Money as an indicator variable for monetary policy when money demand is forward looking

(1)

BANK OF FINLAND

DISCUSSION PAPERS

9

· 2003

Lauri Kajanoja

Research Department

9.4.2003

Money as an indicator variable

for monetary policy

when money demand is

forward looking

Suomen Pankin keskustelualoitteita Finlands Banks diskussionsunderlag

(2)

Suomen Pankki Bank of Finland P.O.Box 160 FIN-00101 HELSINKI Finland ( (( ( + 358 9 1831

(3)

BANK OF FINLAND

DISCUSSION PAPERS

9

· 2003

Lauri Kajanoja*

Research Department

9.4.2003

Money as an indicator variable

for monetary policy when money demand is

forward looking

The views expressed are those of the author and do not necessarily reflect the views of the Bank of Finland.

I am grateful to Martin Ellison, Seppo Honkapohja, Juha Kilponen, Jarmo Kontulainen, David Mayes, Petri Mäki-Fränti, Anssi Rantala, Antti Ripatti, Tuomas Saarenheimo, Eric Schaling, Juha Tarkka, Jouko Vilmunen, and the participants of the Bank of Finland Research Department seminars for useful comments and suggestions.

* Author’s address: Bank of Finland, P.O.Box 160, FIN-00101 Helsinki, Finland. Tel. +358-9-183 2467. Fax +358-9-+358-9-183 2560. E-mail: [email protected].

Suomen Pankin keskustelualoitteita Finlands Banks diskussionsunderlag

(4)

http://www.bof.fi ISBN 952-462-047-2 ISSN 0785-3572 (print) ISBN 952-462-048-0 ISSN 1456-6184 (online)

(5)

3

Money as an indicator variable for monetary policy

when money demand is forward looking

Bank of Finland Discussion Papers 9/2003

Lauri Kajanoja Research Department

Abstract

This paper studies the gain from using money as an indicator when monetary policy in made under data uncertainty. We use a forward and backward looking model, calibrated for the euro area. The policymaker cannot completely observe the state of the economy. Money reveals some of the private sector’s information to the policymaker, especially if there is a forward looking element in money demand. We show that observing money can considerably reduce the loss that is due to incomplete information. However, taking also into account other financial market data could decrease the marginal importance of money as an indicator. Key words: monetary policy, partial information, money, monetary aggregates, euro area

(6)

Raha-aggregaatti indikaattorina rahapolitiikalle, kun

rahan kysyntäyhtälö on eteenpäin katsova

Suomen Pankin keskustelualoitteita 9/2003

Lauri Kajanoja Tutkimusosasto

Tiivistelmä

Tässä tutkimuksessa tarkastellaan rahan määrän hyödyllisyyttä indikaattorina rahapolitiikan päätöksentekijän kannalta, kun taloudellisten muuttujien uusimpia arvoja ei täysin tunneta. Tutkimuksessa käytetään eteenpäin ja taaksepäin katso-vaa makromallia, joka on kalibroitu euroalueen taloudelle. Rahapolitiikan päätök-sentekijä ei havaitse talouden tilaa täydellisesti. Rahan määrä paljastaa rahapolitii-kan päätöksentekijälle osan yksityisten taloudenpitäjien informaatiosta etenkin, jos rahan kysyntäyhtälössä on eteenpäin katsova elementti. Tutkimuksessa näyte-tään, että rahan käyttäminen indikaattorina voi vähentää huomattavasti sitä tappio-ta, jonka informaation epätäydellisyys taloudelle aiheuttaa. Muun rahoitusmark-kinainformaation huomioon ottaminen voisi kuitenkin vähentää rahan määrän merkitystä indikaattorina.

Avainsanat: rahapolitiikka, epätäydellinen informaatio, raha, raha-aggregaatit, euroalue

(7)

5

4 Lqwurgxfwlrq

Prqh| riwhq sod|v qr h{solflw uroh lq wkh prgho iudphzrunv xvhg lq frqwhpsrudu| prqhwdu| srolf| dqdo|vlv1 Prvw qrwdeo|/ wklv lv wkh fdvh iru qhduo| doo dssolfdwlrqv ri wkh Qhz Nh|qhvldq prghoolqj dssurdfk1 Krzhyhu/ vrph vwxglhv kdyh lghqwlhg hhfwv zklfk jlyh d uroh iru prqh|1 Rqh srvvlelolw| lv wr frqvlghu prqh| dv dq lqglfdwru xqghu uhdo wlph gdwd xqfhuwdlqw|1 Wklv sdshu txdqwlwdwlyho| vwxglhv wkdw hhfw1

Prqhwdu| srolf| lv pdgh lq wkh hqylurqphqw ri xqfhuwdlqw|1 Rqh vrxufh ri xqfhuwdlqw| frqfhuqv wkh vwdwh ri wkh hfrqrp|1 Lq whupv ri prqhwdu| srolf| dqdo|vlv/ wkh srolf|pdnhu fdqqrw revhuyh wkh frqwhpsrudqhrxv ydoxhv ri vrph hvvhqwldo yduldeohv/ exw rqo| qrlv| vljqdov ri wkhp1 Lq wkhruhwlfdo whupv/ wkh prvw lpsruwdqw plvvlqj slhfh ri lqirupdwlrq lv/ dujxdeo|/ dq h{dfw phdvxuh ri wkh rxwsxw jds1

Wklv sdshu vwxglhv wkh ydoxh ri prqhwdu| djjuhjdwhv dv lqglfdwru yduldeohv/ dvvxplqj wkdw wkh lqwhuhvw udwh lv xvhg dv wkh prqhwdu| srolf| lqvwuxphqw1 Zh pdnh txdqwlwdwlyh dvvhvvphqwv rq wkh zhdnhqlqj hhfw wkdw gdwd xqfhuwdlqw| kdv rq wkh shuirupdqfh ri wkh hfrqrp|/ dqg rq wkh lpsuryhphqw lq wkh shuirupdqfh eurxjkw derxw e| wkh rswlpdo xvh ri d prqhwdu| djjuhjdwh dv dq lqirupdwlrq yduldeoh1 Zh xvh d Qhz Nh|qhvldq prgho zlwk dgghg edfnzdug orrnlqj ihdwxuhv/ fdoleudwhg iru wkh hxur duhd hfrqrp|1 Wkh ydoxh ri wkh prqhwdu| djjuhjdwh dv dq lqirupdwlrq yduldeoh lv lqfuhdvhg e| wkh h{lvwhqfh ri d iruzdug orrnlqj hohphqw lq wkh ghpdqg iru prqh|1

Lq wkh prgho vhwxs ri wklv sdshu/ wkh srolf|pdnhu fdqqrw revhuyh doo uhohydqw yduldeohv frqwhpsrudqhrxvo|1 Wkhuhiruh/ lw kdv wkh prwlyh iru xvlqj d prqhwdu| djjuhjdwh dv dq dgglwlrqdo lqglfdwru/ zkhq lw owhuv wkh dydlodeoh lqirupdwlrq wr xsgdwh lwv shufhswlrq rq wkh vwdwh ri wkh hfrqrp|1 Wkh rswlpdo zd| wr owhu wkh lqirupdwlrq lv ghulyhg xvlqj wkh phwkrgrorjlfdo iudphzrun vxjjhvwhg e| Vyhqvvrq ) Zrrgirug +5335e,1

Wkh sdshu lv rujdql}hg dv iroorzv1 Wkh prgho lv jlyhq lq Vhfwlrq 5/ zklfk dovr suhvhqwv rswlpdo prqhwdu| srolf|/ jlyhq wkh srolf|pdnhu*v shufhswlrq rq wkh vwdwh ri wkh hfrqrp|1 Vhfwlrq 6 suhvhqwv wkh dvvxpswlrqv frqfhuqlqj wkh lqirupdwlrq dydlodeoh wr wkh srolf|pdnhu1 Vhfwlrq 7 suhvhqwv wkh uhvxowv frqfhuqlqj wkh lqirupdwlrq ydoxh ri prqh|1 Vhfwlrq 8 frqfoxghv1

(10)

5 P rgho

Zh xvh d Qhz Nh|qhvldq prgho zlwk dgghg hqgrjhqrxv lq dwlrq dqg rxwsxw shuvlvwhqfh1 Wkh hfrqrp| lv ghvfulehg lq wkh iroorzlqj htxdwlrqv1

+|' .|+|nn E +|3 # E| .|Z|n n c| +514,

Z|' qw.|Z|nn E w Z|3n E+| +|? n 2c| +515,

Lq wkh deryh htxdwlrqv/ Z| ghqrwhv wkh lq dwlrq udwh dqg | wkh qrplqdo udwh ri

lqwhuhvw1 V|perov +| dqg +?| ghqrwh +wkh orjv ri, rxwsxw dqg srwhqwldo rxwsxw/

uhvshfwlyho|1 V|pero q ghqrwhv wkh glvfrxqw idfwru/ dqg / #/ w/ dqg ghqrwh rwkhu srvlwlyh frqvwdqwv/ zlwk 5 dfc o/ dqg w 5 dfc o1 Wkh huuru whupv c|

dqg 2c| duh l1l1g1 Wkh| kdyh }hur phdqv/ dqg wkhlu yduldqfhv duh ghqrwhg e|

j2

dqg j22/ uhvshfwlyho|1 Wlph shulrgv duh jlyhq lq wkh vxevfulswv/ dqg .| lv

wkh h{shfwdwlrqv rshudwru uhihuulqj wr wkh sulydwh djhqwv* h{shfwdwlrq iruphg lq shulrg |1 Wkh srolf|pdnhu*v h{shfwdwlrqv zloo qrw jhqhudoo| eh lghqwlfdo wr wkh sulydwh djhqwv* h{shfwdwlrqv1

Htxdwlrqv +514, dqg +515, duh/ uhvshfwlyho|/ wkh djjuhjdwh ghpdqg ixqfwlrq dqg wkh djjuhjdwh vxsso| ixqfwlrq1 Wrjhwkhu wkh| irup d prgho zlgho| xvhg lq frqwhpsrudu| prqhwdu| srolf| dqdo|vlv/ vhh hj Uxghexvfk ) Vyhqvvrq +4<<<,/ Fodulgd/ Jdol ) Jhuwohu +4<<<,/ dqg Mhqvhq +5335,1 Zkhq ' dqg w ' wkh prgho lv d yhuvlrq ri wkh vwdqgdug Qhz Nh|qhvldq prgho/ zklfk lv frpsohwho| iruzdug orrnlqj1 Lw fdq eh ghulyhg iurp h{solflw plfurirxqgdwlrqv/ e| olqhdul}lqj d jhqhudo htxloleulxp prgho zlwk vwdjjhuhg sulfh vhwwlqj dqg prqrsrolvwlf frpshwlwlrq1 Wkh prwlydwlrq iru dgglqj wkh odjjhg rxwsxw dqg lq dwlrq whupv lq htxdwlrqv +514, dqg +515, lv wr lpsuryh wkh hpslulfdo shuirupdqfh ri wkh prgho/ dv hpskdvl}hg e| hj Ixkuhu ) Prruh +4<<8, dqg Ixkuhu +4<<:/ 5333,1 Wkh wkhruhwlfdo mxvwlfdwlrqv iru dgglqj wkhvh whupv duh udwkhu ydjxh/ dv glvfxvvhg e| hj Fodulgd/ Jdol ) Jhuwohu +4<<</ Vhfwlrq 9, dqg Mhqvhq +5335,1

