Solving Portfolio Problems
with the Smolyak-Parameterized
Expectations Algorithm
Ángel Gavilán and Juan A. Rojas
Documentos de Trabajo
N.º 0838
2008
SOLVING PORTFOLIO PROBLEMS WITH THE SMOLYAK-PARAMETERIZED
EXPECTATIONS ALGORITHM
SOLVING PORTFOLIO PROBLEMS WITH THE SMOLYAK- PARAMETERIZED
EXPECTATIONS ALGORITHM (*)
Ángel Gavilán and Juan A. Rojas (**) (***)
BANCO DE ESPAÑA
(*) The opinions expressed here are solely those of the authors and do not necessarily reflect the views of the
Bank of Spain or the Eurosystem.
(**)
(***)
Bank of Spain, Research Department. E-mail: [email protected].
Bank of Spain, Research Department. E-mail: [email protected].
Documentos de Trabajo. N.º 0838
2008
The Working Paper Series seeks to disseminate original research in economics and finance. All papers
have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute
to economic analysis and, in particular, to knowledge of the Spanish economy and its international
environment.
The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore,
do not necessarily coincide with those of the Banco de España or the Eurosystem.
The Banco de España disseminates its main reports and most of its publications via the INTERNET at the
following website: http://www.bde.es.
Reproduction for educational and non-commercial purposes is permitted provided that the source is
acknowledged.
© BANCO DE ESPAÑA, Madrid, 2008
ISSN: 0213-2710 (print)
ISSN: 1579-8666 (on line)
Depósito legal: M. 6998-2009
Unidad de Publicaciones, Banco de España
Abstract
We propose a new numerical method to solve stochastic models that combines the parameterized expectations (PEA) and the Smolyak algorithms. This method is especially
convenient to address problems with occasionally binding constraints (a feature inherited
from PEA) and/or a large number of state variables (a feature inherited from Smolyak), i.e.
DSGE models that incorporate portfolio problems and incomplete markets. We describe the
proposed Smolyak-PEA algorithm in the context of a one-country stochastic neoclassical
growth model and compare its accuracy with that of a standard PEA collocation algorithm.
Despite estimating fewer parameters, the former is able to reach the high accuracy levels
of the latter. We further illustrate the working of this algorithm in a two-country neoclassical
model with incomplete markets and portfolio choice. Again, the Smolyak-PEA algorithm
approximates the solution of the problem with a high degree of accuracy. Finally, we show
how this algorithm can efficiently incorporate both occasionally binding constraints and a
partial information approach.
Keywords: Portfolio Choice, Dynamic Macroeconomics, Computational Methods
JEL classification: E2, C68.
4
Lqwurgxfwlrq
