2002 021
WORKING PAPER SERIES
Duration Dependence in Monetary Policy: International Evidence
Michael T. Owyang and Abbigail J. Chiodo Working Paper 2002-021A http://research.stlouisfed.org/wp/2002/2002-021.pdf
October 2002
FEDERAL RESERVE BANK OF ST. LOUIS Research Division 411 Locust Street St. Louis, MO 63102 ______________________________________________________________________________________ The views expressed are those of the individual authors and do not necessarily reflect official positions of the Federal Reserve Bank of St. Louis, the Federal Reserve System, or the Board of Governors. Federal Reserve Bank of St. Louis Working Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to Federal Reserve Bank of St. Louis Working Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Photo courtesy of The Gateway Arch, St. Louis, MO. www.gatewayarch.com
Gxudwlrq Ghshqghqfh lq Prqhwdu| Srolf|= Lqwhuqdwlrqdo Hylghqfh Deeljdlo M1 Fklrgr_dqg Plfkdho W1 Rz|dqjh Ihghudo Uhvhuyh Edqn ri Vdlqw Orxlv Uhvhdufk Ghsduwphqw 744 Orfxvw Vwuhhw Vdlqw Orxlv/ PR 96435 Vhswhpehu 58/ 5335
Devwudfw Zh vwxg| wkh gxudwlrq ri prqhwdu| uhjlphv lq d vlpsoh qhr0fodvvlfdo Skloolsv fxuyh prgho1 Wkh prgho lv dq h{whqvlrq ri Rz|dqj +5334, dqg Rz|dqj dqg Udph| +5334,1 Lq wklv sdshu/ zh frqvlghu wkh uroh ri wkh gxudwlrq ri lq dwlrqdu| uhjlphv rq wkh dyhudjh lq dwlrq udwh lq dq lqwhuqd0 wlrqdo furvv0vhfwlrq1 Zh qg wkdw lq dwlrqdu| uhjlphv lq fhuwdlq frxqwulhv duh gxudwlrq ghshqghqw exw dqwl0lq dwlrqdu| uhjlphv duh qrw1 Lq dggl0 wlrq/ zh qg wkdw frxqwulhv zlwk kljk fhqwudo edqnhu wxuqryhu vzlwfk iurp lq dwlrqdu| wr dqwl0lq dwlrqdu| zlwk orzhu suredelolw|1
4
Lqwurgxfwlrq
Suhylrxv olwhudwxuh rq wkh gl huhqfhv ehwzhhq fhqwudo edqnv dfurvv frxqwulhv kdv lghqwl hg wkdw d fhqwudo edqn*v lqghshqghqfh iurp lq xhqfh e| rwkhu eudqfkhv ri wkh jryhuqphqw fdq d hfw lwv frqgxfw dqg shuirupdqfh +Fxnlhupdq +4 V5 > ===> Vw , dqg ]w @ +]4 > ]5 > ===> ]w ,
lqglfdwh wkh uhdol}hg vwdwhv ri w dqg w / uhvshfwlyho|1 D orzhu w lqglfdwhv d wljkwhu srolf|pdnhu1 Lq d suhylrxv sdshu +Rz|dqj dqg Udph|/ 5334,/ wkh wudq0 vlwlrq suredelolwlhv duh dvvxphg wr eh frqvwdqw ryhu wlph1 Khuh/ wkh wudqvlwlrq suredelolwlhv duh ghshqghqw rq wkh qxpehu ri shulrgv lq wkh fxuuhqw lq dwlrq0 du| hslvrgh1 Iroorzlqj Gxuodqg dqg PfFxug| +4 x w4 > ]w4 ,> 5 . #5 g5 i5 + w4 > x w4 > ]w4 , =
+8,
Wkh sdwk0ghshqghqw wudqvlwlrq suredelolw| wkhq fdq eh gh qhg dv Su^w @ d5 m |w4 > Vw4 > ]w4 ` @ Su^]w A 3m |w4 > Vw4 > ]w4 `=
+9,
Lqfuhdvlqj l lqfuhdvhv wkh xqfrqglwlrqdo olnholkrrg wkdw w @ dl 1 Hdfk gl lv dq sdudphwhu vxppdul}lqj gl huhqw vshfl fdwlrqv1 Hvvhqwldoo|/ +8, doorzv iru irxu srvvleoh dowhuqdwh prghov1 gl @ 4 lpsolhv wkdw wkh vwdwh w @ dl lv gx0 w4 > xw4 > ]w4 , pdsv wkh klvwru| ri wkh udwlrq ghshqghqw1 Hdfk ixqfwlrq il +
hfrqrp| lqwr wkh odwhqw vwdwh yduldeoh ]w 1 Wkh sdudphwhu #l4 lv vlpso| d zhljkw w4 > x w4 > ]w4 ,1 Wklv irupxodwlrq doorzv wkh srvvlelolw| wkdw rqh ri wkh rq il + uhjlphv qrw eh wlph ru gxudwlrq ghshqghqw/ l1h1/ wkh gryh*v wudqvlwlrq sured0 elolwlhv pljkw eh lqyduldqw wr sdwk zkloh wkh kdzn*v wudqvlwlrq suredelolwlhv duh w4 > x w4 > ]w4 , gxudwlrq ghshqghqw1 Wkxv/ zh vxevwlwxwh iru wkh ixqfwlrq il + d phdvxuh ri wkh wlph vlqfh wkh odvw vzlwfk w4 > xw4 > ]w4 , @ Ql +]w4 , > il +