Srwhqwldo rxwsxw lv dvvxphg wr iroorz dq h{rjhqrxv surfhvv +?

| ' 4+?|3n c|c +516,

zkhuh 4 5 dfc o lv d frqvwdqw/ dqg c| lv dq l1l1g1 vwrfkdvwlf huuru whup zlwk

}hur phdq dqg yduldqfh ghqrwhg e| j2 1

Wkh lqvwuxphqw ri prqhwdu| srolf| lv wkh lqwhuhvw udwh1 Wkh srolf|pdnhu vhwv | lq shulrg | vr dv wr plqlpl}h/ lq d glvfuhwlrqdu| pdqqhu/

"

[

'f

q_u

|n|c +517,

zkhuh wkh shulrg orvv ixqfwlrq lv jlyhq e|

(11)

Lq wkh deryh htxdwlrqv/ b+ dqg b ghqrwh srvlwlyh frqvwdqwv1 u|n| ghqrwhv wkh

srolf|pdnhu*v h{shfwdwlrqv ri u|n dv ri shulrg |1

Htxdwlrq +517, lv khuh wdnhq dv wkh vrflhw|*v orvv ixqfwlrq/ iroorzlqj wkh fxvwrpdu| dssurdfk lq wkh olwhudwxuh1 Iru d sxuho| iruzdug orrnlqj prgho/ wkh deryh orvv ixqfwlrq zlwk b ' f lv ghulyhg iurp plfurirxqgdwlrqv wr uhsuhvhqw

wkh +glv,xwlolw| ri d uhsuhvhqwdwlyh frqvxphu lq hj Urwhpehuj ) Zrrgirug +4<<<,1 Wkh lqwhuhvw udwh vprrwklqj remhfwlyh/ uh hfwhg lq b : f/ lpsuryhv

wkh hpslulfdo shuirupdqfh ri wkh prgho lq wkh vhqvh wkdw wkh yduldqfh ri wkh lqwhuhvw udwh zrxog rwkhuzlvh eh lpsodxvleo| odujh lq prgho vlpxodwlrqv/ vhh hj Uxghexvfk ) Vyhqvvrq +4<<<,1

514 Lqglfdwru yduldeohv

Li zh zhuh wr dvvxph ixoo lqirupdwlrq/ wkh prgho zrxog eh frpsohwho| ghqhg lq wkh deryh htxdwlrqv1 Krzhyhu/ zh zloo dvvxph gdwd xqfhuwdlqw|/ dqg wkhuhiruh wkhuh zloo eh urrp iru d prqhwdu| djjuhjdwh dv dq lqglfdwru yduldeoh1 Zh dvvxph wkdw wkh ghpdqg iru prqh| lv jlyhq e| wkh iroorzlqj htxdwlrq1 Lw lqfoxghv d iruzdug orrnlqj hohphqw1

6|' >6|3n >2.|6|nn >+| >e|n ec|c +519,

zkhuh 6|lv ghqhg dv wkh orj ri wkh dprxqw ri prqh| lq uhdo whupv1 V|perov

>/ >₂/ >/ dqg >_e ghqrwh srvlwlyh frqvwdqwv/ dqg ec| lv dq l1l1g1 vwrfkdvwlf

huuru whup zlwk }hur phdq dqg yduldqfh ghqrwhg e| j2 e1

Wkh h{lvwhqfh ri d iruzdug orrnlqj hohphqw lq wkh prqh| ghpdqg htxdwlrq lv qrw d yhu| frpprq dvvxpswlrq1 Krzhyhu/ lw lv vxjjhvwhg e| hj Fxwkehuwvrq ) Wd|oru +4<;:/ 4<<3,/ Qhovrq +5335d/ 5335e,/ dqg Ulsdwwl +4<<;/ Fkdswhu 5,1 Dovr/ vhyhudo hpslulfdo prqh| ghpdqg irupxodwlrqv lqfoxgh d ihdwxuh zklfk lv frqvlvwhqw zlwk wkh deryh prgho/ qdpho|/ wkh ghshqghqfh ri prqh| ghpdqg rq orqj whup lqwhuhvw udwhv14

Wkh prqh| ghpdqg htxdwlrq +519, fdq eh ghulyhg dvvxplqj dgmxvwphqw frvwv lq prqh| kroglqjv/ dv vkrzq lq Dsshqgl{ D1 Hyhq wkrxjk h{solflwo| iruzdug orrnlqj irupxodwlrqv ri hpslulfdo prqh| ghpdqg htxdwlrqv duh uhodwlyho| uduh/ wkh lghd ri dgmxvwphqw frvwv lq prqh| kroglqjv lv zlgho| dfnqrzohgjhg dqg e| qr phdqv qhz/ vhh hj Jrogihog +4<:6,1 Hpslulfdo prqh| ghpdqg prghov sudfwlfdoo| qhyhu lpso| lpphgldwh dgmxvwphqw ri prqh| ghpdqg wr d ohyho jlyhq e| wkh gulylqj h{sodqdwru| yduldeohv1 Wr d odujh h{whqw/ wklv revhuyhg vorz dgmxvwphqw lv vhhq wr vwhp iurp wkh frvwv ri dgmxvwlqj prqh| kroglqjv1 Wkh| duh wkrxjkw wr dulvh iurp ydulrxv frvwv uhodwhg wr sruwirolr dgmxvwphqwv/ lqfoxglqj wkh frvwv ri wkh wlph dqg lqfrqyhqlhqfh lq jdwkhulqj dqg surfhvvlqj lqirupdwlrq1

Xqghu udwlrqdo h{shfwdwlrqv/ wkh h{lvwhqfh ri dgmxvwphqw frvwv gluhfwo| lpsolhv d iruzdug orrnlqj hohphqw lq wkh prqh| ghpdqg htxdwlrq1 Rxu edvholqh sdudphwhu ydoxhv zloo vdwlvi| > ' >₂*q/ lq dffrugdqfh zlwk wkh dgmxvwphqw frvw dssurdfk/ dv vkrzq lq Dsshqgl{ D1

4_{Wr vhh wkdw wklv ihdwxuh lv frqvlvwhqw zlwk wkh fxuuhqw prgho/ vhh htxdwlrq +51:, ehorz/}

dqg uhphpehu wkh h{shfwdwlrqv wkhru| rq wkh whup vwuxfwxuh ri lqwhuhvw udwhv1

(12)

Lq wkh irupxodwlrq ri Dsshqgl{ D/ wkh dgmxvwphqw frvwv duh dvvxphg wr eh uhodwhg wr fkdqjhv lq prqh| kroglqjv lq uhdo whupv/ qrw lq qrplqdo whupv/ dv lq vrph rwkhu vwxglhv1 Lq wklv uhvshfw/ rxu dssurdfk iroorzv wkdw ri Fxwkehuwvrq ) Wd|oru +4<<3, dqg Qhovrq +5335d/ e,/ dprqj rwkhuv1 Erwk irupxodwlrqv kdyh wkhlu phulwv/ wkhruhwlfdoo| dv zhoo dv hpslulfdoo|1 Rxu fkrlfh lv prwlydwhg e| d frqvlghudwlrq ri wkh vrxufh ri wkh dgmxvwphqw frvwv1 Frvwv uhodwhg wr lqirupdwlrq suredeo| grplqdwh gluhfw wudqvdfwlrq frvwv/ zkhq lw frphv wr wkh frvwv ri dgmxvwlqj prqh| kroglqjv e| lqglylgxdo krxvhkrogv dqg upv1

Dgmxvwphqw frvwv lv qrw wkh rqo| dujxphqw wkdw fdq eh suhvhqwhg iru wkh h{lvwhqfh ri d iruzdug orrnlqj hohphqw lq prqh| ghpdqg1 Lw lv dq rog shufhswlrq wkdw wkh ghpdqg iru uhdo edodqfhv fdq eh dhfwhg e| fxuuhqw zhdowk/ dqg qrw mxvw e| fxuuhqw lqfrph1 Iru h{dpsoh/ Iulhgpdq +4<89, lqfoxghv lqfrph lq d prqh| ghpdqg htxdwlrq mxvw dv d sur{| iru zhdowk1 Li lw lv wkh fdvh wkdw zhdowk dhfwv prqh| ghpdqg/ wkhq lw lv reylrxv wkdw rqh vkrxog lqfoxgh d iruzdug orrnlqj hohphqw lq d prqh| ghpdqg ixqfwlrq/ li wkh gulylqj yduldeohv duh lqfrph dqg wkh lqwhuhvw udwh/ dv lq htxdwlrq +519,1 Wklv lv ehfdxvh fxuuhqw zhdowk qdwxudoo| ghshqgv rq wkh h{shfwhg ixwxuh ydoxhv ri wkh gulylqj yduldeohv1 Lq dgglwlrq wr dgmxvwphqw frvwv/ d fdvh iru wkh h{lvwhqfh ri d iruzdug orrnlqj hohphqw lq prqh| ghpdqg fdq eh pdgh dovr rxw ri glhuhqw dujxphqwv1

Zkhq lqwhusuhwlqj vrph uhvxowv suhvhqwhg lq wkh iroorzlqj vhfwlrqv/ lw lv xvhixo wr qrwh wkdw htxdwlrq +519, lpsolhv 6|' 6|3n 2.| % h 6|n 26h|nn ₂ ₂ h 6|n2n & c +51:,

zkhuh h6| >+| >e| n ec|1 Htxdwlrq +51:, suhvhqwv wkh vwdqgdug

iruzdug0edfnzdug orrnlqj vroxwlrq wr wkh vhfrqg rughu glhuhqfh htxdwlrq +519,1 Iru sodxvleoh sdudphwhu ydoxhv/ rqh ri wkh fkdudfwhulvwlf urrwv ri wkh htxdwlrq lv ehwzhhq }hur dqg rqh/ dqg wkh rwkhu lv juhdwhu wkdq rqh1 Khuh/ wkh urrwv duh ghqrwhg e| 5 Efc dqg ₂ 5 Ec 41 Dffruglqj wr htxdwlrq +51:,/ fxuuhqw uhdo prqh| ghpdqg ghshqgv rq wkh h{shfwhg ixwxuh ydoxhv ri rxwsxw dqg wkh lqwhuhvw udwh/ dv zhoo dv rq wkhlu sdvw ydoxhv wkurxjk 6|31

Wkh lqirupdwlrqdo dvvxpswlrqv xvhg lq wklv sdshu zloo qrw eh ghvfulehg dw ixoo xqwlo lq Vhfwlrq 61 Ohw xv/ krzhyhu/ suhvhqw khuh wkh srolf|pdnhu*v qrlv| revhuydwlrq frqfhuqlqj frqwhpsrudqhrxv rxwsxw1 Dv lv zhoo nqrzq/ vl}deoh uhylvlrqv duh riwhq pdgh wr JGS vwdwlvwlfv/ hyhq vhyhudo txduwhuv diwhu wkhlu lqlwldo sxeolfdwlrq1 Wkh iroorzlqj htxdwlrq ghvfulehv wkh lpsolfdwlrqv ri vxfk uhylvlrqv1 Wkh srolf|pdnhu*v shulrg | revhuydwlrq rq vdph shulrg rxwsxw lv jlyhq e|