Pdfurhfrqrplvwv duh ehfrplqj lqfuhdvlqjo| lqwhuhvwhg lq GVJH rshq hfrqrp| prg0
hov wkdw lqfrusrudwh sruwirolr sureohpv dqg lqfrpsohwh pdunhwv1 Wkhvh prghov duh
qhhghg/ iru lqvwdqfh/ wr xqghuvwdqg zkdw lv ehklqg jurvv dqg qhw lqwhuqdwlrqdo fdslwdo
rzv/ zkrvh pdjqlwxghv kdyh vxevwdqwldoo| lqfuhdvhg ryhu wkh odvw ghfdghv/ dqg wr
dqdo|}h wkh pdfurhfrqrplf lpsdfw wkdw vkrfnv pd| kdyh lq dq hfrqrp| dv d ixqfwlrq
ri lwv lqwhuqdwlrqdo sruwirolr vwuxfwxuh +l1h1 lq whupv ri ydoxdwlrq hhfwv,1
Vroylqj wklv nlqg ri prghov/ krzhyhu/ lw lv qrw dq hdv| wdvn1 Lq idfw/ rqo| uhfhqwo|
ihz phwkrgv kdyh ehhq sursrvhg lq wkh olwhudwxuh zklfk fdq ghdo zlwk wkhvh prghov1
Ghyhuhx{ dqg Vxwkhuodqg +533:, dqg ydq Zlqfrrs dqg Wlooh +533:, duh h{dpsohv
ri wklv olwhudwxuh1 Wkh| kdyh erwk ghyhorshg shuwxuedwlrq phwkrgv zkhuh/ sxw vlp0
so|/ wkh vroxwlrq lv edvhg rq dssur{lpdwlrqv ri wkh sureohp*v htxloleulxp frqglwlrqv
durxqg wkh vwhdg| vwdwh/ zlwk d vshfldo hpskdvlv rq wkh sruwirolr htxdwlrqv14 Wkhvh
phwkrgv duh wkhq frqyhqlhqw wr dqdo|}h wkh hhfw ri glhuhqw vkrfnv rq d frxqwu|*v
htxloleulxp sruwirolr dqg rq sulfhv durxqg wkh vwhdg| vwdwh1 Krzhyhu/ vlqfh wkhvh
duh orfdo phwkrgv/ wkh| fdq qrw eh xvhg wr vwxg| wkh rswlpdo sdwk ri dgmxvwphqw
ri dq hfrqrp| klw e| shupdqhqw vkrfnv wkdw fkdqjh lwv vwhdg| vwdwh1 Rqh h{dpsoh
ri wklv nlqg ri vkrfnv zrxog eh wkh zruogzlgh ghprjudsklf fkdqjh1 Wkh olwhudwxuh
kdv h{whqvlyho| vwxglhg wkh lpsdfw ri wklv vkrfn rq qhw lqwhuqdwlrqdo fdslwdo rzv
xvlqj rqh0dvvhw pxowl0frxqwu| prghov15 Krzhyhu/ li rqh zhuh wr dqdo|}h wkh hhfw ri
wklv vkrfn rq jurvv fdslwdo rzv ru dvvhw sulfhv zlwk d prgho lqfrusrudwlqj sruwirolr
ghflvlrqv/ wkhq d joredo vroxwlrq phwkrg zrxog eh qhhghg1
Lq wklv sdshu zh sursrvh d joredo vroxwlrq phwkrg wkdw lv vxlwdeoh iru vroylqj vwr0
fkdvwlf prghov fkdudfwhul}hg e| d odujh qxpehu ri vwdwh yduldeohv dqg/ lq sduwlfxodu/
GVJH rshq hfrqrp| prghov zlwk sruwirolr sureohpv dqg lqfrpsohwh pdunhwv1 Zh
frpelqh wzr h{lvwlqj dojrulwkpv lq wkh olwhudwxuh1 Rq wkh rqh kdqg/ zh frqvlghu wkh
sdudphwhul}hg h{shfwdwlrq dojrulwkp +SHD,/ d zhoo0nqrzq surmhfwlrq phwkrg16 Lq
vkruw/ wklv phwkrg*v dssurdfk wr vroyh vwrfkdvwlf sureohpv frqvlvwv rq sdudphwhul}lqj
wkh frqglwlrqdo h{shfwdwlrqv dsshdulqj lq wkh sureohpv* uvw0rughu frqglwlrqv1 Pruh
vshflfdoo|/ zh frqvlghu d yduldqw ri SHD fdoohg SHD froorfdwlrq1 Wklv dssurdfk edvl0
fdoo| dssur{lpdwhv wkrvh frqglwlrqdo h{shfwdwlrqv xvlqj sro|qrpldov zkrvh xqnqrzq
frh!flhqwv duh ghwhuplqhg e| lpsrvlqj wkdw wkh dssur{lpdwlrq h{dfwo| krogv dw wkh
vr0fdoohg froorfdwlrq srlqwv1
Rq wkh rwkhu kdqg/ zh pdnh xvh ri wkh Vpro|dn dojrulwkp/ uvw lqwurgxfhg lq
4
Hydqv dqg Kqdwnryvnd +5338, dovr ghyhors d qxphulfdo phwkrg wr vroyh sureohpv zlwk sruw0
irolr ghflvlrqv dqg lqfrpsohwh pdunhwv xvlqj shuwxuedwlrq phwkrgv/ dowkrxjk lq frpelqdwlrq zlwk