+:,
zkhuh Ql +]w4 , frxqwv wkh qxpehu ri shulrgv wkdw wkh hfrqrp| kdv uhpdlqhg lq wkh dl vwdwh1 Wkh srolf|pdnhu vhwv dq lq dwlrq wdujhw w hdfk shulrg/ exw wkh uhdol}hg
9
lq dwlrq udwh lv vxemhfw wr zklwh qrlvh/ %6w 1 Wkh uhdol}hg udwh ri lq dwlrq fdq wkhq eh zulwwhq
w . %6w > w @
+;,
zkhuh %6w Q +3> 5%6 ,1 Jlyhq wkh Skloolsv fxuyh/ +4,/ dqg wkh srolf|pdnhu*v orvv ixqfwlrq/ +6,/ wkh srolf|pdnhu*v fxuuhqw0shulrg/ h{shfwhg orvv0plqlpl}lqj lq dwlrq ghflvlrq lv w @ "w +n hw . w ,
+
+44,
zkhuh %5w Q +3> 5%5 ,16 Lw lv dvvxphg wkdw wkh srolf|pdnhu fdq revhuyh wkh djhqwv* h{shfwdwlrqv exw qrw wkh phwkrg e| zklfk wkh| duh ghwhuplqhg1 Wkh lpsolfdwlrq ri dgglqj wkh wlph ydu|lqj wudqvlwlrq suredelolwlhv wr wkh prgho lv wkdw wkh vzlwfk ehwzhhq uhjlphv lq wkh srolf|pdnhu*v suhihuhqfhv fdq 6 Lw
lv srvvleoh wr prgli| +44, wr doorz iru iruzdug0orrnlqj h{shfwdwlrqv1
:
uhvsrqg wr wkh frqglwlrqv lq wkh hfrqrp|1
6
Uhvxowv
614
X1V1 Uhvxowv
Hpsor|lqj wkh hvwlpdwlrq whfkqltxh rxwolqhg lq wkh dsshqgl{/ hvwlpdwhv ri wkh sdudphwhuv jryhuqlqj wkh prgho +4,/ +7,/ +8,/ +9,/ +44,/ dqg +43, zhuh rewdlqhg1 Wkhvh uhvxowv duh frpsduhg wr wkh uhvxowv ri Rz|dqj dqg Udph| +5334, lq zklfk d vlplodu vwuxfwxudo prgho lv hpsor|hg zlwk frqvwdqw wudqvlwlrq suredelolwlhv iru wkh srolf|pdnhu*v uhodwlyh wrohudqfhv iru lq dwlrq yhuvxv xqhpsor|phqw1 Wdeoh 4 glvsod|v wkh uhvxowlqj sdudphwhu hvwlpdwhv1 Wkh ydoxhv d4 dqg d5 lpsolhg e| wkh hvwlpdwhv ri n/ {4 / dqg {5 duh d4 @ 44;=48 dqg d5 @ 663=:41 Wkh odujhu ydoxhv ri d4 dqg d5 uhodwlyh wr wkrvh ri Rz|dqj dqg Udph| +5334, duh gxh wr wkh orzhu hvwlpdwh ri n17 Wklv lqglfdwhv wkdw wkh dffrpprgdwlyh srolf|pdnhu lv zloolqj wr vhw dq lq dwlrq wdujhw wkuhh wlphv kljkhu wkdq wkh lq dwlrq kdzn1 Wkh sdudphwhuv k4 dqg k5 duh wkh vwhdg| vwdwh xqhpsor|phqw udwhv lpsolhg e| wkh prgho18 Wkh dgyhuvh vkrfn whqgv wr lqfuhdvh wkh h{shfwhg xqhpsor|phqw udwh derxw 516(1 Wkh vdpsoh shulrg vkrzv lqfuhdvhv lq h{fhvv ri 516(/ h{sodlqhg e| wkh prgho dv whpsrudu| vkrfnv1 Vrph lqfuhdvhv duh odujh exw wkh| duh qrw shuvlvwhqw1 Wkh Pdunry surfhvv jryhuqlqj wkh Skloolsv fxuyh h{sodlqv wkh shuvlvwhqw fkdqjh lq xqhpsor|phqw fdxvhg e|/ iru h{dpsoh/ wkh orqj0odvwlqj h hfwv ri dq rlo vkrfn1 Wkh wudqvlwru| h hfwv duh dwwulexwhg wr %w vkrfnv1 Qrwh dovr wkdw wkh hvwlpdwhg ydoxh ri n lv vpdoohu iru wkh GG vshfl fdwlrq wkdq iru d QGG vshfl fdwlrq1 Wkhuh duh wzr lpsolfdwlrqv ri wklv qglqj1 Iluvw/ wkh h hfw ri d prqhwdu| vxusulvh rq wkh xqhpsor|phqw udwh lv vpdoohu1 Vhfrqg/ 7 Pd|eh
lw*v qhfhvvdu| wr fdoleudwh nB wkh uhvxowv zlwk Nrhqlj +5334, dqg Jrugrq +4 Vw.4 , @ s+Vw mh h hw > s+Vw.4 mh |w > ] hw > * h, @S
]w
s+Vw.4 mVw ,s+Vw mh |w > ]hw > hhw > * h, hw > hw > s+Vw.4 mh | w > Vw > ] hhw > * h ,s+Vw mh |w > ] hhw > * h, hw > s+Vw.4 mVw ,s+Vw mh |w > ] hhw > * h, @S > h hw > s+Vw.4 mVw ,s+Vw mh |w > ] hw > * h, ]w