+f

| ' +|n | +51;,

zkhuh |lv dq l1l1g1 vwrfkdvwlf huuru whup zlwk }hur phdq dqg yduldqfh ghqrwhg

e| j2

1 Zkloh wkh revhuydwlrq rq wkh vdph |hdu rxwsxw lv qrlv|/ lw zloo eh

dvvxphg wkdw wkh srolf|pdnhu ohduqv wkh wuxh ydoxh ri +| lq |hdu | n 1 Wkh

ydoxh ri j2

xvhg lq wkh vlpxodwlrqv lv edvhg rq wkh vwxg| e| Frhqhq/ Ohylq )

Zlhodqg +5334,/ dqg lw lv fdofxodwhg xvlqj wkh dfwxdo uhylvlrqv wkdw kdyh wdnhq sodfh lq wkh hxur duhd JGS vwdwlvwlfv1

(13)

515 Sdudphwhu ydoxhv

Wkh ydoxhv ri wkh sdudphwhuv wkdw dsshdu lq wkh deryh htxdwlrqv duh fkrvhq vr dv wr uhsuhvhqw frpsurplvh ydoxhv ri wkh olwhudwxuh1 Wkh yduldqfhv iru nh| yduldeohv wkdw wkh sdudphwhuv surgxfh lq wkh vlpxodwlrqv duh uhdvrqdeo| forvh wr wkrvh revhuyhg lq wkh gdwd1 Wkh edvholqh ydoxhv duh suhvhqwhg lq Wdeoh 41 Zkhq frqvlghulqj wkhp/ rqh vkrxog nhhs lq plqg wkdw khuh wkh wlph iuhtxhqf| lv wdnhq wr eh dqqxdo udwkhu wkdq txduwhuo|/ dv lq vhyhudo rwkhu sdshuv1

Yduldeohv +?

|/ +|/ +|f/ dqg 6|/ duh ghqhg dv shufhqwdjh ghyldwlrqv iurp

ydoxhv1 Wkhuhiruh/ qr frqvwdqw whupv dsshdu lq htxdwlrqv +514,/ +515,/ +516,/ ru +519,1 Frqvhtxhqwo|/ iru h{dpsoh/ wkh lq dwlrq yduldeoh Z| fdq eh ghqhg dv d

ghyldwlrq ri vrph wdujhw ydoxh uhdfkhg lq wkh vwhdg| vwdwh1

w # q 4 b+ b

fDS fD2 ffS fH fbS fbD f

> >₂ > >_e j j j2 j je

fH f. f2D f f2e f.b fSD fS f2Dc f2 Wdeoh 4= Edvholqh sdudphwhu ydoxhv1

Wkh ydoxhv ri wkh sdudphwhuv lq htxdwlrqv +514,/ +515,/ +516,/ +518,/ dqg +51;, duh vlplodu wr wkrvh xvhg e| Hkupdqq ) Vphwv +5334, iru wkh hxur duhd/ h{fhsw iru wkh dsshdudqfh ri q lq htxdwlrq +515,1 Wkh sdudphwhuv ri wkh prqh| ghpdqg htxdwlrq duh fkrvhq vr wkdw lq wkh orqj uxq wkh| lpso| xqlw vfdoh hodvwlflw| dqg dq lqwhuhvw udwh vhplhodvwlflw| ri fD1 Wkhvh duh frqyhqwlrqdo ydoxhv1 Wkh sdudphwhuv wkdw ghwhuplqh wkh g|qdplfv ri wkh prqh| ghpdqg htxdwlrq/ > dqg >₂/ duh vhw dffruglqj wr wkh ydoxhv hvwlpdwhg dqg xvhg e| Qhovrq +5335d/ e,15 _{Wkh edvholqh ydoxhv vdwlvi| >}

' >2*q/ dv glvfxvvhg lq Vhfwlrq 514 dqg

Dsshqgl{ D1 Xvlqj wkh hvwlpdwhv ri Fxwkehuwvrq ) Wd|oru +4<;:/ 4<<3, zrxog uhvxow lq vrphzkdw juhdwhu ydoxhv iru > dqg >₂ wkdq wkrvh jlyhq lq Wdeoh 41

Zkdw uhpdlqv wr eh ghvfulehg lv wkh fkrlfh ri wkh ydoxh iru wkh sdudphwhu wkdw lv wkh prvw fulwlfdo rqh iru wkh uhvxowv ri wklv vwxg|/ wkdw lv/ wkh prqh| ghpdqg vkrfn vwdqgdug huuru je1 Khuh/ prvw ri wkh uhvxowv duh suhvhqwhg iru

wkh wzr glhuhqw ydoxhv suhvhqwhg lq Wdeoh 41 Lq fkrrvlqj je rqh fdq xvh

hpslulfdo prqh| ghpdqg vwxglhv dv d srlqw ri uhihuhqfh16 _{Iru j}

e frqfhuqlqj

dqqxdo hxur duhd P6/ wkh| lpso| d udqjh ri f2 wkurxjk f2D1 Vlqfh wkhvh duh lq0vdpsoh hvwlpdwhv/ d ydoxh forvh wr f2D vhhpv wr eh d jrrg fdqglgdwh1

5_{Qhovrq xvhv txduwhuo| iuhtxhqf|1 Zh lqihu Qhovrq*v ydoxh iru sdudphwhu / zklfk dsshduv}

lq Dsshqgl{ D/ dqg wkhq xvh wkdw ydoxh wr fdofxodwh 4 dqg 5 iru dqqxdo iuhtxhqf|1 6_{Wkh hpslulfdo vwxglhv xvhg dv uhihuhqfhv khuh duh Eudqg ) Fdvvrod +5333,/ Eu jjhpdq}

+5333,/ Fdo}d/ Jhughuphlhu ) Ohy| +5334,/ Frhqhq ) Yhjd +4<<<,/ Ghgrod/ Jdlrwwl ) Vlolsr +5334,/ Jhuodfk ) Vyhqvvrq +5335,/ dqg hduolhu vwxglhv vxuyh|hg e| Eurzqh/ Idjdq ) Khqu| +4<<:,1

(14)

Krzhyhu/ wr vhh wkh odujhvw sodxvleoh lqirupdwlrq ydoxh ri 6| lq wkh fxuuhqw

frqwh{w/ zh frqvlghu wkh fdvh ri je' f2/ lq dgglwlrq wr je ' f2D17

Wr fkhfn urexvwqhvv/ zh h{shulphqw zlwk d odujh vfdoh ri qrq0edvholqh sdudphwhu ydoxhv iru hyhu| sdudphwhu ri wkh prgho1 Wklv zloo eh grqh lq Iljxuhv 7 dqg 81

516 Rswlpdo prqhwdu| srolf|

Wkh srolf|pdnhu vhwv wkh lqwhuhvw udwh lq shulrg | vr dv wr plqlpl}h wkh orvv suhvhqwhg lq htxdwlrq +517,1 Zh frqvlghu rswlpdo wlph0frqvlvwhqw srolf|/ phdqlqj wkdw wkh srolf|pdnhu uh0rswlpl}hv hdfk shulrg/ wdnlqj wkh sulydwh vhfwru h{shfwdwlrqv dv jlyhq1

Vyhqvvrq ) Zrrgirug +5335e, vkrz wkdw wkh vr fdoohg fhuwdlqw| htxlydohqfh uhvxow krogv lq wklv vhwxs/ hyhq wkrxjk wkh srolf|pdnhu grhv qrw kdyh frpsohwh lqirupdwlrq rq wkh vwdwh ri wkh hfrqrp|/ dqg hyhq wkrxjk wkh lqirupdwlrq vwuxfwxuh lv dv|pphwulf1 Wkh fhuwdlqw| htxlydohqfh phdqv wkdw wkh rswlpdo srolf| lv wkh vdph dv li wkh vwdwh ri wkh hfrqrp| zhuh ixoo| revhuydeoh/ h{fhsw wkdw wkh srolf|pdnhu qrz uhvsrqgv wr wkh rswlpdo hvwlpdwh ri wkh vwdwh ri wkh hfrqrp| udwkhu wkdq wr wkh dfwxdo vwdwh1 Wklv phdqv wkdw wkh rswlpdo glvfuhwlrqdu| prqhwdu| srolf| fdq eh uhsuhvhqwhg e| d uhdfwlrq ixqfwlrq zklfk jlyhv | dv d olqhdu frpelqdwlrq ri wkh vwdwh yduldeohv ri wkh prgho/ dv wkh|

Wkh prgho lv suhvhqwhg lq d vwdwh0vsdfh irup lq Dsshqgl{ E1 Wkhuh/ wkh yhfwru ri vwdwh yduldeohv lv ghqhg dv

f| ' +|3 Z|3 +|? c| 2c| |3 6|3 ec| +51<,

Wkh rswlpdo glvfuhwlrqdu| prqhwdu| srolf| uhdfwlrq ixqfwlrq lv ghulyhg xvlqj wkh dojrulwkp ghvfulehg lq Vyhqvvrq ) Zrrgirug +5335e/ Dsshqgl{ D,1 Rqh fdq zulwh wkh rswlpdo srolf| lq whupv ri d yhfwru 8 dv

|' 8 f||

' f+|3|n fS.Z|3| f+||? n 22bc|| +5143,

n b2c||n feD|3|c

zkhuh +r| ghqrwhv wkh srolf|pdnhu*v shulrg | shufhswlrq ri +r/ dqg vlploduo|

iru doo rwkhu yduldeohv1 Qrwh wkdw wkh rqo| vwdwh yduldeohv wkdw gr qrw hqwhu

7_{Wkh hpslulfdo prghov ri hxur duhd P6 frqvlghuhg khuh xvh txduwhuo| gdwd1 Wkh hvwlpdwh}

iru txduwhuo| x7lv wudqviruphg wr dqqxdo iuhtxhqf| wdnlqj wkh dqqxdo huuru dv dq dyhudjh ri

wkh txduwhuo| huuruv/ dqg dvvxplqj wkdw wkh txduwhuo| huuruv duh vhuldoo| xqfruuhodwhg1 Wkh odwwhu dvvxpswlrq phdqv wkdw wkh hqgsrlqwv ri wkh udqjh iru dqqxdo x7 vkrxog suredeo| eh

voljkwo| juhdwhu wkdq wkh 3=45 dqg 3=58 jlyhq lq wkh wh{w1 Lq dgglwlrq/ rqh vkrxog qrwh wkdw wkh ydoxhv ri x7 frqvlghuhg khuh qhhg qrw eh sodxvleoh iru rwkhu prqhwdu| djjuhjdwhv ru

rwkhu frxqwulhv1 Lqghhg/ lq rwkhu vlpxodwlrq vwxglhv zlwk prqh| ghpdqg htxdwlrqv/ ydoxhv frqvlghudeo| juhdwhu wkdq rqh duh w|slfdoo| xvhg iru x7/ zkhq wudqviruphg lqwr dqqxdo