surmhfwlrq phwkrgv dqg frqwlqxrxv wlph dssur{lpdwlrqv whfkqltxhv1
5
Vhh/ iru lqvwdqfh/ Grphlm dqg Iorghq +5339, dqg Nuxhjhu dqg Oxgzlj +533:,1
6
Vhh/ iru lqvwdqfh/ Ghq Kddq dqg Pdufhw +4<<3,/ Pdufhw dqg Oruhq}rql +4<<<,/ Pdufhw dqg
Vlqjohwrq +4<<<,/ Fkulvwldqr dqg Ilvkhu +5333, dqg Pdoldu dqg Pdoldu +5336,1
BANCO DE ESPAÑA
9
DOCUMENTO DE TRABAJO N.º 0838
wkh hfrqrplf olwhudwxuh e| Nuxhjhu dqg Nxeohu +5337,1 Wklv lv d vsduvh julg phwkrg
wkdw lv sduwlfxoduo| frqyhqlhqw wr dgguhvv kljk0glphqvlrqdo sureohpv1 Dq xqghvludeoh
ihdwxuh ri wkh vwdqgdug SHD froorfdwlrq dojrulwkp lv wkdw lw ehfrphv frpsxwdwlrqdoo|
xqihdvleoh lq sureohpv zlwk d odujh qxpehu ri vwdwh yduldeohv1 Wkh xqghuo|lqj uhdvrq
lv wkdw wkh qxpehu ri froorfdwlrq srlqwv dqg xqnqrzq frh!flhqwv wr eh irxqg zlwk
wklv phwkrg jurz h{srqhqwldoo| zlwk wkh glphqvlrq ri wkh sureohp1 Wkh Vpro|dn
dojrulwkp euhdnv wklv fxuvh ri glphqvlrqdolw| edvlfdoo| e| fkrrvlqj fduhixoo| erwk wkh
dssur{lpdwlqj ixqfwlrq dqg wkh froorfdwlrq srlqwv1
E| frpelqlqj wkh SHD froorfdwlrq dqg wkh Vpro|dn dojrulwkpv/ lq zkdw zh fdoo
wkh Vpro|dn0SHD dojrulwkp/ zh wkxv kdyh d joredo surmhfwlrq phwkrg wkdw lv vxlw0
deoh wr dgguhvv kljk0glphqvlrqdo sureohpv/ dv wkrvh lqyroylqj sruwirolr ghflvlrqv lq
d pxowl0frxqwu| vhw xs1 Zh looxvwudwh wkh zrunlqj ri wklv phwkrg lq wkh frqwh{w ri
d vwrfkdvwlf wzr0frxqwu| qhrfodvvlfdo prgho zlwk lqfrpsohwh pdunhwv dqg sruwirolr
ghflvlrqv dqg qg wkdw wklv phwkrg*v dffxudf| hdvlo| phhwv xvxdo vwdqgdugv lq wkh
olwhudwxuh1
Lq dgglwlrq/ zh dsso| wkh dojrulwkp wr wzr h{whqvlrqv ri wkh edvlf vhw xs1 Iluvw/
zh frqvlghu d sduwldo lqirupdwlrq vfhqdulr zkhuh lqglylgxdov/ lq rughu wr wdnh wkhlu
sruwirolr ghflvlrqv/ gr qrw nqrz wkh zkroh dvvhw glvwulexwlrq lq wkh hfrqrp| exw rqo|
vrph prphqwv ri lw1 Wkh sduwldo lqirupdwlrq dssurdfk/ pdgh srsxodu e| Nuxvhoo dqg
Vplwk +4<<;,/ lw lv frpprqo| dgrswhg lq wkh olwhudwxuh wr vroyh prghov lq zklfk vrph
nlqg ri glvwulexwlrq hqwhuv lqwr wkh lqglylgxdov lqirupdwlrqdo vhw1 Wkh dgydqwdjh ri
wklv dssurdfk lv wkdw lw uhgxfhv wkh glphqvlrqdolw| ri wkhvh sureohpv vxevwdqwldoo|1
Krzhyhu/ lw reylrxvo| hqwdlov dq dffxudf| orvv uhodwlyh wr wkh ixoo lqirupdwlrq fdvh1
Zlwk wkh h{fhswlrq ri Nuxhjhu dqg Nxeohu +5337,/ wklv dffxudf| orvv kdv qrw ehhq
txdqwlhg1 Lq wklv vhqvh/ zh vkrz krz wkh Vpro|dn0SHD dojrulwkp fdq eh lpsoh0
phqwhg lq d sduwldo lqirupdwlrq vfhqdulr dqg surylgh d phdvxuh ri wkh dffxudf| orvv
fdxvhg e| wkh ghsduwxuh iurp ixoo lqirupdwlrq1
Vhfrqg/ zh dgg rffdvlrqdoo| elqglqj frqvwudlqwv wr wkh sruwirolr sureohp idfhg
e| lqglylgxdov1 Vxfk uhvwulfwlrqv rq sruwirolr ghflvlrqv duh frpprq erwk lq qdq0
fldo pdunhwv duh lq wkhruhwlfdo prghov1 Zh vkrz krz wkh Vpro|dn0SHD dojrulwkp