zkhuh s+Vw.4 mVw , lv wkh wudqvlwlrq suredelolw| dqg wkh owhu ghwhuplqhv wkh ghq0 hw > |w > ] hhw > * h ,1 Wkh uvw lqhtxdolw| lv vlpso| dq dssolfdwlrq ri Ed|hv* vlw| s+Vw mh Odz1 Wkh qdo wzr lqhtxdolwlhv dulvh iurp wkh Pdunry surshuw| ri Vw = lq ghwhu0 plqlqj wkh ghqvlw| iru Vw.4 / wkh rqo| uhohydqw lqirupdwlrq lq wkh dydlodeoh vhw lv wkh suhylrxv vwdwh Vw 1 Wkh qxphudwru lq +45, lv fdofxodwhg iurp wkh Kdplowrq owhu dv hw > i+|w mh |w4 > Vw > ]hw > hhw > * h ,s+Vw mh |w4 > ] hhw > * h, hw > |w > ] hhw > * h, @ s+Vw mh h h i +|w mh |w4 > ]w > hw > * h,
56
hw > hw > i+|w mh |w4 > Vw > ] hhw > * h ,s+Vw mh |w4 > ] hhw > * h, @S > h h h h i +|w mh |w4 > Vw > ]w > hw > * h ,s+Vw mh |w4 > ]w > hw > * h, ]w zkhuh [ hw > hw > |w4 > ] hhw > * h, @ s+Vw mVw4 ,s+Vw4 mh |w4 > ] hhw > * h ,= s+Vw mh ]w4 hw > |w4 > ] hhw > * h , lv wdnhq iurp wkh suhylrxv lwhudwlrq1 Wkh ghqvlw| s+Vw4 mh
:17
Frqglwlrqdo Ghqvlw| ri wkh Sdudphwhu Yhfwru1
Wkh frqglwlrqdo ghqvlwlhv iru wkh hohphqwv ri wkh sdudphwhu yhfwru duh jhqhu0 dwhg e| hpsor|lqj Ed|hvldq ROV1 Wkh Ed|hvldq srvwhulru glvwulexwlrq iru hdfk hohphqw ri * h / frqglwlrqdo rq doo rwkhu hohphqwv ri * h / fdq eh ghwhuplqhg jlyhq d sulru glvwulexwlrq1 Li prgho sdudphwhuv/ h{foxglqj wkh yduldqfhv/ kdyh sulru glv0 wulexwlrqv ri wkh irup s3 +*l , Q+dl > Dl 5 ,/ wkhq wkhlu fruuhvsrqglqj srvwhulru frqglwlrqdo glvwulexwlrqv duh jlyhq e| hW > s+*l m*l > |hW > VhW > ] hhW , Q +dl > Dl 5 ,>
+46,
4 4 3 4 4 3 3 4 zkhuh dl @ +D l . [l [l , +Dl dl . [l \l ,/ Dl @ +Dl . [l [l , / [l lv wkh dssursuldwh uhjuhvvru/ dqg \l lv d iruhfdvw huuru1 Iru h{dpsoh/ frqvlghu eW hvwlpdwlqj wkh ydoxhv ri wkh yhfwru $ d lq +4,1 Lq wklv fdvh/ gh qh [ @ E ] $ dqg \ @ hW / zkhuh wkh urz w rq0gldjrqdo hohphqw ri E lv W Vw4 . n hw dqg }4W > }h5W `/ dqg }hlW lv d W 4 yhfwru wkh r 0gldjrqdo hohphqwv duh }hur/ ]eW @ ^h zlwk uhsuhvhqwdwlyh hohphqw }lw @ 4 l ]l @ l1 Rwkhu hohphqwv ri wkh sdudphwhu $ yhfwru fdq eh hvwlpdwhg vlploduo|1 Wkh ydoxhv iru wkh yhfwru k fdq eh jhqhudwhg
57
iurp d srvwhulru qrupdo vlplodu wr +46, lq zklfk 5 9 [ @ 7
VeW $ hW VeW d]
6 : 8>
6 h W hW , : hW n+h 9 x \ @ 7 8> $ hW hW n d] hhW 5
eW 1 zkhuh VeW lv gh qhg vlploduo| wr ]
Jlyhq frqglwlrqdo sulruv iru wkh yduldqfhv ri wkh irup 5%4 LJ+ 53 > U53 , dqg
5%l LJ+ 53 > O53 ,/ l @ 5> 6/ wkh srvwhulru frqglwlrqdo suredelolwlhv duh jlyhq e| . W U3 . 4 hW > * 5%4 mh > ,> |W > hhW > VhW > ] h LJ+ 3 5 5 zkhuh $ $ 4 @ +h xW n+h hW hW , k VhW ,3 +h xW n+h hW hW , k VhW ,> dqg/ iru l @ 5> 6= 3 . W O3 . l 5%l mh > ,> |W > hhW > VhW > ]hW > * h LJ+ 5 5 zkhuh
:18
$h $ $ $ hw + w d] k Vw . nh hw ,,3 +h w d ]hw + k Vhw . nh hw ,,= l @ +h
Wudqvlwlrq Suredelolwlhv
Jlyhq d sulru suredelolw| glvwulexwlrq iru Wlld ri wkh irup Wlld +xll > xml , iru m 9@ l/ wkh glvwulexwlrq iru wkh wudqvlwlrq suredelolwlhv Wlld lv ghwhuplqhg e| Wlld @ Su^w @ dl mw4 @ dl ` +xll . qll > xml . qml ,>
58
zkhuh qll lv wkh qxpehu ri shulrgv wkdw w uhpdlqhg lq vwdwh l/ dqg qml lv wkh qxpehu ri shulrgv wkdw w vzlwfkhg wr vwdwh m 9@ l diwhu ehjlqqlqj lq vwdwh l1 @ 4 W 1 D vlplodu Wkhq wkh rwkhu hohphqwv ri W fdq eh ghwhuplqhg e| Wml ll surfhgxuh lv xvhg wr jhqhudwh wkh hohphqwv ri W 1 4 dqg 5 1