(15)

wkh uhdfwlrq ixqfwlrq duh wkh odvw wzr hohphqwv ri yhfwru f|/ wkdw lv/ wkrvh

dvvrfldwhg zlwk prqh| ghpdqg1 Wklv lv gxh wr wkh fhuwdlqw| htxlydohqfh1 Wkh uhdfwlrq ixqfwlrq jlyhq lq htxdwlrq +5143, kdv vwdqgdug surshuwlhv iru d Qhz Nh|qhvldq prgho zlwk dgghg edfnzdug orrnlqj ihdwxuhv1 Wkh srolf|pdnhu udlvhv wkh lqwhuhvw udwh lq uhvsrqvh wr d shufhswlrq wkdw wkhuh kdv ehhq d srvlwlyh djjuhjdwh ghpdqg vkrfn ru d srvlwlyh frvw0sxvk vkrfn1 Wklv lv uh hfwhg lq wkh frh!flhqwv ri c|| dqg 2c||1 Odjjhg rxwsxw dqg lq dwlrq dhfw wkh lqwhuhvw

udwh ehfdxvh ri wkh odjjhg whupv lq htxdwlrqv +514, dqg +515,1 Li srwhqwldo rxwsxw +?

| lv lqfuhdvhg/ dffruglqj wr wkh srolf|pdnhu*v shufhswlrq/ lw zloo orzhu

wkh lqwhuhvw udwh/ vlqfh qrz wkhuh lv pruh urrp iru surgxfwlrq wr lqfuhdvh zlwkrxw dq lqfuhdvh lq lq dwlrq18 _{Wkh odvw whup ri htxdwlrq +5143, iroorzv}

iurp wkh lqwhuhvw udwh vprrwklqj prwlyh dvvxphg lq htxdwlrq +518,1

6 Lqirupdwlrq

Lq wklv vhfwlrq/ zh ghvfuleh wkh lqirupdwlrq vhwv wkdw wkh djhqwv duh dvvxphg wr kdyh1 Zh wkhq vkrz krz wkdw lqirupdwlrq lv rswlpdoo| owhuhg/ xvlqj wkh phwkrgrorjlfdo iudphzrun ghyhorshg e| Vyhqvvrq ) Zrrgirug +5335d/ e/ f/ 5336, lq d vhulhv ri sdshuv1 Wkh| exlog rq wkh zrun e| Shduopdq/ Fxuulh ) Ohylqh +4<;9, dqg Shduopdq +4<<5,/ dprqj rwkhuv1 Suhylrxv dssolfdwlrqv ri wkh iudphzrun lqfoxgh Drnl +5334/ 5335,/ Frhqhq/ Ohylq ) Zlhodqg +5334,/ Hkupdqq ) Vphwv +5334,/ Grwvh| ) Kruqvwhlq +5335,/ dqg Qhovrq +5335e,1

Wkh irup ri wkh prgho dqg wkh sdudphwhu ydoxhv duh dvvxphg wr eh nqrzq wr hyhu|erg|1 Lq frqwudvw/ wkh ydoxhv ri doo yduldeohv duh qrw dozd|v nqrzq wr wkh fhqwudo edqn1

614 Lqirupdwlrqdo dvvxpswlrqv

Wkh prgho vhwxs ri wklv sdshu lv dqdo|}hg xqghu wkuhh glhuhqw vhwv ri lqirupdwlrqdo dvvxpswlrqv1 Wkh| duh ghvfulehg ehorz1 Wkh uvw fdvh lv rqh zkhuh wkh srolf|pdnhu dqg wkh sulydwh djhqwv kdyh wkh vdph lqirupdwlrq vhw1 Lq frqwudvw/ lqirupdwlrq lv dv|pphwulf lq wkh odwwhu wzr fdvhv1 Lq wkh odvw fdvh/ wzr glhuhqw ydoxhv zloo eh xvhg iru wkh prqh| ghpdqg huuru whup yduldqfh j2

e1

Frpsohwh lqirupdwlrq1 Lq shulrg |/ erwk wkh srolf|pdnhu dqg wkh sulydwh djhqwv revhuyh wkh frpsohwh vwdwh yduldeoh yhfwru fr iru doo r |1

Htxlydohqwo|/ rqh fdq dvvxph wkdw lq shulrg | wkh| revhuyh wkh yduldeoh vhw i+?

|c +|c Z|c 6|j dqg doo lwv hduolhu ydoxhv/ ehfdxvh revhuylqj wklv vhw ri yduldeohv

jlyhv wkhp hqrxjk lqirupdwlrq vr wkdw wkh| fdq lqihu fr iru doo r |1

8_{Lw fdq eh qrwhg wkdw wkh frh!flhqw ri |}q

wmw zrxog eh }hur xqghu d Qhz Nh|qhvldq prgho

vhwxs zlwk ! @ @ 3 dqg @ 4/ vlqfh rxwsxw zrxog lpphgldwho| dgmxvw wr d fkdqjh lq lwv srwhqwldo ohyho zlwkrxw dq lqwhuhvw udwh uhdfwlrq1

(16)

yduldeohv ru/ htxlydohqwo|/ lw uhphpehuv lwv rzq suhylrxv hvwlpdwhv fr|3 iru

doo r |1 Wkh sulydwh djhqwv djdlq revhuyh fr iru doo r |1

Lqfrpsohwh lqirupdwlrq/ prqh| revhuyhg1 Lq shulrg | wkh

srolf|pdnhu revhuyhv i+|3c +f|c Z|c 6|j dqg dovr uhphpehuv fr|3 iru doo r |1

Wkh sulydwh djhqwv djdlq revhuyh fr iru doo r |1 Lq wklv fdvh/ uhvxowv zloo eh

suhvhqwhg iru wzr glhuhqw dvvxpswlrqv frqfhuqlqj wkh prqh| ghpdqg huuru whup yduldqfh j2

e1 Iru wkhvh ydoxhv/ vhh Wdeoh 41

615 Ilowhulqj ri lqirupdwlrq

Ghqrwlqj wkh revhuydwlrq yhfwru e| ~|/ wkh owhulqj htxdwlrqv fdq eh zulwwhq

dv

f|| ' xf||3n [~| +614,

zkhuh x dqg [ duh pdwulfhv zlwk wkh dssursuldwh glphqvlrqv1

Wr qg rxw wkh rswlpdo x dqg [/ zh dsso| wkh iudphzrun ri Vyhqvvrq dqg Zrrgirug +5335e,1 Dsshqgl{ E suhvhqwv wkh ghulydwlrq ri wkh rswlpdo x dqg [ lq whupv ri wkh sdudphwhuv ri wkh prgho1

Xvlqj wkh rswlpdo owhu/ rqh fdq h{suhvv wkh lqwhuhvw udwh gluhfwo| dv d ixqfwlrq ri wkh srolf|pdnhu*v revhuydwlrqv1 Dffruglqj wr htxdwlrq +5143,/ wkh rswlpdo srolf| fdq eh zulwwhq dv | ' 8 f||1 Wkh rswlpdo owhulqj surfhvv ri

htxdwlrq +614, fdq/ lq wxuq/ dovr eh zulwwhq dv f|| ' Kf|3|3n[~|zkhuh K lv

d pdwul{ zlwk wkh dssursuldwh glphqvlrqv/ vhh Dsshqgl{ E1 Lwhudwlqj iruzdug wkh odwwhu htxdwlrq/ wkh lqwhuhvw udwh fdq eh zulwwhq dv

|' " [ 'f 8 K_[u_~ | +615,

zkhuh u lv wkh odj rshudwru1 Wkh orqj uxq hhfwv ri wkh revhuydwlrqv rq | fdq

eh irxqg e| vhwwlqj u ' f lq htxdwlrq +615,1

61514 Wkh fdvh zkhuh prqh| lv revhuyhg

Dv dq h{dpsoh/ ohw xv frqvlghu vrph ihdwxuhv ri wkh rswlpdo owhu lq wkh fdvh zkhuh wkh srolf|pdnhu kdv lqfrpsohwh lqirupdwlrq exw revhuyhv prqh|/ dqg je' f2D1

Wkh revhuydwlrq yhfwru ri wkh srolf|pdnhu lv

(17)

Wkh rswlpdo owhu lv jlyhq e| htxdwlrq +614, zkhuh [ ' 5 9 9 9 9 9 9 9 9 9 9 7 f f f f f f f ffS ff ff fDf fb. f... fbD ffb ffS fHS fS.H ffHD f f f f f f f f ff2f ff f2HD fDHb 6 : : : : : : : : : : 8 c +617,

dqg wkh fruuhvsrqglqj pdwul{ x lv suhvhqwhg lq htxdwlrq +E5149, ri Dsshqgl{ E151

Wkh odvw froxpq ri wkh deryh pdwul{ [ suhvhqwv wkh frh!flhqwv ri wkh revhuyhg uhdo edodqfhv 6| lq wkh rswlpdo owhulqj htxdwlrqv1 Qrwh wkdw wkh

juhdwhu lv 6| wkh juhdwhu lv +_||?/ dv wkh fruuhvsrqglqj frh!flhqw htxdov d[o_ce '

fDf : f1 Wkh irxuwk froxpq ri [ lv reylrxvo| devhqw lq wkh fdvh zkhuh prqh| lv qrw revhuyhg1 Krzhyhu/ wkh h{lvwhqfh ri wkh irxuwk froxpq ri [ lv qrw wkh rqo| glhuhqfh ehwzhhq wkhvh wzr fdvhv1 Reylrxvo|/ dgglqj prqh| lq wkh vhw ri revhuydeoh yduldeohv dovr fkdqjhv wkh hohphqwv lq wkh uvw wkuhh froxpqv ri [1

Ohw xv qrz frqvlghu wkh rswlpdo lqwhuhvw udwh h{suhvvhv gluhfwo| lq

whupv ri wkh srolf|pdnhu*v revhuydwlrqv1 Iru vrph vhw ri frqvwdqwv

Wkh odwwhu h{suhvvlrqv jlyh xv wkh orqj uxq hhfwv ri revhuydwlrqv +|3/ +f|/

Z| dqg 6| rq wkh lqwhuhvw udwh1 Wkh fruuhvsrqglqj vdph shulrg hhfwv duh

@cf ' f2/ @2cf ' H/ @cf ' H/ dqg @ecf ' fSD1 Wkh qhjdwlyh hhfw

ri uhdo edodqfhv rq wkh lqwhuhvw udwh iroorzv iurp wkh idfw wkdw prqh| lv dq lqglfdwru ri srwhqwldo rxwsxw/ dv vhhq lq htxdwlrq +617,1 Wkh vdph shulrg hhfw ri prqh| rq wkh lqwhuhvw udwh lv rqo| fSD/ zkloh wkh orqj uxq hhfw lv S.1 Wklv uh hfwv wkh idfw wkdw wkh srolf|pdnhu fdqqrw h{dfwo| lghqwli| wkh vrxufh ri wkh odwhvw fkdqjh lq uhdo edodqfhv1 Wkh| duh qrw rqo| dhfwhg e| srwhqwldo rxwsxw vkrfnv exw dovr e| djjuhjdwh ghpdqg vkrfnv/ dprqj rwkhu wklqjv1

Wkh idfw wkdw wkh dprxqw ri prqh| lq uhdo whupv lv d ydoxdeoh lqglfdwru ri srwhqwldo rxwsxw uhvxowv iurp wkh irup ri wkh prqh| ghpdqg htxdwlrq1 Wklv fdq eh vhhq e| frqvlghulqj htxdwlrq +51:,/ qrwlqj wkdw fxuuhqw prqh| ghpdqg lv dhfwhg e| wkh sulydwh djhqwv* h{shfwdwlrqv frqfhuqlqj ixwxuh rxwsxw1 Uhphpehu wkdw xqghu wkh fxuuhqw dvvxpswlrqv wkh sulydwh djhqwv revhuyh wkh frqwhpsrudqhrxv srwhqwldo rxwsxw/ exw wkh srolf|pdnhu grhv qrw1 48