fdq hdvlo| ghdo zlwk wklv w|sh ri frqvwudlqwv1 Lq idfw/ dv dujxhg e| Fkulvwldqr dqg
Ilvkhu +5333,/ lw lv suhflvho| lq wkh suhvhqfh ri wkhvh frqvwudlqwv zkhq sdudphwhul}lqj
frqglwlrqdo h{shfwdwlrqv lv fohduo| pruh frqyhqlhqw wkdq sdudphwhul}lqj srolf| ixqf0
wlrqv/ vlqfh wkh odwwhu dssurdfk pxvw xqghuwdnh d fxpehuvrph gluhfw vhdufk iru erwk
wkh srolf| dqg pxowlsolhu ixqfwlrqv1 Lq dgglwlrq/ zh vwxg| krz wkh rswlpdo srolf|
ixqfwlrqv fkdqjh/ lq rxu vhw xs/ ghshqglqj rq wkh vhyhulw| ri sruwirolr frqvwudlqwv1
Ehiruh dqdo|}lqj wkh shuirupdqfh ri wkh Vpro|dn0SHD dojrulwkp lq d wzr0frxqwu|
prgho zlwk sruwirolr ghflvlrqv/ wkh iroorzlqj vhfwlrq dssolhv lw wr wkh vwdqgdug rqh0
frxqwu| vwrfkdvwlf qhrfodvvlfdo jurzwk prgho1 Wklv prgho/ wkdw doorzv xv wr lqwurgxfh
BANCO DE ESPAÑA
10
DOCUMENTO DE TRABAJO N.º 0838
lq d vlpsoh zd| erwk wkh SHD dqg wkh Vpro|dn dojrulwkpv/ kdv ehhq wudglwlrqdoo|
xvhg lq wkh olwhudwxuh dv wkh ehqfkpdun prgho lq zklfk wr looxvwudwh wkh zrunlqj ri
glhuhqw frpsxwdwlrqdo phwkrgv dqg wr vhw frpsdulvrqv dprqj wkhp1 Zh iroorz
wklv olqh dqg frpsduh wkh shuirupdqfh ri wkh Vpro|dn0SHD dojrulwkp zlwk wkdw ri
d vwdqgdug SHD froorfdwlrq1 Zh qg wkdw/ iru wklv vlpsoh prgho/ wkh Vpro|dn0SHD
dojrulwkp dfklhyhv dffxudf| ohyhov vlplodu wr wkrvh ri d vwdqgdug SHD froorfdwlrq/
ghvslwh wkh idfw wkdw lw xvhv d vxevwdqwldoo| vpdoohu qxpehu ri froorfdwlrq srlqwv dqg
ri frh!flhqwv lq wkh dssur{lpdwlqj ixqfwlrq1 Wklv lv suhflvho| wkh juhdw dgydqwdjh ri
wklv phwkrg dqg zkdw pdnhv lw vxlwdeoh wr ghdo zlwk kljk0glphqvlrqdo sureohpv/ dv
wkrvh lqyroylqj pxowl0frxqwu| sruwirolr ghflvlrqv1
5
Rqh0frxqwu| vwrfkdvwlf prgho
Frqvlghu wkh vwdqgdug rqh0frxqwu| vwrfkdvwlf qhrfodvvlfdo jurzwk prgho=
4@
Hf
"
[
w'f
vxemhfw wr
w
f3
w

fw n nwn ' E nw n w nw ;w
+4,
zkhuh *L} w ' *L} w3 n w / w Q Ef> 2 dqg wkh lqlwldo frqglwlrqv Enf > f duh
jlyhq1
Wkh uvw rughu frqglwlrq ri wklv sureohp lv=
3
' Hw f3
f3
w
wn E wn nwn n  +5,
Li wkh xwlolw| ixqfwlrq lv orjdulwkplf + ' , dqg fdslwdo frpsohwho| ghsuhfl0
dwhv lq rqh shulrg + ' ,/ wklv sureohp kdv dq dqdo|wlfdo vroxwlrq1 Qdpho|/ fw '
E w nw 1 Iru jhqhudo sdudphwhu ydoxhv/ krzhyhu/ d forvhg0irup vroxwlrq wr wklv
sureohp grhv qrw h{lvw dqg qxphulfdo phwkrgv duh qhhghg wr vroyh lw1
514
SHD froorfdwlrq
Wkh sdudphwhul}hg h{shfwdwlrq dojrulwkp +SHD, lv rqh ri wkh dydlodeoh phwkrgv
lq wkh olwhudwxuh wkdw fdq kdqgoh wklv w|sh ri vwrfkdvwlf sureohpv1 Lwv dssurdfk lv
vlpsoh1 Lw qrwhv wkdw wkh frqglwlrqdo h{shfwdwlrq lq htxdwlrq +5, lv d ixqfwlrq ri wkh
vwdwh yduldeohv/ Enw > w / dqg uhsodfhv lw zlwk d ixqfwlrq [Enw > w ( x/ zkhuh erwk wkh
ixqfwlrqdo irup +[, dqg wkh sdudphwhuv +x, duh fkrvhq vr wkdw wklv dssur{lpdwlrq
BANCO DE ESPAÑA
11
DOCUMENTO DE TRABAJO N.º 0838
lv dv dffxudwh dv srvvleoh17 Glhuhqw zd|v ri lpsohphqwlqj wklv lghd ghshqg rq wkh
