89 51584 f@3 @ 3=3 W@ 43 0<1<48 0;1398 0918<7 071<3: 414:4 41966 51444 51::: W@ 58 0441;55 0<173; 0:16<: 08167; 41345 4173; 41:;; 51654 W@ 83 04518:: 0<1<59 0:1986 081856 31<:5 4166: 419:; 5143; W@433 0461697 0431588 0:1<5; 0819;3 31<8< 41645 41985 513;7 @ 3=8 W@ 43 0:1;<5 0916<5 081563 061;<: 31;97 414;; 4183; 41<48 W@ 58 0<18;< 0:18:; 081;<: 07167: 31:<4 413<; 416:5 41:98 W@ 83 0<1<87 0:1<:6 09145; 071787 31:<: 413<5 4167; 419;; W@433 043178: 0;148< 09174: 0718;: 31::5 41385 41646 4193; @ 4=3 W@ 43 081586 07153< 061697 05179< 31967 31;76 41334 41554 W@ 58 081;<5 07196; 0619:: 051986 318<: 31:<6 31<94 4149: W@ 83 09188: 08145; 061<<8 051:;; 318<; 31:<; 31<88 41489 W@433 09195< 0814:; 071396 051;<4 318;; 31::: 31<68 41458 Q

(26)

Wdeoh 4 +frqw*g, = Fulwlfdo ydoxhv ri wkh h{dfw glvwulexwlrq ri W +A , 4( 518( 8( 43( <3( <8( <:18( <<( f@3 @ 5=3 W@ 43 051766 041;:6 041795 04138: 31763 3187< 3197; 31:8; W@ 58 0518;< 041<<9 04189< 041435 3174: 31864 31963 31:6: W@ 83 051:48 051388 0418;5 041464 31749 31867 3195< 31:6; W@433 051:54 05143: 041946 041484 31744 3185: 3194: 31:4; f@5 @ 3=3 W@ 43 0431389 0:1<74 09149; 0715;3 31:73 41436 417:< 5143: W@ 58 0441549 0;1756 09163; 0715<9 318<3 31;:: 414;3 41935 W@ 83 0441:99 0;1:69 09185: 0716:3 31886 31;3: 413:; 41763 W@433 0441:73 0;1:98 091884 0716;8 31878 31;35 41398 41756 @ 3=8 W@ 43 0:18<; 081:<9 071794 051;65 3178< 31988 31;9: 41496 W@ 58 0;1756 091635 0719;9 051<95 31739 318;6 31::3 41387 W@ 83 0;1<86 0917<6 071:84 061349 31734 31898 31:78 4133; W@433 0<153: 0918<8 071;34 061333 316:: 31883 31:4; 31<8: @ 4=3 W@ 43 07168< 051897 041754 031:;; 315<3 316;9 317;4 31944 W@ 58 071639 051;:3 04194: 031;66 31596 31689 31778 3188; W@ 83 071:36 061369 0419:9 031;93 31595 3167; 31765 31876 W@433 08136: 061496 041:68 031;;: 31585 31673 3174; 31859 @ 5=3 W@ 43 031895 03176: 03166; 03157: 31498 31547 3158: 31644 W@ 58 0318;8 031765 031667 031574 31487 31533 3156: 315;; W@ 83 0318<9 031778 031668 031575 31484 314<9 31568 315:< W@433 0318<: 031757 031658 03156; 3147< 314<5 31559 315:5

(27)

Wdeoh 5= Fulwlfdo ydoxhv iurp wkh Hgjhzruwk dssur{lpdwlrq 4( 518( 8( 43( <3( <8( <:18( <<( f@08 @ 3=3 W@ 43 0:1676 091:3< 081<36 071:84 515:9 51;53 61593 61<38 W@ 58 0441;6; 0<1<;3 0;1596 091643 516;3 51<9; 61765 71334 W@ 83 0471577 0441859 0<1649 091<;3 5174; 6134: 617;3 7134< W@433 0481848 045167< 0<1;;4 0:1677 5176: 61373 61836 7135: @ 3=8 W@ 43 09174; 081<63 0815:; 071635 51436 51895 51;;9 614<3 W@ 58 0<1399 0:1<<6 091;53 081669 514;: 519<5 6138: 61747 W@ 83 0431886 0<1343 0:184; 081::8 5154; 51:6; 61446 617;4 W@433 04417:; 0<1933 0:1<3< 091349 51566 51:94 61474 61846 @ 4=3 W@ 43 061687 0614<7 051<7; 051844 41:68 5136; 51547 51666 W@ 58 0714;5 061<85 061945 06136: 41::8 51437 515<< 51766 W@ 83 07188: 0715;< 061;<: 061588 41:<3 5145< 51663 5179< W@433 071;4< 071853 0713<4 061734 41:<; 51474 51679 517;; @ 5=3 W@ 43 0413<< 041388 031<;5 031;6< 41456 4157< 41647 41687 W@ 58 0414;; 041473 041396 031<44 4147< 415;5 41684 416<6 W@ 83 041553 0414:4 0413<4 031<6: 4148; 415<7 41697 4173: W@433 041569 0414;: 041439 031<83 41496 41633 416:4 41748

(28)

Wdeoh 5 +frqw*g,= Fulwlfdo ydoxhv iurp wkh Hgjhzruwk dssur{lpdwlrq 4( 518( 8( 43( <3( <8( <:18( <<( f@313 @ 3=3 W@ 43 0<1753 0:1<53 091833 071;:3 41473 418:3 41<;3 517;3 W@ 58 0441::3 0<1653 0:1693 081683 41343 41733 41::3 51583 W@ 83 0451:43 0<1;93 0:19<3 081853 31<:3 41673 419<3 51483 W@433 0461543 0431473 0:1;93 081943 31<83 41643 41983 513<3 @ 3=8 W@ 43 0918<: 081;35 071<37 061:9; 31;<5 4157< 41944 51457 W@ 58 0;1368 091:75 081834 0713<; 31:;< 413:: 4166: 419:; W@ 83 0;1:5; 0:1478 081:73 071556 31:8; 4135: 4158: 41865 W@433 0<1463 0:169: 081;9; 0715;; 31:76 41336 4154< 4179: f@5 @ 3=3 W@ 43 0<1367 0:178: 081;48 071379 31:35 41347 41675 41;33 W@ 58 0431;5; 0;157: 091566 071549 318<4 31;98 41485 41894 W@ 83 0441755 0;1885 0916<5 0715:; 31888 31;45 413;4 41796 W@433 0441:67 0;1:44 0917:7 071644 3186: 31:;8 41377 41743 @ 3=8 W@ 43 091367 081334 061<57 05194: 317:6 31:45 31<<5 41765 W@ 58 091;;6 081793 071484 0519:; 316;: 3187< 31:4; 31<9< W@ 83 0:1599 081979 07156; 051:34 31697 31839 31975 31;53 W@433 0:17;7 081:7; 0715;6 051:46 31686 317;9 3193< 31:8: @ 4=3 W@ 43 031958 0318;5 031853 031757 31538 31577 3159: 315;4 W@ 58 03194: 0318:8 031848 031754 314;: 31556 31575 31588 W@ 83 031939 031898 03183: 031749 314;5 31549 31568 3157: W@433 031934 031894 031837 031746 314:< 31546 31565 31576 A

(29)

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Wdeoh 7= Fulwlfdo ydoxhv iurp wkh vwrfkdvwlf h{sdqvlrq

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