(18)

Dovr uhphpehu wkh vwurqj shuvlvwhqfh lq srwhqwldo rxwsxw1 Wkh sulydwh djhqwv nqrz wkdw wkh srolf|pdnhu zloo ryhu wlph ohduq derxw wkh wuxh ydoxh ri srwhqwldo rxwsxw dqg dfw vr dv wr eulqj rxwsxw forvh wr srwhqwldo1 Wkhuhiruh/ wkh srolf|pdnhu fdq ohduq derxw shulrg | srwhqwldo rxwsxw e| frqvlghulqj shulrg | ghpdqg iru prqh|1

7 Ydoxh ri prqh| dv dq lqglfdwru

Wdeoh 5 suhvhqwv wkh hfrqrplf rxwfrph xqghu irxu glhuhqw lqirupdwlrqdo dvvxpswlrqv1 Wkh ydoxhv uhsruwhg lq wkh wdeoh duh edvhg rq vlpxodwlrqv1 Wkh| duh wkh phdq ydoxhv ryhu 4333 uhsolfdwlrqv/ zlwk hdfk ehlqj d vdpsoh ri 833 |hduv1 Lq hdfk uhsolfdwlrq/ wkh prgho zdv vlpxodwhg iru 933 |hduv/ dqg wkh uvw 433 |hduv zhuh ljqruhg lq rughu wr devwudfw iurp wkh vwduw0xs ghsduwxuhv iurp vwhdg| vwdwh frqglwlrqv1 Wkh vdph vhwv ri vkrfn vhulhv fuhdwhg e| d udqgrp qxpehu jhqhudwru zhuh xvhg iru doo wkh lqirupdwlrqdo dvvxpswlrqv1

Vwg Phdq orvv Ydu ri Ydu ri Ydu ri

Lqirupdwlrq ri PG +Lqgh{/ frpsohwh rxwsxw lq d0 lqw1udwh

+L@lqfrpsohwh, vkrfn lqirupdwlrq @ 433, jds wlrq fkdqjh Frpsohwh ff fbb2 Se DD L/ qr prqh| f 2bf e D.. L/ prqh| f2D fH 2. Df DS2 L/ prqh| f2 fS Sf Se DeHf Wdeoh 5= Hfrqrplf rxwfrphv1

Dv vkrzq e| Wdeoh 5/ wkh pruh lqirupdwlrq wkh srolf|pdnhu kdv/ wkh ehwwhu

lv wkh shuirupdqfh ri wkh hfrqrp|1 Iru hdfk dvvxpswlrq frqfhuqlqj b+ dqg

b/ wkh hfrqrp| shuirupv ehvw lq wkh fdvh ri frpsohwh lqirupdwlrq1 Wkh zruvw

slhfh ri lqirupdwlrq1 Xvlqj prqh| dv dq lqglfdwru lpsuryhv wkh shuirupdqfh ri wkh hfrqrp|1 Wkh vpdoohu lv wkh prqh| ghpdqg huuru whup yduldqfh j2

e wkh

juhdwhu lv wkh lpsuryhphqw1

Lq dgglwlrq wr Wdeoh 5/ wkh surshuwlhv ri wkh prgho xqghu glhuhqw lqirupdwlrqdo dvvxpswlrqv fdq eh vwxglhg e| frqvlghulqj wkh hhfwv ri sduwlfxodu vkrfnv1 Iljxuhv 4 wkurxjk 6 vkrz wkh g|qdplf uhvsrqvhv ri wkh hfrqrp| wr rqh vwdqgdug ghyldwlrq lqqrydwlrqv lq wkh vkrfn whupv1

Iljxuh 4 suhvhqwv lpsxovh uhvsrqvhv vkrzlqj irxu fdvhv lq hdfk jxuh1 Wkh wkuhh fdvhv duh 4, frpsohwh lqirupdwlrq/ 5, lqfrpsohwh lqirupdwlrq zlwk prqh| qrw revhuyhg/ 6, lqfrpsohwh lqirupdwlrq zlwk prqh| revhuyhg dqg je ' f2D/

(19)

dqg 7, lqfrpsohwh lqirupdwlrq zlwk prqh| revhuyhg dqg je' f21 Wkh ohvv

wkh srolf|pdnhu kdv lqirupdwlrq/ wkh vwurqjhu whqg wr eh wkh uhdfwlrqv ri wkh rxwsxw jds dqg wkh lq dwlrq udwh wr wkh vkrfnv lq c|/ 2c|/ dqg c|1 Wklv

lv ehfdxvh zlwk ohvv lqirupdwlrq wkh lqwhuhvw udwh uhdfwlrqv whqg wr eh ixuwkhu dzd| iurp rswlpdo1 Krzhyhu/ dgglwlrqdo lqirupdwlrq fdq dovr ohdg wr d zruvh rxwfrph lq wkh fdvh ri vrph sduwlfxodu vkrfnv/ dv vkrzq lq wkh uhvsrqvh ri wkh hfrqrp| wr dq rxwsxw revhuydwlrq vkrfn1

Iljxuh 5 suhvhqwv wkh dfwxdo lpsxovh uhvsrqvh dqg wkh uhvsrqvhv dv frqwhpsrudulo| shufhlyhg e| wkh srolf|pdnhu1 Doo olqhv lq doo judskv fruuhvsrqg wr wkh lqirupdwlrq dvvxpswlrq fdvh qxpehu 6 ri wkh suhylrxv sdudjudsk1 Wkdw lv/ zh vwxg| wkh fdvh zkhuh lqirupdwlrq lv lqfrpsohwh exw prqh| lv revhuyhg/ zlwk wkh vwdqgdug ghyldwlrq ri wkh prqh| ghpdqg huuru whup ehlqj je ' f2D1

Wkh judskv lq wkh wklug froxpq suhvhqw wkh uhvsrqvhv ri lq dwlrq dqg wkh lqwhuhvw udwh/ zklfk duh frqwhpsrudqhrxvo| nqrzq wr wkh srolf|pdnhu1 Wkh uvw froxpq ri Iljxuh 5 vkrzv krz idvw wkh srolf|pdnhu ohduqv wkh wuxh ydoxh ri wkh rxwsxw jds/ zkhq wkh hfrqrp| idfhv glhuhqw vkrfnv1 Vlqfh khuh lw lv dvvxphg wkdw prqh| lv xvhg dv dq lqglfdwru/ d vkrfn lq wkh prqh| ghpdqg ixqfwlrq lv vkrzq wr ohdg wr d plvwdnhq uhvsrqvh e| wkh srolf|pdnhu1 Xqghu frpsohwh lqirupdwlrq/ wkh srolf|pdnhu qdwxudoo| grhv qrw uhdfw wr vxfk d vkrfn/ dv vkrzq lq Iljxuh 41

Iljxuh 6 vkrzv wkh huuruv ri wkh srolf|pdnhu lq lwv frqwhpsrudqhrxv shufhswlrqv frqfhuqlqj +?

| dqg +|1 Lq hdfk jxuh wkh shufhswlrq huuru lv

suhvhqwhg iru wkuhh fdvhv= 4, lqfrpsohwh lqirupdwlrq zlwk prqh| qrw revhuyhg/ 5, lqfrpsohwh lqirupdwlrq zlwk prqh| revhuyhg/ dvvxplqj je ' f2D/ dqg 6,

wkh vdph dv wkh suhylrxv fdvh h{fhsw wkdw je ' f21 Dq huuru deryh }hur

phdqv wkdw wkh srolf|pdnhu*v shufhswlrq ri wkh yduldeoh lv juhdwhu wkdq wkh wuxh ydoxh1 Iru h{dpsoh/ wkh uvw judsk lq wkh uvw froxpq vkrzv krz pxfk idvwhu wkh srolf|pdnhu ohduqv wkh wuxh ydoxh ri d srwhqwldo rxwsxw vkrfn li lw xvhv prqhwdu| djjuhjdwh dv dq lqglfdwru1

714 Urexvwqhvv ri wkh lqirupdwlrq ydoxh ri prqh|

Iljxuh 7 vwxglhv wkh urexvwqhvv ri wkh uhvxowv wr glhuhqw sdudphwhu ydoxhv1 Lq Iljxuh 7 hdfk judsk ghslfwv/ iru ydulrxv ydoxhv ri rqh sdudphwhu/ wkh hhfw ri revhuylqj prqh| rq wkh h{shfwhg glvfrxqwhg orvvhv ri htxdwlrq +517,19 _Zkdw

lv vkrzq lv krz pxfk/ lq shufhqwdjh whupv/ prqh| uhgxfhv wkh orvv wkdw lv fdxvhg e| wkh lqfrpsohwhqhvv ri lqirupdwlrq1 D uhgxfwlrq ri 433( zrxog phdq wkdw xvlqj prqh| dv dq lqglfdwru ohdgv wr frpsohwh lqirupdwlrq1 Wkdw lv/ wkh judskv ghslfw ff EuUU uUU*EuUU uU/ zkhuh uU ghqrwhv wkh

ydoxh ri htxdwlrq +517, xqghu frpsohwh lqirupdwlrq/ uUU wkh fruuhvsrqglqj

orvv xqghu lqfrpsohwh lqirupdwlrq zlwkrxw prqh|/ dqg uUU xqghu lqfrpsohwh

lqirupdwlrq zlwk prqh| revhuyhg1

9_{Wkh ydoxh ri htxdwlrq +517, fdq eh fdofxodwhg dv lq Hkupdqq ) Vphwv +5334/ Dsshqgl{}

D16,/ zlwk vpdoo prglfdwlrqv gxh wr wkh dvvxpswlrq ri dv|pphwulf lqirupdwlrq1 Wkh prglfdwlrqv frqfhuq wkh whup zklfk wkh| ghqrwh e| Fk5₁

(20)

Iljxuh 7 orrnv dw rqh sdudphwhu dw d wlph/ vhwwlqj wkh uhvw dw wkhlu edvholqh ydoxhv/ zlwk je ' f2D1 Wkh yhuwlfdo gdvkhg olqh glvsod|v wkh edvholqh ydoxh1

Wkh judskv duh lq wkh iroorzlqj rughu= q/ / / w/ #/ j/ j2/ j/ j/ 4/

> ' >₂*q/ >*E > >₂/ >_e*E > >₂/ je/ b+/ b1 Dprqj wkhvh/

>*E > >₂ lv wkh orqj uxq vfdoh hodvwlflw| ri prqh| ghpdqg/ dqg >_e*E > >₂ lv wkh orqj uxq lqwhuhvw udwh vhpl hodvwlflw| ri prqh| ghpdqg1