ixqfwlrqdo irup fkrvhq iru wkh dssur{lpdwlqj ixqfwlrq dqg2ru rq wkh surfhgxuh xvhg
wr qg wkh ehvw vhw ri sdudphwhuv ri wkdw ixqfwlrq1 Fkulvwldqr dqg Ilvkhu +5333,
ghvfuleh vhyhudo ri wkhvh SHD dojrulwkpv lq ghwdlo dqg frpsduh wkhp zlwk rwkhu qrq0
SHD dojrulwkpv lq wkh frqwh{w ri wkh vwdqgdug rqh0frxqwu| vwrfkdvwlf qhrfodvvlfdo
jurzwk prgho1 Wkh frpsdulvrq idyruv wkh vr0fdoohg SHD froorfdwlrq dojrulwkp/ d
vshfldo fdvh ri Fkhe|vkhy SHD1 Rq zkdw iroorzv/ zh irfxv rq wklv phwkrg/ wkdw
zrunv lq wkh iroorzlqj idvklrq=
D410 Irup d wzr0glphqvlrqdo julg +vlqfh wkhuh duh wzr vwdwh yduldeohv lq wklv
h{dpsoh,/ K 2 / frpelqlqj qn glhuhqw ydoxhv ri n dqg q glhuhqw ydoxhv ri 18 Wkh
qn +q , glhuhqw ydoxhv ri n +, duh fkrvhq wr eh wkh qn +q , }hurv ri d Fkhe|vkhy
sro|qrpldo ri rughu qn +q ,1 Wkh q }hurv ri d Fkhe|vkhy sro|qrpldo ri rughu q/
q ' i  > ===> q j/ duh ghqhg e|=
2m 
m ' ULt
> m ' > ===> q
2q
Wkh glhuhqw ydoxhv lq hdfk glphqvlrq duh wkhq frpelqhg wr fuhdwh wkh julg xvlqj
wkh whqvru surgxfw1 Qdpho|/
K 2 ' qn q
D510 Frqvwuxfw dq dssur{lpdwlqj ixqfwlrq frpelqlqj Fkhe|vkhy sro|qrpldov lq
wkh iroorzlqj idvklrq=
[En> ( x '
q
qn [
[
!lm Wl E* En Wm E* E
+6,
l'f m'f
zkhuh Wq E{ ghqrwhv d Fkhe|vkhy sro|qrpldo ri rughu q hydoxdwhg dw { 5 d> o19
Fkhe|vkhy sro|qrpldov ri glhuhqw rughuv duh ghqhg lq wkh iroorzlqj zd|=
Wf E{ ' 
W E{ ' {
Wq E{ ' 2{Wq3 E{ Wq32 E{ ;q 2
7
Lq sulqflsoh/ fkrrvlqj fduhixoo| wkh dssur{lpdwlqj ixqfwlrq lw vkrxog eh srvvleoh wr dwwdlq
kljk ohyhov ri dffxudf|1 Iru lqvwdqfh/ li lv d sro|qrpldo/ duelwudu| ohyhov ri dffxudf| frxog eh
rewdlqhg lq wkh dssur{lpdwlrq e| surjuhvvlyho| lqfuhdvlqj wkh rughu ri wkh sro|qrpldo1
8
Lq jhqhudo/ irup d g0glphqvlrqdo julg/ K g / ri wkh g vwdwh yduldeohv ri wkh sureohp1
9
Vwdwh yduldeohv gr qrw xvxdoo| olh rq wkh lqwhuydo ^4> 4`/ exw lw lv srvvleoh wr vhw xsshu dqg
orzhu erxqgv iru wkhp1 Wkh ixqfwlrq * +, lv wkhq xvhg wr fkdqjh wkh vfdoh ri wkrvh yduldeohv1 Iru
lqvwdqfh/ lq wkh fdvh ri n/ * = ^n> n` $ ^4> 4`1 Dq h{dpsoh ri vxfk dq vfdolqj ixqfwlrq zrxog eh
* +n, @ 5 nn
4=
nn
BANCO DE ESPAÑA
12
DOCUMENTO DE TRABAJO N.º 0838
D610 Lq wkh dssur{lpdwlqj ixqfwlrq +6, wkhuh duh qn q xqnqrzq frh!flhqwv
!lm wkdw qhhg wr eh ghwhuplqhg1 Wkh dssurdfk xvhg e| wklv dojrulwkp wr qg wkrvh
frh!flhqwv lv vlpsoh1 Lw lpsrvhv wkdw/ dw doo julg srlqwv/ dovr nqrzq dv froorfdwlrq
srlqwv/ [En> ( x dssur{lpdwhv h{dfwo| wkh frqglwlrqdo h{shfwdwlrq lq +5,/ zkhuh
wkh odwwhu lv frpsxwhg xvlqj vwdqgdug txdgudwxuh whfkqltxhv +l1h1/ Jdxvv0Khuplwh
txdgudwxuh,1 Wkh idfw wkdw wkhuh duh h{dfwo| dv pdq| xqnqrzq frh!flhqwv dv julg
srlqwv pdnhv lw srvvleoh wr uhdfk vxfk h{dfw dssur{lpdwlrq1 Lq wklv vhqvh/ vlqfh wkhuh
duh dv pdq| htxdwlrqv [E> x ' Hw do +rqh iru hdfk julg srlqw, dv xqnqrzqv +wkh !lm *v,/