Wkh srvwhulru frqglwlrqdo ghqvlw| iurp zklfk wkh sdudphwhuv
4 dqg 5 fdq eh jhqhudwhg lv gh qhg dv iroorzv1 Jlyhq wkh sulru ghqvlw| @ +4 > 5 , Q +o3 > 3 , wkh srvwhulru ghqvlw| lv jlyhq e| hhW > !Q +o> , m]W > #4 > #5 > g4 > g5 @ 4> VhW > ]W > |hW > zkhuh
4 3 4 4 3 o @ + 3 . [ [, +3 o3 . [ \ , dqg
4 3 4 @ + 3 . [ [, Lq wklv fdvh/ wkh l0wk hohphqwv ri wkh W0urz yhfwruv [ dqg \ duh gh qhg dv
[ @ ]w4 dqg \ @ ]w *]w4 zkhuh ]w zdv frpsxwhg lq wkh suhylrxv lwhudwlrq dqg * @ +# 4 g4 i4 > #5 g5 i5 ,1
59
#l frqglwlrqdo rq gl @ 41 Wkh srvwhulru frqglwlrqdo ghqvlw| lv s+#l m]W > #l > gl > gl @
hhW > !, 4> VhW > ]W > |hW >
l hhW > !Q z l > Z # l m]W > #l > gl > gl @ 4> VhW > ]W > |hW > zkhuh
z l @ +Z3>l4 . [ 3 [,4 +Z3>l4 z3>l . [ 3 \ ,>
l @ +Z 4 . [ 3 [,4 > Z 3>l dqg z3>l dqg Z3>l duh sdudphwhuv wdnh iurp wkh sulru glvwulexwlrq #l Q +z3 > Z3 , = Wkh l0wk hohphqwv ri wkh W0urz yhfwruv [ dqg \ duh gh qhg e|
[ @ ]w4 dqg \ @ ]w *]w4 Lq wklv fdvh/ * @ +*4 > *5 ,/ zkhuh w4 > x w4 > ]w4 , *l @ l . #l gl il + dqg
*l @ l 5:
dqg @ + 4 > 5 ,/ zkhuh w4 > x w4 > ]w4 , l @ il + dqg
l @ 3=
:19
Jhqhudwlqj wkh gl *v1
Uhfdoo wkdw wkh srvwhulru suredelolw| ri hdfk gl uhsuhvhqwv wkh srvwhulru sure0 delolw| ri wlph ydu|lqj wudqvlwlrq suredelolwlhv ru gxudwlrq ghshqghqfh iru wkh vwdwh l1 Jlyhq wkdw wkh sulru suredelolw| ri wlph ydu|lqj wudqvlwlrq suredelolwlhv lq vwdwh lv sl / Ed|hv Odz jlyhv wkh srvwhulru suredelolw| dv @ hhW > ! s gl m]W > #l > #l > gl > VhW > ]W > |hW >
sl > sl . l +4 sl ,
zkhuh l @ 5 h{s
:1:
z l z3>l 5Zl 5Z3>l
# $& v % Zl z l 4 s = l Z3>l Z
Jhqhudwlqj wkh srvwhulru suredelolwlhv iru ]W 1
Lq wklv prgho/ wkh wudqvlwlrq pdwul{ wkdw jryhuqv wkh srolf|pdnhu*v suhihuhqfhv lv ghshqghqw rq wkh klvwru| ]w @ +]4 > ]5 > ===> ]w4 > ]w.4 > ===> ]W ,1 Vlqfh hdfk vwdwh lv jhqhudwhg frqglwlrqdo rq lwv klvwru|/ hdfk ]w pxvw eh jhqhudwhg lqglylgxdoo|1 dqg wkh sdudphwhuv lq +45,/ wkh srvwhulru Vxssuhvvlqj wkh frqglwlrqlqj rq VhW / !/ suredelolw| iru ]w frqglwlrqdo rq wkh vhw ri revhuydeohv |W @ +|4 > |5 > ===> |W , dqg ]w @ +]4 > ]5 > ===> ]w4 > ]w.4 > ===> ]W , fdq eh zulwwhq dv
5;
Su^]w m |W > ]w ` @ Su^]w m |w > ]w `
@
@
Su^]w.4 m]w > |w4 ` Su^|w > ]w.4 > ===> ]W m]w > |w4 > ]w4 ` Su^|w > ]w.4 > ===> ]W m |w4 > ]w4 `
@
@
Su^]w > |w > ]w.4 > ===> ]W m |w4 > ]w4 ` Su^|w > ]w.4 > ===> ]W m |w4 > ]w4 `
Su^]w.4 m]w ` Su^|w > ]w.4 > ===> ]W m]w > |w4 > ]w4 ` Su^|w > ]w.4 > ===> ]W m |w4 > ]w4 `
Su^]w.4 m]w ` Su^|w m]w > |w4 ` Su^]w m]w4 ` Su^]w.5 > ===> ]W m |w > ]w.4 ` Su^|w > ]w.4 > ===> ]W m |w4 > ]w4 `
zklfk vlpsol hv wr
Su^]w.4 m]w ` Su^|w m]w > |w4 ` @S Su^]w m |W > ]w > VhW > !` Su^] w.4 m]w ` Su^|w m]w > |w4 ` Su^]w m]w4 ` ]w zkhuh Su^]w.4 m]w ` dqg Su^]w m]w4 ` duh wudqvlwlrq suredelolwlhv jhqhudwhg
iurp ]w dqg Su^]w m]w4 ` lv wkh frqglwlrqdo iruhfdvw huuru1 Wkh uvw htxdolw|
dulvhv iurp wkh idfw wkdw gdwd sdvw shulrg w lv qrq0lqirupdwlyh1 Wkh irxuwk htxdolw| frphv iurp wkh Pdunry surshuwlhv ri ]w 1