Dv Iljxuh 7 vkrzv/ wkh vl}h ri wkh lpsuryhphqw lq wkh shuirupdqfh ri wkh hfrqrp| eurxjkw derxw e| xvlqj prqh| dv dq lqglfdwru vhhpv wr eh idluo| urexvw wr fkdqjhv lq wkh prgho sdudphwhuv1 Lq wklv uhvshfw/ wkh sdudphwhu ydoxh wkdw lv wkh prvw fuxfldo lv wkdw ri je/ dv qrwhg deryh1 Wkh fkrlfh ri je

zdv glvfxvvhg lq ghwdlo lq Vhfwlrq 5151

Wkh g|qdplfv ri prqh| ghpdqg duh dovr fuxfldo wr wkh uhvxowv1 Wkh hhfw ri fkdqjlqj wkh ydoxh ri > ' >₂*q lv vkrzq lq Iljxuh 7/ wklug urz/ wklug froxpq1 Xqghu wkh edvholqh > dqg >₂/ wkh fxuuhqw ydoxhv ri wkh gulylqj yduldeohv kdyh d zhljkw ri > >₂ ' f2D lq htxdwlrq +519,1 Dw wkh ohiw hqg ri wkh judsk > ' >₂*q ' f/ phdqlqj wkdw prqh| ghpdqg htxdwlrq lv ri d vwdwlf irup1 Vxfk d irup lv iuhtxhqwo| xvhg lq vlpsolhg prghov/ hyhq wkrxjk lw lv qrw hpslulfdoo| sodxvleoh1 Li > ' >2*q ' f wkh ydoxh ri prqh| dv dq lqglfdwru lv sudfwlfdoo|

}hur1 Wkh uhdvrq iru wklv zdv glvfxvvhg dw wkh hqg ri Vhfwlrq 61514= lw lv wkh ohdg whup lq wkh ghpdqg iru prqh| wkdw pdnhv prqh| dq lqglfdwru ri srwhqwldo rxwsxw1

Wr vhh wkh lpsruwdqfh ri wkh iruzdug orrnlqj hohphqw lq prqh| ghpdqg/ ohw xv vwxg| zkdw kdsshqv li >₂ fkdqjhv uhodwlyh wr >1 Vr idu/ lqfoxglqj Iljxuh 7/ zh kdyh dvvxphg wkdw > ' >2*q1 Wklv lv qrw wkh fdvh lq Iljxuh 81 Khuh/

lq rughu qrw wr fkdqjh wkh orqj uxq frh!flhqwv ri +| dqg | zkhq zh fkdqjh

>₂/ zh vhw > ' >W

n >W2*q >2*q/ zkhuh >W dqg >W2 ghqrwh wkh edvholqh ydoxhv1

Dffruglqj wr Iljxuh 8/ wkh ydoxh ri >2 kdv d vwurqj hhfw rq wkh jdlq iurp

revhuylqj prqh|1 Dw wkh ohiw hqg ri Iljxuh 8 wkh prqh| ghpdqg htxdwlrq lv sxuho| edfnzdug orrnlqj/ zlwk >₂ ' f dqg > ' f.D1 Dv lq wkh fdvh ri vwdwlf prqh| ghpdqg/ wkh lqirupdwlrq ydoxh ri prqh| lv qrz forvh wr }hur1

Lq dgglwlrq wr >₂ : f/ dqrwkhu fuxfldo dvvxpswlrq frqfhuqlqj wkh ydoxh ri prqh| dv lqglfdwru lv wkdw wkh sulydwh djhqwv srvvhvv vrph lqirupdwlrq qrw nqrzq wr wkh srolf|pdnhu1 Li zh fkdqjh wkh vhwxs e| dvvxplqj v|pphwulf lqirupdwlrq/ prqh| lv qr orqjhu dq lpsruwdqw slhfh ri lqirupdwlrq wr wkh srolf|pdnhu1 Vxfk dq dvvxpswlrq zrxog phdq wkdw wkh vhw ri yduldeohv wkdw wkh sulydwh djhqwv revhuyh lv lghqwlfdo wr wkh rqh revhuyhg e| wkh srolf|pdnhu/ ghvfulehg lq Vhfwlrq 6141

Lq wklv sdshu/ prqhwdu| srolf| lv dvvxphg wr eh glvfuhwlrqdu|/ dqg dvvxplqj frpplwphqw frxog pdnh d glhuhqfh1 Krzhyhu/ wkh uhvxowv ri Hkupdq ) Vphwv +5334, zlwklq d urxjko| vlplodu prgho iudphzrun duh idluo| urexvw wr d fkdqjh lq wklv dvvxpswlrq/ vxjjhvwlqj wkdw wkh vdph frxog eh wuxh khuh1

Wkh uhvxowv ri wklv vwxg| fdq eh frpsduhg wr wkrvh ri Frhqhq/ Ohylq ) Zlhodqg +5334,/ Grwvh| ) Kruqvwhlq +5335, dqg Qhovrq +5335e,/ zkr dovr vwxg| wkh uroh ri d prqhwdu| djjuhjdwh dv dq lqglfdwru/ dvvxplqj wkdw wkh srolf|pdnhu grhv qrw kdyh frpsohwh lqirupdwlrq1 Wkh ydoxh ri prqh| dv dq lqglfdwru lv irxqg wr eh idluo| vpdoo lq wkh iruphu wzr vwxglhv/ uh hfwlqj wkh idfw wkdw prqh| ghpdqg htxdwlrq lv qrw iruzdug orrnlqj dv lw lv lq wklv vwxg|1 Dovr/ wkh sulydwh djhqwv* lqirupdwlrq vhw grhv qrw lqfoxgh dq|wklqj wkdw wkh

(21)

Frqfhuqlqj wkh lqwhusuhwdwlrq ri wkh uhvxowv ri wklv vwxg|/ uhfdoo wkdw wkh odujhu lv wkh prqhwdu| djjuhjdwh/ wkh orzhu vkrxog wkh lqwhuhvw udwh eh/ dv vkrzq lq htxdwlrq +618,1 Wklv zrxog vhhp wr eh lq frqwudglfwlrq zlwk wkh zd| fhqwudo edqnv w|slfdoo| lqwhusuhw prqhwdu| djjuhjdwhv dv lqglfdwruv ri ixwxuh lq dwlrqdu| suhvvxuhv1 Krzhyhu/ rqh vkrxog qrwh wkdw wklv uhvxow rqo| uhihuv wr wkh lqirupdwlrq ydoxh ri wkh prqhwdu| yduldeoh lq dgglwlrq wr doo rwkhu lqirupdwlrq wkdw wkh srolf|pdnhu kdv/ lqfoxglqj wkh udwh ri lq dwlrq1 Wkh uhvxow rqo| krogv iru wkh vwrfn ri prqh| lq uhdo whupv/ qrw lq qrplqdo whupv1 Qrplqdo prqh| jurzwk fdq eh srvlwlyho| fruuhodwhg zlwk lq dwlrq1 Zkdw wkh prgho iudphzrun lpsolhv lv wkdw li uhdo edodqfhv duh odujhu wkdq suhglfwhg e| vwdqgdug edfnzdug orrnlqj prqh| ghpdqg htxdwlrqv/ wklv lv d srvvleoh lqglfdwlrq ri dq lqfuhdvh lq srwhqwldo rxwsxw1 Vxfk d frqfoxvlrq qdwxudoo| ghshqgv rq wkh dvvxpswlrq wkdw wkh srolf|pdnhu grhv qrw wklqn wkdw wkh sulydwh djhqwv duh pdnlqj d plvwdnh lq wkhlu fxuuhqw dvvhvvphqw ri srwhqwldo rxwsxw1

8 Frqfoxvlrqv

Wklv sdshu suhvhqwv d txdqwlwdwlyh dvvhvvphqw rq wkh xvhixoqhvv ri prqh| dv dq lqglfdwru ri wkh vwdwh ri wkh hfrqrp|/ zkhq prqhwdu| srolf| lv frqvlghuhg1 Zh xvh d edfnzdug0iruzdug orrnlqj prgho/ fdoleudwhg iru wkh hxur duhd hfrqrp|1 Wkh prgho dqg lwv sdudphwhuv duh dvvxphg wr eh shuihfwo| nqrzq/ exw wkh ydoxhv ri vrph yduldeohv duh qrw1 Wkh srolf|pdnhu grhv qrw kdyh frpsohwh lqirupdwlrq rq wkh vwdwh ri wkh hfrqrp|1 Zh xvh wkh iudphzrun ri Vyhqvvrq dqg Zrrgirug +5335e, wr qg rxw krz wkh srolf|pdnhu rswlpdoo| owhuv wkh lqirupdwlrq lw uhfhlyhv1

Uhjduglqj wkh uhvxowv/ dq lpsruwdqw dvvxpswlrq lv wkdw wkh sulydwh djhqwv srvvhvv lqirupdwlrq zklfk wkh srolf|pdnhu fdqqrw gluhfwo| revhuyh1 Dqrwkhu lpsruwdqw ihdwxuh ri wkh prgho lv wkh iruzdug orrnlqj qdwxuh ri wkh prqh| ghpdqg htxdwlrq1 Zlwkrxw wkhvh ihdwxuhv/ wkh ydoxh ri prqh| dv dq lqglfdwru yduldeoh wxuqv rxw wr eh yhu| vpdoo1 Krzhyhu/ wkh fruh uhvxowv frqfhuqlqj

(22)

wkh ydoxh ri prqh| dv dq lqglfdwru duh idluo| urexvw wr rwkhu fkdqjhv lq wkh sdudphwhuv ri wkh prgho1

Wkh lqirupdwlrq ydoxh ri prqh| lq wklv iudphzrun odujho| vwhpv iurp wkh idfw wkdw wkh ghpdqg iru uhdo edodqfhv lv dhfwhg e| wkh sulydwh djhqwv* lqirupdwlrq frqfhuqlqj srwhqwldo rxwsxw1 Li wkh srolf|pdnhu revhuyhv wkh jurzwk lq uhdo edodqfhv ehlqj juhdwhu wkdq wkdw suhglfwhg e| d vwdqgdug edfnzdug orrnlqj prqh| ghpdqg htxdwlrq/ wkh srolf|pdnhu uhylvhv xszdug lwv hvwlpdwh ri srwhqwldo rxwsxw1 Wklv phdqv wkdw idvwhu rxwsxw jurzwk orrnv wr eh srvvleoh zlwkrxw lqfuhdvhg lq dwlrqdu| suhvvxuhv/ ohdglqj d orzhu lqwhuhvw udwh xqghu rswlpdo prqhwdu| srolf|1 Vxfk d srolf| uhvsrqvh wr prqh| jurzwk fdq eh vhhq dv xqfrqyhqwlrqdo1 Qdwxudoo|/ wkh lqihuhqfh e| wkh srolf|pdnhu iroorzv iurp wkh dvvxpswlrq wkdw wkh srolf|pdnhu wklqnv wkdw wkh sulydwh djhqwv duh pdnlqj d fruuhfw dvvhvvphqw rq wkh odwhvw ghyhorsphqwv lq srwhqwldo rxwsxw1