lw zrxog eh srvvleoh wr qg wkrvh xqnqrzq frh!flhqwv xvlqj dq| vwdqgdug qrq0olqhdu
htxdwlrqv vroyhu1 Wklv dssurdfk/ krzhyhu/ pd| eh frpsxwdwlrqdoo| fxpehuvrph/
hvshfldoo| li wkh qxpehu ri julg srlqwv ru xqnqrzqv lv odujh1 Wklv lv zk|/ lq sudfwlfh/
uhvhdufkhuv kdyh ydvwo| suhihu dq lwhudwlyh surfhgxuh ri wkh iroorzlqj w|sh=
D6l10 Pdnh dq lqlwldo dvvxpswlrq/ xf / derxw wkh xqnqrzq frh!flhqwv1
D6ll10 Iru hdfk julg srlqw vroyh iru fw vxfk wkdw
f3
' [En> ( xf w
dqg xvh wkh exgjhw frqvwudlqw +4, wr qg nwn 1
3
D6lll10 Iru hdfk julg srlqw frpsxwh Hw f3
wn E wn nwn n  xvlqj vwdqgdug
txdgudwxuh whfkqltxhv1 Wkh ydoxh wr nwn kdv ehhq rewdlqhg lq vwhs D6ll dqg fwn lv
frpsxwhg vr wkdw f3
wn ' [Enwn > wn ( xf iru wkh glhuhqw ydoxhv ri wn frqvlghuhg
lq wkh txdgudwxuh surfhgxuh1
3
D6ly10 Rewdlq d qhz vhw ri frh!flhqwv/ x / uhjuhvvlqj Hw f3
wn E wn nwn n  rq wkh uljkw kdqg vlgh ri htxdwlrq +6,1
D6y10 Li xf dqg x duh forvh hqrxjk +iru d jlyhq ghjuhh ri suhflvlrq,/ vwrs wkh
lwhudwlyh surfhgxuh dqg dssur{lpdwh wkh frqglwlrqdo h{shfwdwlrq lq +5, xvlqj wkh
dssur{lpdwlqj ixqfwlrq +6, zlwk wkhvh frh!flhqwv1 Rwkhuzlvh/ xsgdwh wkh frh!flhqwv
dqg uhshdw vwhsv D6ll0D6ly xqwlo frqyhujhqfh ehwzhhq xl dqg xln lv dfklhyhg1
515
Vpro|dn0SHD
Wkh SHD froorfdwlrq dojrulwkp ghvfulehg lq vhfwlrq 514 zrunv idluo| zhoo iru orz
glphqvlrqdo sureohpv1 Krzhyhu/ dv wkh glphqvlrq ri wkh sureohp lqfuhdvhv wkh lp0
sohphqwdwlrq ri wklv dojrulwkp ehfrphv h{srqhqwldoo| pruh frpsolfdwhg1 Wr jhw d
vhqvh ri wkh fxuvh ri glphqvlrqdolw| idfhg e| wklv dojrulwkp vlpso| qrwh wkdw/ zkloh
lq d 50glphqvlrqdo sureohp xvlqj sro|qrpldov ri rughu 7 lq wkh dssur{lpdwlrq lp0
solhv qglqj 58 xqnqrzq frh!flhqwv/ lq sureohpv zlwk 9 ru < vwdwh yduldeohv lw zrxog
lpso| vroylqj iru 48/958 dqg 4/<86/458 xqnqrzqv/ uhvshfwlyho|1 Wkh odwwhu lv fohduo|
xqihdvleoh dqg lw lv zkhuh wkh Vpro|dn dojrulwkp ehfrphv h{wuhpho| frqyhqlhqw1
BANCO DE ESPAÑA
13
DOCUMENTO DE TRABAJO N.º 0838
Wkh Vpro|dn dojrulwkp zdv uvw lqwurgxfhg lq wkh hfrqrplf olwhudwxuh e| Nuxhjhu
dqg Nxeohu +5337, dqg lw lv sduwlfxoduo| lqglfdwhg wr ghdo zlwk kljk0glphqvlrqdo sure0
ohpv1 Wkh uhdvrq lv vlpsoh1 E| fkrrvlqj fduhixoo| erwk wkh dssur{lpdwlqj ixqfwlrq
dqg wkh froorfdwlrq srlqwv/ wkh qxpehu ri xqnqrzq frh!flhqwv wr eh irxqg lq wklv
dojrulwkp grhv qrw jurz h{srqhqwldoo| zlwk wkh glphqvlrq ri wkh sureohp1 Pruh lq0
whuhvwlqjo|/ dv zh zloo vkrz/ ghvslwh hpsor|lqj ohvv frh!flhqwv lq wkh dssur{lpdwlrqv
uhodwlyh wr rwkhu phwkrgv/ wkhuh lv qr vxevwdqwldo dffxudf| orvv lq xvlqj wklv dssurdfk1
Nuxhjhu dqg Nxeohu +5337, vwxg| dq ryhuodsslqj jhqhudwlrqv hfrqrp| zlwk vwr0
fkdvwlf djjuhjdwh surgxfwlrq dqg/ e| xvlqj wkh Vpro|dn dojrulwkp/ pdqdjh wr vroyh
sureohpv zlwk 53063 vwdwh yduldeohv1 Lq d glhuhqw dssolfdwlrq ri wklv dojrulwkp/