5
Duration Dependence in Monetary Policy: International Evidence
Michael T. Owyang and Abbigail J. Chiodo Working Paper 2002-021A http://research.stlouisfed.org/wp/2002/2002-021.pdf
October 2002
FEDERAL RESERVE BANK OF ST. LOUIS Research Division 411 Locust Street St. Louis, MO 63102 ______________________________________________________________________________________ The views expressed are those of the individual authors and do not necessarily reflect official positions of the Federal Reserve Bank of St. Louis, the Federal Reserve System, or the Board of Governors. Federal Reserve Bank of St. Louis Working Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to Federal Reserve Bank of St. Louis Working Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Photo courtesy of The Gateway Arch, St. Louis, MO. www.gatewayarch.com
Gxudwlrq Ghshqghqfh lq Prqhwdu| Srolf|= Lqwhuqdwlrqdo Hylghqfh Deeljdlo M1 Fklrgr_dqg Plfkdho W1 Rz|dqjh Ihghudo Uhvhuyh Edqn ri Vdlqw Orxlv Uhvhdufk Ghsduwphqw 744 Orfxvw Vwuhhw Vdlqw Orxlv/ PR 96435 Vhswhpehu 58/ 5335
Devwudfw Zh vwxg| wkh gxudwlrq ri prqhwdu| uhjlphv lq d vlpsoh qhr0fodvvlfdo Skloolsv fxuyh prgho1 Wkh prgho lv dq h{whqvlrq ri Rz|dqj +5334, dqg Rz|dqj dqg Udph| +5334,1 Lq wklv sdshu/ zh frqvlghu wkh uroh ri wkh gxudwlrq ri lq dwlrqdu| uhjlphv rq wkh dyhudjh lq dwlrq udwh lq dq lqwhuqd0 wlrqdo furvv0vhfwlrq1 Zh qg wkdw lq dwlrqdu| uhjlphv lq fhuwdlq frxqwulhv duh gxudwlrq ghshqghqw exw dqwl0lq dwlrqdu| uhjlphv duh qrw1 Lq dggl0 wlrq/ zh qg wkdw frxqwulhv zlwk kljk fhqwudo edqnhu wxuqryhu vzlwfk iurp lq dwlrqdu| wr dqwl0lq dwlrqdu| zlwk orzhu suredelolw|1
4
Lqwurgxfwlrq
Suhylrxv olwhudwxuh rq wkh gl huhqfhv ehwzhhq fhqwudo edqnv dfurvv frxqwulhv kdv lghqwl hg wkdw d fhqwudo edqn*v lqghshqghqfh iurp lq xhqfh e| rwkhu eudqfkhv ri wkh jryhuqphqw fdq d hfw lwv frqgxfw dqg shuirupdqfh +Fxnlhupdq +4 V5 > ===> Vw , dqg ]w @ +]4 > ]5 > ===> ]w ,
lqglfdwh wkh uhdol}hg vwdwhv ri w dqg w / uhvshfwlyho|1 D orzhu w lqglfdwhv d wljkwhu srolf|pdnhu1 Lq d suhylrxv sdshu +Rz|dqj dqg Udph|/ 5334,/ wkh wudq0 vlwlrq suredelolwlhv duh dvvxphg wr eh frqvwdqw ryhu wlph1 Khuh/ wkh wudqvlwlrq suredelolwlhv duh ghshqghqw rq wkh qxpehu ri shulrgv lq wkh fxuuhqw lq dwlrq0 du| hslvrgh1 Iroorzlqj Gxuodqg dqg PfFxug| +4 x w4 > ]w4 ,> 5 . #5 g5 i5 + w4 > x w4 > ]w4 , =
+8,
Wkh sdwk0ghshqghqw wudqvlwlrq suredelolw| wkhq fdq eh gh qhg dv Su^w @ d5 m |w4 > Vw4 > ]w4 ` @ Su^]w A 3m |w4 > Vw4 > ]w4 `=
+9,
Lqfuhdvlqj l lqfuhdvhv wkh xqfrqglwlrqdo olnholkrrg wkdw w @ dl 1 Hdfk gl lv dq sdudphwhu vxppdul}lqj gl huhqw vshfl fdwlrqv1 Hvvhqwldoo|/ +8, doorzv iru irxu srvvleoh dowhuqdwh prghov1 gl @ 4 lpsolhv wkdw wkh vwdwh w @ dl lv gx0 w4 > xw4 > ]w4 , pdsv wkh klvwru| ri wkh udwlrq ghshqghqw1 Hdfk ixqfwlrq il +
hfrqrp| lqwr wkh odwhqw vwdwh yduldeoh ]w 1 Wkh sdudphwhu #l4 lv vlpso| d zhljkw w4 > x w4 > ]w4 ,1 Wklv irupxodwlrq doorzv wkh srvvlelolw| wkdw rqh ri wkh rq il + uhjlphv qrw eh wlph ru gxudwlrq ghshqghqw/ l1h1/ wkh gryh*v wudqvlwlrq sured0 elolwlhv pljkw eh lqyduldqw wr sdwk zkloh wkh kdzn*v wudqvlwlrq suredelolwlhv duh w4 > x w4 > ]w4 , gxudwlrq ghshqghqw1 Wkxv/ zh vxevwlwxwh iru wkh ixqfwlrq il + d phdvxuh ri wkh wlph vlqfh wkh odvw vzlwfk w4 > xw4 > ]w4 , @ Ql +]w4 , > il +