Lq wklv vwxg| zh txdqwli| rqh srvvleoh uroh wkdw prqh| fdq sod| lq prqhwdu| srolf|1 Zh orrn dw prqh| dv dq lqglfdwru/ zh irfxv rq wkh fdvh zkhuh prqh| rqo| pdwwhuv wkurxjk wkh prqh| ghpdqg htxdwlrq/ dqg zh ghdo zlwk d idluo| olplwhg vhw ri revhuydeoh yduldeohv1 Reylrxvo|/ wkh ydoxh ri prqh| dv dq lqglfdwru ghshqgv rq wkh h{lvwhqfh ri rwkhu srvvleoh vrxufhv ri lqirupdwlrq1 Lq sudfwlfh/ fhqwudo edqnv revhuyh d odujh qxpehu ri lqglfdwruv1 Lw lv olnho| wkdw vrph lqglfdwruv qrw wdnhq lqwr dffrxqw lq wklv vwxg| uhyhdo slhfhv ri wkh vdph lqirupdwlrq dv prqh|1 D orw ri uhvhdufk uhpdlqv wr eh grqh rq wkh rswlpdo xvh ri prqh| dv dq lqglfdwru zkhq dqdo|}hg lq frpelqdwlrq zlwk rwkhu lqglfdwruv/ hvshfldoo| rwkhu qdqfldo pdunhw gdwd1 Lw zrxog vhhp wkdw wkh dqdo|vlv frqfhuqlqj prqh| lv lghdoo| grqh dv d sduw ri dq dqdo|vlv frqfhuqlqj dovr htxlw| sulfhv/ erqg |lhogv/ dqg fuhglw/ dprqj rwkhu wklqjv1

(23)

D Dsshqgl{= Ghpdqg iru prqh|

Wkh irup ri wkh prqh| ghpdqg htxdwlrq +519, fdq eh prwlydwhg e| dvvxplqj wkdw wkhuh duh txdgudwlf dgmxvwphqw frvwv lq prqh| kroglqjv lq uhdo whupv1 Dvvxph wkdw d sulydwh djhqw vhwv khu ghpdqg iru prqh| vr dv wr plqlpl}h

[" 'fq _\ |n| +D414, zlwk shulrglf orvv \|' E6| 6W|2n 1 E6| 6|32 +D415, zkhuh 6W | ' )++| )|n |c +D416, Lq htxdwlrq +D416,/ )₊ dqg )ghqrwh srvlwlyh frqvwdqwv/ dqg |lv dq l1l1g1 phdq

}hur vkrfn1 Lw lv vwudljkwiruzdug wr vkrz wkdw htxdwlrq +519, iroorzv iurp wklv pd{lpl}dwlrq sureohp/ zkhq > ' )₊V/ >_e ' )V/ > ' 1V/ >₂ ' q1V/ dqg ec|' |V/ zkhuh V _n1nq1 1

Htxdwlrq +D416, kdv wkh irup ri d vwdqgdug orqj uxq prqh| ghpdqg ixqfwlrq1 Wkh ydoxhv ri > dqg >_e ri htxdwlrq +519, wkdw dsshdu lq Wdeoh 4 duh vhw vr dv wr lpso| )₊ ' dqg ) ' fD1

(24)

E Dsshqgl{= Ilowhulqj

Wklv dsshqgl{ vkrzv krz wkh srolf|pdnhu rswlpdoo| xvhv wkh lqirupdwlrq lw kdv wr irup d shufhswlrq rq wkh vwdwh ri wkh hfrqrp|1 Iluvw/ wkh prgho lv suhvhqwhg lq d vwdwh0vsdfh irup1

E14 Prgho lq vwdwh0vsdfh irup

Wkh prgho ri Vhfwlrq 5 fdq eh zulwwhq lq vwdwh0vsdfh irup dv f|n l.|%|n ' f| %| n |n |n f c +E414,

zkhuh f| lv wkh yhfwru ri suhghwhuplqhg yduldeohv/ ghqhg dv

f| ' +|3 Z|3 +|? c| 2c| |3 6|3 ec| +E415,

%| lv wkh yhfwru ri iruzdug0orrnlqj yduldeohv/ ghqhg dv

%|' +| Z| 6| c +E416,

dqg | lv wkh iroorzlqj yhfwru ri huuru whupv1

|' f f c| c| 2c| f f ec| +E417,

Wkh pdwulfhv wkdw dsshdu lq htxdwlrq E414 fdq eh zulwwhq dv

' 5 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 7 f f f f f f f f f f f f f f f f f f f f f f 4 f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f E f f f f f f f f f E w f f f f f f f f f f f f > > f 6 : : : : : : : : : : : : : : : : 8 c +E418, l ' 5 7 #_{f qw} f_f f f >₂ 6 8 c +E419, dqg ' f f f f f f f # f >_e +E41:, Htxdwlrq +518, fdq eh zulwwhq dv u|' t|` t|c +E41;,

(25)

zkhuh ` ' 5 7 b_{f f}+ f f f f b 6 8 _+E41<, dqg t|' f| %| n |c +E4143, zkhuh ' 5 7 f f f f f f f f f_{f f f f f f f f f f} f f f f f f f f f f 6 8 _+E4144, dqg ' f f +E4145,

Dv dq h{dpsoh ri dq revhuydwlrq yhfwru/ ohw xv suhvhqw ~| ri Vhfwlrq 615141

Lw lv ~|' +|3 +|f Z| 6| dqg fdq eh zulwwhq dv iroorzv1: ~|' ( f| %| n |c +E4146, zkhuh | ' f | f f +E4147, dqg ( ' 5 9 9 7 f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f 6 : : 8 +E4148,

E15 Rswlpdo owhu

Wklv vhfwlrq iroorzv Vyhqvvrq ) Zrrgirug +5335e,1 Lw ghvfulehv krz pdwulfhv x dqg [ fdq eh ghulyhg/ zkhq wkh sdudphwhuv ri wkh prgho duh nqrzq1

Dv lq htxdwlrq +5143,/ wkh rswlpdo prqhwdu| srolf| fdq eh ghvfulehg dv

|' 8 f|| +E514,

zkhuh 8 lv wkh yhfwru jlyhq lq htxdwlrq +5143,1 Lq dgglwlrq/ wkh srolf|pdnhu*v hvwlpdwh ri wkh iruzdug orrnlqj yduldeohv ri wkh prgho fdq eh zulwwhq dv

%||' Cf|| +E515,

:_]_w _{frxog dovr eh ghqhg wr lqfoxgh odjv ri |}_w4_{/ |}3

w/ w/ dqg pw1 Lq wkdw fdvh/ wkh lqyhuvh

rshudwruv lq htxdwlrqv +E5146, dqg +E5147, duh ghqhg dv d jhqhudol}hg lqyhuvhv1

(26)

zkhuh C lv d pdwul{ ri wkh dssursuldwh glphqvlrqv1 Wkh vdph dojrulwkp wkdw lv xvhg wr vroyh iru 8 dovr |lhogv C1 Wkh dfwxdo %| ghshqgv qrw rqo| rq wkh

vwdwh ri wkh hfrqrp| f| exw dovr rq wkh srolf|pdnhu*v hvwlpdwh ri lw1 Rqh fdq

zulwh

%|' Cf|nC Cf||c +E516,

iru pdwul{ C _{wr eh ghwhuplqhg1}

Htxdwlrqv +E414,/ +E4146,/ +E514,/ dqg +E516, lpso| wkdw

f|n ' Mf|n af||n |nc +E517, ~|' uf|n f||n |c +E518, zkhuh M n 2Cc +E519, a 8 n 2C Cc +E51:, u (n (2Cc +E51;, (2C Cc +E51<,

zkhuh zh xvh wkh iroorzlqj ghfrpsrvlwlrqv ri pdwulfhv / / dqg (/ pdgh dffruglqj wr f| dqg %|1 ' 2 2 22 c ' 2 c ( ' ( (2 +E5143,

Vyhqvvrq ) Zrrgirug +5335e, vkrz wkdw wkh rswlpdo pdwulfhv x dqg [ ri htxdwlrq +614, fdq eh zulwwhq dv

x ' EU n g3EU gu +E5144,

[ ' EU n g3gc +E5145,

zkhuh g dqg duh ghqhg dv iroorzv1 g ' u_{Eu u}_{n P} 23c +E5146, zkhuh P2 SJ E|/ dqg SJf| f||3/ vdwlvi|lqj ' Mk u_{Eu u}_{n P} 23u l M_{n P} c +E5147,

zkhuh P SJ E|1 Xvlqj htxdwlrq +E517,/ pdwul{ K wkdw dsshduv lq htxdwlrq

+615, fdq eh zulwwhq dv x EM n a1 Ilqdoo|/ lw fdq eh vkrzq wkdw pdwul{ C _vdwlvhv C _'3 22 2n lCnC CguM +E5148,

Wkh qxphulfdo fdofxodwlrqv uhtxluhg wr vroyh iru x dqg [ duh pdgh pruh frpsolfdwhg e| wkh idfw wkdw wkh pdwulfhv M/ u/ g/ / dqg C _{duh ghwhuplqhg}

vlpxowdqhrxvo| e| wkh v|vwhp ri htxdwlrqv +E519,/ +E51;,/ +E5146,/ +E5147,/ dqg +E5148,/ sduw ri zklfk duh qrq0olqhdu1

(27)

Dq h{dpsoh ri pdwul{ [ lv jlyhq lq htxdwlrq +617, lq Vhfwlrq 615141 Lw vkrzv wkh rswlpdo [ lq wkh fdvh ri lqfrpsohwh lqirupdwlrq/ zkhq prqh| lv revhuyhg

dqg je' f2D1 Wkh fruuhvsrqglqj rswlpdo x fdq eh vkrzq wr eh x ' 5 9 9 9 9 9 9 9 9 9 9 7 f f f f f f f f f f f f f f f ff. fS fS2D ffeD f2e ffD. f2f fSf2

fffH ff.f f2b. ffHf feD ffe2 ffe f2

fffe feeH fD ff2S ffS. ff2f ffb ff f f f f f f f f f f f f f f fffb ffDb fe ff2S f2 ffS. f2. f2b 6 : : : : : : : : : : 8 +E5149, 58

(28)

Uhihuhqfhv

^4` Drnl/ N1 +5334, Rq wkh rswlpdo prqhwdu| srolf| uhvsrqvh wr qrlv| lqglfdwruv1 Mrxuqdo ri Prqhwdu| Hfrqrplfv/ iruwkfrplqj1

^5` Drnl/ N1 +5335, Rswlpdo frpplwphqw srolf| xqghu qrlv| lqirupdwlrq1 FHSU Glvfxvvlrq Sdshu Qr1 66:31

^6` Eudqg/ N1 Fdvvrod/ Q1 +5333, D Prqh| Ghpdqg V|vwhp iru Hxur Duhd P61 Hxurshdq Fhqwudo Edqn Zrunlqj Sdshu Qr1 6<1

^7` Eurzqh/ I1[1 Idjdq/ J1 Khqu|/ M1 +4<<:, Prqh| ghpdqg lq HX frxqwulhv= D vxuyh|1 Hxurshdq Prqhwdu| Lqvwlwxwh Vwd sdshu Qr1 :24<<:1

^9` Fdo}d/ D1 Jhughvphlhu/ G1 Ohy|/ M1 +5334, Hxur duhd prqh| ghpdqg= Phdvxulqj wkh rssruwxqlw| frvwv dssursuldwho|1 LPI Zrunlqj Sdshu Qr1 3424:<1

^:` Fodulgd/ U1 Jdol/ M1 0 Jhuwohu/ P1 +4<<<, Wkh vflhqfh ri prqhwdu| srolf|= D Qhz Nh|qhvldq dssurdfk1 Mrxuqdo ri Hfrqrplf Olwhudwxuh/ Yro1 6;/ 49944:3:1

^;` Frhqhq/ J1 Yhjd/ M10O1 +4<<<, Wkh Ghpdqg iru P6 lq wkh Hxur Duhd1 Hxurshdq Fhqwudo Edqn Zrunlqj Sdshu Qr1 91