Pdolq/ Nuxhjhu dqg Nxeohu +533:, frqvlghu d pxowl0frxqwu| vwrfkdvwlf qhrfodvvlfdo
jurzwk prgho dqg duh deoh wr vroyh sureohpv zlwk xs wr 45 vwdwh yduldeohv1 Lq erwk
dssolfdwlrqv/ krzhyhu/ wkh Vpro|dn dojrulwkp lv xvhg wr dssur{lpdwh wkh srolf| ixqf0
wlrqv1: Wkh dssurdfk lq wklv sdshu/ lqvwhdg/ lv wr xvh wkh Vpro|dn dojrulwkp wr
dssur{lpdwh wkh frqglwlrqdo h{shfwdwlrqv wkdw dsshdu lq vwrfkdvwlf prghov/ wkdw lv/
lq frpelqdwlrq zlwk wkh SHD dojrulwkp1 Dv dujxhg e| Fkulvwldqr dqg Ilvkhu +5333,/
wklv dowhuqdwlyh urxwh lv sduwlfxoduo| lqglfdwhg li wkh sureohp ehlqj dqdo|}hg ihdwxuhv
rffdvlrqdoo| elqglqj frqvwudlqwv/ dv lw lv xvxdoo| wkh fdvh/ iru lqvwdqfh/ lq sruwirolr
sureohpv1 Lq wkrvh fdvhv/ dssur{lpdwlqj wkh frqglwlrqdo h{shfwdwlrqv lv pruh frq0
yhqlhqw wkdq dssur{lpdwlqj wkh srolf| ixqfwlrqv/ vlqfh wkh odwwhu dssurdfk qhhgv wr
mrlqwo| sdudphwhul}h wkh srolf| dqg wkh pxowlsolhu ixqfwlrqv/ zkdw pd| eh d fxpehu0
vrph dqg frpsolfdwhg wdvn1
Zh qrz ghvfuleh krz wr frpelqh wkh Vpro|dn dqg wkh SHD froorfdwlrq dojrulwkpv1
Lqfrusrudwlqj wkh iruphu lqwr wkh surfhgxuh ghvfulehg lq vhfwlrq 514 edvlfdoo| hqwdlov
wzr fkdqjhv=
E410 Wkh julg1 Wkhuh duh wzr edvlf glhuhqfhv ehwzhhq wkh julg lq SHD froorfd0
wlrq dqg wkh julg lq Vpro|dn1 Rq wkh rqh kdqg/ lq Vpro|dn/ wkh julg srlqwv lq hdfk
glphqvlrq duh qrw wkh }hurv ri Fkhe|vkhy sro|qrpldov exw wkh h{wuhpd ri wkrvh sro|0
qrpldov1 Wkh q h{wuhpd ri d Fkhe|vkhy sro|qrpldo ri rughu q/ Jq ' iJ > ===> Jq j/
duh ghqhg e|=
m
Jm ' ULt
> m ' > ===> q
q
Wklv glhuhqfh lv lpsruwdqw ehfdxvh wkh h{wuhpd ri d Fkhe|vkhy sro|qrpldo ri d
jlyhq rughu duh qhvwhg lq wkh h{wuhpd ri kljkhu rughu sro|qrpldov/ l1h1 Jq Jqn
;q ' > 2> === Wklv lv fuxfldo lq uhgxflqj wkh glphqvlrqdolw| ri wkh sureohp1
Rq wkh rwkhu kdqg/ lq Vpro|dn/ wkh pxowl0glphqvlrqdo julg lv qrw fuhdwhg e| vlpso|
:
BANCO DE ESPAÑA
Erwk sdshuv surylgh d ghwdlohg ghvfulswlrq ri wkh Vpro|dn dojrulwkp1
14
DOCUMENTO DE TRABAJO N.º 0838
dsso|lqj wkh whqvru surgxfw wr wkh vhw ri julg srlqwv lq hdfk glphqvlrq1 Lqvwhdg/ lw
lv frqvwuxfwhg dffruglqj wr wkh iroorzlqj h{suhvvlrq=
^
JpEl === JpElg K g> '
l= ƒlƒ'gn
g
]nn
zkhuh l ' El > ===> lg 5
lv d yhfwru ri srvlwlyh lqwhjhuv/ mlm ' l n === n lg /
g ghqrwhv wkh glphqvlrq ri wkh sureohp +wkh qxpehu ri vwdwh yduldeohv, dqg lv d
phdvxuh ri wkh ghvluhg glphqvlrq ri wkh julg +dqg/ e| h{whqvlrq/ ri wkh dffxudf|
sxuvxhg lq wkh zkroh dssur{lpdwlrq surfhgxuh,1 Ilqdoo|/ wkh pE*v duh lqwhjhuv wkdw
ghqh wkh qxpehu ri Fkhe|vkhy h{wuhpd wr frqvlghu lq hdfk srvvleoh glphqvlrq1 Wkh|
duh frqvwuxfwhg lq wkh iroorzlqj zd|=
pE ' 
pEl ' 2l3 n  iru l ' 2> > ===
E510 Wkh dssur{lpdwlqj ixqfwlrq1 Lq wkh Vpro|dn dojrulwkp/ wkh dssur{lpdwlqj
ixqfwlrq lv frqvwuxfwhg lq wkh iroorzlqj idvklrq=
[
g
g>
gn3ƒlƒ