+:,
zkhuh Ql +]w4 , frxqwv wkh qxpehu ri shulrgv wkdw wkh hfrqrp| kdv uhpdlqhg lq wkh dl vwdwh1 Wkh srolf|pdnhu vhwv dq lq dwlrq wdujhw w hdfk shulrg/ exw wkh uhdol}hg
9
lq dwlrq udwh lv vxemhfw wr zklwh qrlvh/ %6w 1 Wkh uhdol}hg udwh ri lq dwlrq fdq wkhq eh zulwwhq
w . %6w > w @
+;,
zkhuh %6w Q +3> 5%6 ,1 Jlyhq wkh Skloolsv fxuyh/ +4,/ dqg wkh srolf|pdnhu*v orvv ixqfwlrq/ +6,/ wkh srolf|pdnhu*v fxuuhqw0shulrg/ h{shfwhg orvv0plqlpl}lqj lq dwlrq ghflvlrq lv w @ "w +n hw . w ,
+
+44,
zkhuh %5w Q +3> 5%5 ,16 Lw lv dvvxphg wkdw wkh srolf|pdnhu fdq revhuyh wkh djhqwv* h{shfwdwlrqv exw qrw wkh phwkrg e| zklfk wkh| duh ghwhuplqhg1 Wkh lpsolfdwlrq ri dgglqj wkh wlph ydu|lqj wudqvlwlrq suredelolwlhv wr wkh prgho lv wkdw wkh vzlwfk ehwzhhq uhjlphv lq wkh srolf|pdnhu*v suhihuhqfhv fdq 6 Lw
lv srvvleoh wr prgli| +44, wr doorz iru iruzdug0orrnlqj h{shfwdwlrqv1
:
uhvsrqg wr wkh frqglwlrqv lq wkh hfrqrp|1
6
Uhvxowv
614
X1V1 Uhvxowv
Hpsor|lqj wkh hvwlpdwlrq whfkqltxh rxwolqhg lq wkh dsshqgl{/ hvwlpdwhv ri wkh sdudphwhuv jryhuqlqj wkh prgho +4,/ +7,/ +8,/ +9,/ +44,/ dqg +43, zhuh rewdlqhg1 Wkhvh uhvxowv duh frpsduhg wr wkh uhvxowv ri Rz|dqj dqg Udph| +5334, lq zklfk d vlplodu vwuxfwxudo prgho lv hpsor|hg zlwk frqvwdqw wudqvlwlrq suredelolwlhv iru wkh srolf|pdnhu*v uhodwlyh wrohudqfhv iru lq dwlrq yhuvxv xqhpsor|phqw1 Wdeoh 4 glvsod|v wkh uhvxowlqj sdudphwhu hvwlpdwhv1 Wkh ydoxhv d4 dqg d5 lpsolhg e| wkh hvwlpdwhv ri n/ {4 / dqg {5 duh d4 @ 44;=48 dqg d5 @ 663=:41 Wkh odujhu ydoxhv ri d4 dqg d5 uhodwlyh wr wkrvh ri Rz|dqj dqg Udph| +5334, duh gxh wr wkh orzhu hvwlpdwh ri n17 Wklv lqglfdwhv wkdw wkh dffrpprgdwlyh srolf|pdnhu lv zloolqj wr vhw dq lq dwlrq wdujhw wkuhh wlphv kljkhu wkdq wkh lq dwlrq kdzn1 Wkh sdudphwhuv k4 dqg k5 duh wkh vwhdg| vwdwh xqhpsor|phqw udwhv lpsolhg e| wkh prgho18 Wkh dgyhuvh vkrfn whqgv wr lqfuhdvh wkh h{shfwhg xqhpsor|phqw udwh derxw 516(1 Wkh vdpsoh shulrg vkrzv lqfuhdvhv lq h{fhvv ri 516(/ h{sodlqhg e| wkh prgho dv whpsrudu| vkrfnv1 Vrph lqfuhdvhv duh odujh exw wkh| duh qrw shuvlvwhqw1 Wkh Pdunry surfhvv jryhuqlqj wkh Skloolsv fxuyh h{sodlqv wkh shuvlvwhqw fkdqjh lq xqhpsor|phqw fdxvhg e|/ iru h{dpsoh/ wkh orqj0odvwlqj h hfwv ri dq rlo vkrfn1 Wkh wudqvlwru| h hfwv duh dwwulexwhg wr %w vkrfnv1 Qrwh dovr wkdw wkh hvwlpdwhg ydoxh ri n lv vpdoohu iru wkh GG vshfl fdwlrq wkdq iru d QGG vshfl fdwlrq1 Wkhuh duh wzr lpsolfdwlrqv ri wklv qglqj1 Iluvw/ wkh h hfw ri d prqhwdu| vxusulvh rq wkh xqhpsor|phqw udwh lv vpdoohu1 Vhfrqg/ 7 Pd|eh
lw*v qhfhvvdu| wr fdoleudwh nB wkh uhvxowv zlwk Nrhqlj +5334, dqg Jrugrq +4 Vw.4 , @ s+Vw mh h hw > s+Vw.4 mh |w > ] hw > * h, @S
]w
s+Vw.4 mVw ,s+Vw mh |w > ]hw > hhw > * h, hw > hw > s+Vw.4 mh | w > Vw > ] hhw > * h ,s+Vw mh |w > ] hhw > * h, hw > s+Vw.4 mVw ,s+Vw mh |w > ] hhw > * h, @S > h hw > s+Vw.4 mVw ,s+Vw mh |w > ] hw > * h, ]w