^<` Frhqhq/ J1 Ohylq/ D1 Zlhodqg/ Y1 +5334, Gdwd xqfhuwdlqw| dqg wkh uroh ri prqh| dv dq lqirupdwlrq yduldeoh iru prqhwdu| srolf|1 HFE Zrunlqj Sdshu Qr1 ;71

^43` Fxwkehuwvrq/ N1 Wd|oru/ P1S1 +4<;:, Wkh Ghpdqg iru Prqh|= D G|qdplf Udwlrqdo H{shfwdwlrqv Prgho1 Hfrqrplf Mrxuqdo/ Yro1 <:/ 98:91

^44` Fxwkehuwvrq/ N1 Wd|oru/ P1S1 +4<<3, Prqh| Ghpdqg/

H{shfwdwlrqv/ dqg wkh Iruzdug0Orrnlqj Prgho1 Mrxuqdo ri Srolf| Prgholqj/ Yro1 5;<6481

^45` Ghgrod/ O1 Jdlrwwl/ H1 Vlolsr/ O1 +5334, Prqh| ghpdqg lq wkh hxur duhd= Gr qdwlrqdo glhuhqfhv pdwwhuB Edqfd g*Lwdold/ Whpl gl glvfxvvlrqh Qr1 7381

^47` Hkupdqq/ P1 Vphwv/ I1 +5334, Xqfhuwdlq srwhqwldo rxwsxw= Lpsolfdwlrqv iru prqhwdu| srolf|1 Hxurshdq Fhqwudo Edqn Zrunlqj Sdshu Qr1 8<1

(29)

^49` Ixkuhu/ M1F1 +4<<:, Wkh +xq,lpsruwdqfh ri iruzdug0orrnlqj ehkdylru lq sulfh vshflfdwlrqv1 Mrxuqdo ri Prqh|/ Fuhglw/ dqg Edqnlqj/ Yro1 5</ 66;6831

^4:` Ixkuhu/ M1F1 +5333, Kdelw Irupdwlrq lq Frqvxpswlrq dqg Lwv Lpsolfdwlrqv iru Prqhwdu|0Srolf| Prghov1 Dphulfdq Hfrqrplf Uhylhz1 Yro1 <3/ 69:6<31

^4;` Ixkuhu/ M1 Prruh/ J1 +4<<8, Lq dwlrq Shuvlvwhqfh1 Txduwhuo| Mrxuqdo ri Hfrqrplfv/ Yro1 443/ 45:48<1

^4<` Jhuodfk/ V1 Vyhqvvrq/ O1H1R1 +5335, Prqh| dqg lq dwlrq lq wkh hxur duhd= D fdvh iru prqhwdu| lqglfdwruvB FHSU Glvfxvvlrq Sdshu Qr1 66<51

^53` Jrogihog/ V1P1 +4<:6, Wkh Ghpdqg iru Prqh| Uhylvlwhg1 Eurrnlqjv Sdshuv rq Hfrqrplf Dfwlylw|/ Yro1 6/ 8::96;1

^54` Mhqvhq/ K1 +5335, Wdujhwlqj Qrplqdo Lqfrph Jurzwk ru Lq dwlrqB Dphulfdq Hfrqrplf Uhylhz/ Yro1 <5/ <5;<891

^55` Qhovrq/ H1 +5335d, Gluhfw hhfwv ri edvh prqh| rq djjuhjdwh ghpdqg= Wkhru| dqg hylghqfh1 Mrxuqdo ri Prqhwdu| Hfrqrplfv/ Yro1 7</ 9;::3;1

^56` Qhovrq/ H1 +5335e, Wkh ixwxuh ri prqhwdu| djjuhjdwhv lq prqhwdu| srolf| dqdo|vlv1 Plphr1 Edqn ri Hqjodqg1

^57` Shduopdq/ M1J1 +4<<5, Uhsxwdwlrqdo dqg qrquhsxwdwlrqdo srolflhv xqghu Sduwldo lqirupdwlrq1 Mrxuqdo ri Hfrqrplf G|qdplfv dqg Frqwuro/ Yro1 49/ 66<68:1

^58` Shduopdq/ M1J1 Fxuulh/ G1 Ohylqh/ S1 +4<;9, Udwlrqdo h{shfwdwlrqv prghov zlwk sduwldo lqirupdwlrq1 Hfrqrplf Prghoolqj/ Yro1 6/ <3 4381

^59` Ulsdwwl/ D1 +4<<;, Ghpdqg iru prqh| lq lq dwlrq0wdujhwlqj prqhwdu| srolf|1 Edqn ri Ilqodqg Vwxglhv H=461 Edqn ri Ilqodqg1 ^5:` Urwhpehuj/ M1 Zrrgirug/ P1 +4<<<, Lqwhuhvw0udwh uxohv lq dq

^5<` Vyhqvvrq/ O1H1R1 Zrrgirug/ P1 +5335d, Lqglfdwru yduldeohv iru rswlpdo srolf|1 Mrxuqdo ri Prqhwdu| Hfrqrplfv/ iruwkfrplqj1 ^63` Vyhqvvrq/ O1H1R1 Zrrgirug/ P1 +5335e, Lqglfdwru yduldeohv iru

rswlpdo srolf| xqghu dv|pphwulf lqirupdwlrq1 Mrxuqdo ri Hfrqrplf G|qdplfv dqg Frqwuro/ iruwkfrplqj1

^64` Vyhqvvrq/ O1H1R1 Zrrgirug/ P1 +5335f, Lqglfdwru yduldeohv iru rswlpdo srolf|1 +D orqjhu yhuvlrq ri Vyhqvvrq hw do 5335d,1 Plphr/ Sulqfhwrq Xqlyhuvlw|/ Mdqxdu| 53351

(30)

^65` Vyhqvvrq/ O1H1R1 Zrrgirug/ P1 +5336, Rswlpdo srolf| zlwk sduwldo lqirupdwlrq lq d iruzdug0orrnlqj prgho= Fhuwdlqw|0 htxlydohqfh uhgx{1 QEHU Zrunlqj Sdshu Qr1 <7631

(31)

Iljxuh 4= Dfwxdo lpsxovh uhvsrqvhv1

Vrolg olqh= frpsohwh lqirupdwlrq1 Gdvkhg olqh= lqfrpsohwh lqirupdwlrq/ zlwkrxw prqh|1 Vkruw gdvkhv= lqfrpsohwh lqirupdwlrq/ zlwk prqh|/

vwg+PG,@31581 Grwwhg olqh= lqfrpsohwh lqirupdwlrq/ zlwk prqh|/ vwg+PG,@31451

(32)

(33)

Iljxuh 5= Dfwxdo dqg shufhlyhg lpsxovh uhvsrqvhv1 Fdvh ri lqfrpsohwh lqirupdwlrq zlwk prqh|/ vwg+PG,@31581

(34)

(35)

Iljxuh 6= Shufhswlrq huuruv1

Vrolg olqh= zlwkrxw prqh|1 Gdvkhg olqh= zlwk prqh|/ vwg+PG,@31581 Vkruw gdvkhv= zlwk prqh|/ vwg+PG,@31451

(36)

Iljxuh 7= Qrq0edvholqh sdudphwhu ydoxhv1 Wkh jdlq iurp revhuylqj prqh|= wkh shufhqwdjh uhgxfwlrq lq wkh orvv fdxvhg e| lqfrpsohwh lqirupdwlrq1

(37)

Iljxuh 8= Qrq0edvholqh prqh| ghpdqg1 Wkh jdlq iurp revhuylqj prqh|= wkh shufhqwdjh uhgxfwlrq lq wkh orvv fdxvhg e| lqfrpsohwh lqirupdwlrq1

(38)

BANK OF FINLAND DISCUSSION PAPERS

ISSN 0785-3572, print; ISSN 1456-6184, online

1/2003 Tanai Khiaonarong Payment systems efficiency, policy approaches, and the

role of the central bank. 2003. 69 p. ISBN 952-462-025-1, print;

ISBN 952-462-026-X, online. (TU)

2/2003 Iftekhar Hasan – Heiko Schmiedel Do networks in the stock exchange

industry pay off? European evidence. 2003. 44 p. ISBN 952-462-027-8, print;

ISBN 952-462-028-6, online. (TU)

3/2003 Johanna Lukkarila Comparison between Asian, Russian and Turkish

financial crises. (In finnish). 2003. 57 p. ISBN 952-462-029-4, print;

ISBN 952-462-030-8, online. (KT)

4/2003 Samu Peura – Esa Jokivuolle Simulation-based stress testing of banks’

regulatory capital adequacy. 2003. 41 p. ISBN 952-462-035-9, print;

ISBN 952-462-036-7, online. (RM)

5/2003 Peik Granlund Economic evaluation of bank exit regimes in US, EU and

Japanese financial centres. 2003. 60 p. ISBN 952-462-037-5, print;

ISBN 952-462-038-3, online. (TU)

6/2003 Tuomas Takalo – Otto Toivanen Equilibrium in financial markets with

adverse selection. 2003. 45 p. ISBN 952-462-039-1, print;

ISBN 952-462-040-5, online (TU)

7/2003 Harry Leinonen Restructuring securities systems processing – a blue print

proposal for real-time/t+0 processing. 2003. 90 p. ISBN 952-462-041-3,

print; ISBN 952-462-042-1, online (TU)

8/2003 Hanna Jyrkönen – Heli Paunonen Card, Internet and mobile payments in

Finland. 2003. 45 p. ISBN 952-462-043-X, print; ISBN 952-462-044-8, online

(RM)

9/2003 Lauri Kajanoja Money as an indicator variable for monetary policy when

money demand is forward looking. 2003. 35 p. ISBN 952-462-047-2, print;

Money as an indicator variable for monetary policy when money demand is forward looking

BANK OF FINLAND

DISCUSSION PAPERS

9

·

2003

Lauri Kajanoja

Research Department

9.4.2003

Money as an indicator variable

for monetary policy

when money demand is

forward looking

BANK OF FINLAND

DISCUSSION PAPERS

9

·

2003

Lauri Kajanoja*

Research Department

9.4.2003

Money as an indicator variable

for monetary policy when money demand is

forward looking

Money as an indicator variable for monetary policy

when money demand is forward looking

Bank of Finland Discussion Papers 9/2003

Abstract

Raha-aggregaatti indikaattorina rahapolitiikalle, kun

rahan kysyntäyhtälö on eteenpäin katsova

Suomen Pankin keskustelualoitteita 9/2003

Tiivistelmä

Contents

4 Lqwurgxfwlrq

5 P rgho

514 Lqglfdwru yduldeohv

515 Sdudphwhu ydoxhv

516 Rswlpdo prqhwdu| srolf|

6 Lqirupdwlrq

614 Lqirupdwlrqdo dvvxpswlrqv

615 Ilowhulqj ri lqirupdwlrq

7 Ydoxh ri prqh| dv dq lqglfdwru

714 Urexvwqhvv ri wkh lqirupdwlrq ydoxh ri prqh|

8 Frqfoxvlrqv

D Dsshqgl{= Ghpdqg iru prqh|

E Dsshqgl{= Ilowhulqj

E14 Prgho lq vwdwh0vsdfh irup

E15 Rswlpdo owhu

Uhihuhqfhv

E15 Rswlpdo owhu