zkhuh s+Vw.4 mVw , lv wkh wudqvlwlrq suredelolw| dqg wkh owhu ghwhuplqhv wkh ghq0 hw > |w > ] hhw > * h ,1 Wkh uvw lqhtxdolw| lv vlpso| dq dssolfdwlrq ri Ed|hv* vlw| s+Vw mh Odz1 Wkh qdo wzr lqhtxdolwlhv dulvh iurp wkh Pdunry surshuw| ri Vw = lq ghwhu0 plqlqj wkh ghqvlw| iru Vw.4 / wkh rqo| uhohydqw lqirupdwlrq lq wkh dydlodeoh vhw lv wkh suhylrxv vwdwh Vw 1 Wkh qxphudwru lq +45, lv fdofxodwhg iurp wkh Kdplowrq owhu dv hw > i+|w mh |w4 > Vw > ]hw > hhw > * h ,s+Vw mh |w4 > ] hhw > * h, hw > |w > ] hhw > * h, @ s+Vw mh h h i +|w mh |w4 > ]w > hw > * h,
56
hw > hw > i+|w mh |w4 > Vw > ] hhw > * h ,s+Vw mh |w4 > ] hhw > * h, @S > h h h h i +|w mh |w4 > Vw > ]w > hw > * h ,s+Vw mh |w4 > ]w > hw > * h, ]w zkhuh [ hw > hw > |w4 > ] hhw > * h, @ s+Vw mVw4 ,s+Vw4 mh |w4 > ] hhw > * h ,= s+Vw mh ]w4 hw > |w4 > ] hhw > * h , lv wdnhq iurp wkh suhylrxv lwhudwlrq1 Wkh ghqvlw| s+Vw4 mh
:17
Frqglwlrqdo Ghqvlw| ri wkh Sdudphwhu Yhfwru1
Wkh frqglwlrqdo ghqvlwlhv iru wkh hohphqwv ri wkh sdudphwhu yhfwru duh jhqhu0 dwhg e| hpsor|lqj Ed|hvldq ROV1 Wkh Ed|hvldq srvwhulru glvwulexwlrq iru hdfk hohphqw ri * h / frqglwlrqdo rq doo rwkhu hohphqwv ri * h / fdq eh ghwhuplqhg jlyhq d sulru glvwulexwlrq1 Li prgho sdudphwhuv/ h{foxglqj wkh yduldqfhv/ kdyh sulru glv0 wulexwlrqv ri wkh irup s3 +*l , Q+dl > Dl 5 ,/ wkhq wkhlu fruuhvsrqglqj srvwhulru frqglwlrqdo glvwulexwlrqv duh jlyhq e| hW > s+*l m*l > |hW > VhW > ] hhW , Q +dl > Dl 5 ,>
+46,
4 4 3 4 4 3 3 4 zkhuh dl @ +D l . [l [l , +Dl dl . [l \l ,/ Dl @ +Dl . [l [l , / [l lv wkh dssursuldwh uhjuhvvru/ dqg \l lv d iruhfdvw huuru1 Iru h{dpsoh/ frqvlghu eW hvwlpdwlqj wkh ydoxhv ri wkh yhfwru $ d lq +4,1 Lq wklv fdvh/ gh qh [ @ E ] $ dqg \ @ hW / zkhuh wkh urz w rq0gldjrqdo hohphqw ri E lv W Vw4 . n hw dqg }4W > }h5W `/ dqg }hlW lv d W 4 yhfwru wkh r 0gldjrqdo hohphqwv duh }hur/ ]eW @ ^h zlwk uhsuhvhqwdwlyh hohphqw }lw @ 4 l ]l @ l1 Rwkhu hohphqwv ri wkh sdudphwhu $ yhfwru fdq eh hvwlpdwhg vlploduo|1 Wkh ydoxhv iru wkh yhfwru k fdq eh jhqhudwhg
57
iurp d srvwhulru qrupdo vlplodu wr +46, lq zklfk 5 9 [ @ 7
VeW $ hW VeW d]
6 : 8>
6 h W hW , : hW n+h 9 x \ @ 7 8> $ hW hW n d] hhW 5
eW 1 zkhuh VeW lv gh qhg vlploduo| wr ]
Jlyhq frqglwlrqdo sulruv iru wkh yduldqfhv ri wkh irup 5%4 LJ+ 53 > U53 , dqg
5%l LJ+ 53 > O53 ,/ l @ 5> 6/ wkh srvwhulru frqglwlrqdo suredelolwlhv duh jlyhq e| . W U3 . 4 hW > * 5%4 mh > ,> |W > hhW > VhW > ] h LJ+ 3 5 5 zkhuh $ $ 4 @ +h xW n+h hW hW , k VhW ,3 +h xW n+h hW hW , k VhW ,> dqg/ iru l @ 5> 6= 3 . W O3 . l 5%l mh > ,> |W > hhW > VhW > ]hW > * h LJ+ 5 5 zkhuh
:18
$h $ $ $ hw + w d] k Vw . nh hw ,,3 +h w d ]hw + k Vhw . nh hw ,,= l @ +h
Wudqvlwlrq Suredelolwlhv
Jlyhq d sulru suredelolw| glvwulexwlrq iru Wlld ri wkh irup Wlld +xll > xml , iru m 9@ l/ wkh glvwulexwlrq iru wkh wudqvlwlrq suredelolwlhv Wlld lv ghwhuplqhg e| Wlld @ Su^w @ dl mw4 @ dl ` +xll . qll > xml . qml ,>
58
zkhuh qll lv wkh qxpehu ri shulrgv wkdw w uhpdlqhg lq vwdwh l/ dqg qml lv wkh qxpehu ri shulrgv wkdw w vzlwfkhg wr vwdwh m 9@ l diwhu ehjlqqlqj lq vwdwh l1 @ 4 W 1 D vlplodu Wkhq wkh rwkhu hohphqwv ri W fdq eh ghwhuplqhg e| Wml ll surfhgxuh lv xvhg wr jhqhudwh wkh hohphqwv ri W 1 4 dqg 5 1
Wkh srvwhulru frqglwlrqdo ghqvlw| iurp zklfk wkh sdudphwhuv
4 dqg 5 fdq eh jhqhudwhg lv gh qhg dv iroorzv1 Jlyhq wkh sulru ghqvlw| @ +4 > 5 , Q +o3 > 3 , wkh srvwhulru ghqvlw| lv jlyhq e| hhW > !Q +o> , m]W > #4 > #5 > g4 > g5 @ 4> VhW > ]W > |hW > zkhuh
4 3 4 4 3 o @ + 3 . [ [, +3 o3 . [ \ , dqg
4 3 4 @ + 3 . [ [, Lq wklv fdvh/ wkh l0wk hohphqwv ri wkh W0urz yhfwruv [ dqg \ duh gh qhg dv
[ @ ]w4 dqg \ @ ]w *]w4 zkhuh ]w zdv frpsxwhg lq wkh suhylrxv lwhudwlrq dqg * @ +# 4 g4 i4 > #5 g5 i5 ,1
59
#l frqglwlrqdo rq gl @ 41 Wkh srvwhulru frqglwlrqdo ghqvlw| lv s+#l m]W > #l > gl > gl @
hhW > !, 4> VhW > ]W > |hW >
l hhW > !Q z l > Z # l m]W > #l > gl > gl @ 4> VhW > ]W > |hW > zkhuh
z l @ +Z3>l4 . [ 3 [,4 +Z3>l4 z3>l . [ 3 \ ,>
l @ +Z 4 . [ 3 [,4 > Z 3>l dqg z3>l dqg Z3>l duh sdudphwhuv wdnh iurp wkh sulru glvwulexwlrq #l Q +z3 > Z3 , = Wkh l0wk hohphqwv ri wkh W0urz yhfwruv [ dqg \ duh gh qhg e|
[ @ ]w4 dqg \ @ ]w *]w4 Lq wklv fdvh/ * @ +*4 > *5 ,/ zkhuh w4 > x w4 > ]w4 , *l @ l . #l gl il + dqg
*l @ l 5:
dqg @ + 4 > 5 ,/ zkhuh w4 > x w4 > ]w4 , l @ il + dqg
l @ 3=
:19
Jhqhudwlqj wkh gl *v1
Uhfdoo wkdw wkh srvwhulru suredelolw| ri hdfk gl uhsuhvhqwv wkh srvwhulru sure0 delolw| ri wlph ydu|lqj wudqvlwlrq suredelolwlhv ru gxudwlrq ghshqghqfh iru wkh vwdwh l1 Jlyhq wkdw wkh sulru suredelolw| ri wlph ydu|lqj wudqvlwlrq suredelolwlhv lq vwdwh lv sl / Ed|hv Odz jlyhv wkh srvwhulru suredelolw| dv @ hhW > ! s gl m]W > #l > #l > gl > VhW > ]W > |hW >
sl > sl . l +4 sl ,
zkhuh l @ 5 h{s
:1:
z l z3>l 5Zl 5Z3>l
# $& v % Zl z l 4 s = l Z3>l Z
Jhqhudwlqj wkh srvwhulru suredelolwlhv iru ]W 1
Lq wklv prgho/ wkh wudqvlwlrq pdwul{ wkdw jryhuqv wkh srolf|pdnhu*v suhihuhqfhv lv ghshqghqw rq wkh klvwru| ]w @ +]4 > ]5 > ===> ]w4 > ]w.4 > ===> ]W ,1 Vlqfh hdfk vwdwh lv jhqhudwhg frqglwlrqdo rq lwv klvwru|/ hdfk ]w pxvw eh jhqhudwhg lqglylgxdoo|1 dqg wkh sdudphwhuv lq +45,/ wkh srvwhulru Vxssuhvvlqj wkh frqglwlrqlqj rq VhW / !/ suredelolw| iru ]w frqglwlrqdo rq wkh vhw ri revhuydeohv |W @ +|4 > |5 > ===> |W , dqg ]w @ +]4 > ]5 > ===> ]w4 > ]w.4 > ===> ]W , fdq eh zulwwhq dv
5;
Su^]w m |W > ]w ` @ Su^]w m |w > ]w `
@
@
Su^]w.4 m]w > |w4 ` Su^|w > ]w.4 > ===> ]W m]w > |w4 > ]w4 ` Su^|w > ]w.4 > ===> ]W m |w4 > ]w4 `
@
@
Su^]w > |w > ]w.4 > ===> ]W m |w4 > ]w4 ` Su^|w > ]w.4 > ===> ]W m |w4 > ]w4 `
Su^]w.4 m]w ` Su^|w > ]w.4 > ===> ]W m]w > |w4 > ]w4 ` Su^|w > ]w.4 > ===> ]W m |w4 > ]w4 `
Su^]w.4 m]w ` Su^|w m]w > |w4 ` Su^]w m]w4 ` Su^]w.5 > ===> ]W m |w > ]w.4 ` Su^|w > ]w.4 > ===> ]W m |w4 > ]w4 `
zklfk vlpsol hv wr
Su^]w.4 m]w ` Su^|w m]w > |w4 ` @S Su^]w m |W > ]w > VhW > !` Su^] w.4 m]w ` Su^|w m]w > |w4 ` Su^]w m]w4 ` ]w zkhuh Su^]w.4 m]w ` dqg Su^]w m]w4 ` duh wudqvlwlrq suredelolwlhv jhqhudwhg
iurp ]w dqg Su^]w m]w4 ` lv wkh frqglwlrqdo iruhfdvw huuru1 Wkh uvw htxdolw|
dulvhv iurp wkh idfw wkdw gdwd sdvw shulrg w lv qrq0lqirupdwlyh1 Wkh irxuwk htxdolw| frphv iurp wkh Pdunry surshuwlhv ri ]w 1
5