E
+7,
[EY ( x '
sl EY g n mlm
t$ƒlƒ$gn
zkhuh
pEl l
s EY '
[
o '
pElg ===
[
!o ===og Wo E* Ey ===Wog E* Eyg og '
dqg t ' 4@ Eg> n 1 Y ' Ey > ===> yg lv d g0glphqvlrqdo yhfwru ri wkh vwdwh
yduldeohv dqg wkh vfdolqj ixqfwlrq * lv xvhg wr sodfh wkh ydoxhv wkhvh yduldeohv rq wkh
lqwhuydo d> o1
Wkxv/ zkloh lq wkh vwdqgdug froorfdwlrq phwkrg wkh dssur{lpdwlqj ixqfwlrq lv
mxvw wkh whqvru surgxfw ri Fkhe|vkhy sro|qrpldov lq hdfk glphqvlrq/ lq Vpro|dn wklv
ixqfwlrq lv frqvwuxfwhg xvlqj olqhdu frpelqdwlrqv ri whqvru surgxfwv ri Fkhe|vkhy
sro|qrpldov lq fhuwdlq glphqvlrqv1 Lq dq| fdvh/ qrwh wkdw lq wklv dojrulwkp/ dv lq
wkh vwdqgdug SHD froorfdwlrq/ wkh qxpehu ri sro|qrpldo frh!flhqwv wr eh hvwlpdwhg
lv h{dfwo| htxdo wr wkh qxpehu ri julg srlqwv lq K g> 1 Lq wklv vhqvh/ wkh qxpehu ri
julg srlqwv lq K g> +htxdo wr wkh qxpehu ri xqnqrzq frh!flhqwv lq +7,, lv jlyhq e|
htxdwlrq +8, iru dq dffxudf| ohyho ' 2 dqg e| htxdwlrq +9, iru dq dffxudf| ohyho
' 1
_4EK g>2 '  n eg n 2g Eg 
_4EK g> '  n Hg n Sg Eg  n H
BANCO DE ESPAÑA
15
DOCUMENTO DE TRABAJO N.º 0838
g Eg  Eg 2
S
+8,
+9,
D vlpsoh h{sorudwlrq ri wkhvh irupxodv pdnhv hylghqw krz wkh Vpro|dn dojrulwkp
lv deoh wr dyrlg wkh fxuvh ri glphqvlrqdolw| idfhg e| wkh SHD froorfdwlrq dojrulwkp1
Iru lqvwdqfh/ lq d sureohp zlwk < vwdwh yduldeohv/ xvlqj Vpro|dn zlwk dq dffxudf|
ohyho ' 2 E ' zrxog lpso| vroylqj iru 4;4 +4/4::, xqnqrzqv1 Wklv lv ihdvleoh
dqg lw frqwudvwv zlwk wkh qxpehu ri xqnqrzqv wkdw d SHD froorfdwlrq dojrulwkp
zrxog qhhg wr hvwlpdwh lq d sureohp zlwk wkh vdph glphqvlrq1 Pruhryhu/ zh zloo
vkrz ehorz +dqg wklv kdv dovr ehhq vkrzq e| Nuxhjhu dqg Nxeohu +5337, dqg Pdolq/
Nuxhjhu dqg Nxeohu +533:,, wkdw xvlqj Vpro|dn zlwk dffxudf| ohyhov ' 2 ru ' lv hqrxjk wr uhdfk wkh suhflvlrq ohyhov lq wkh dssur{lpdwlrqv w|slfdoo| frqvlghuhg lq
wkh olwhudwxuh1
516
Txdqwlwdwlyh hydoxdwlrq
Wklv vhfwlrq hydoxdwhv/ lq wkh frqwh{w ri wkh rqh0frxqwu| vwrfkdvwlf qhrfodvvlfdo jurzwk
prgho/ wkh shuirupdqfh ri wkh Vpro|dn0SHD dojrulwkp suhvhqwhg lq vhfwlrq 515 dqg
frpsduhv lw zlwk wkdw ri wkh vwdqgdug SHD froorfdwlrq1 Wkh sxusrvh ri wklv frp0
sdulvrq lv wr vkrz wkdw dsso|lqj wkh Vpro|dn phwkrg wr wkh sdudphwhul}dwlrq ri
frqglwlrqdo h{shfwdwlrqv gholyhuv vlplodu ru hyhq ehwwhu uhvxowv lq whupv ri wkh txdolw|
ri dssur{lpdwlrqv wkdq wkh vwdqgdug SHD froorfdwlrq dojrulwkp1 Dqg wklv lv gh0
vslwh wkh idfw wkdw wkh iruphu uhtxluhv wkh frpsxwdwlrq ri d pxfk orzhu qxpehu ri
sro|qrpldo frh!flhqwv1
Wr ehjlq zlwk/ zh frqvlghu wkh iroorzlqj vwdqgdug sdudphwhul}dwlrq ri wkh prgho=
Wdeoh 41 Sdudphwhu ydoxhv
3166 31<8 418 3135 31<
Uhjduglqj yrodwlolw|/ zh iroorz Pdolq/ Nuxhjhu dqg Nxeohu +533:, dqg hydoxdwh
wkh shuirupdqfh ri wkh wzr dojrulwkpv lq wzr glhuhqw vfhqdulrv1 D orz yrodwlolw|
vfhqdulr/ zkhuh ' f=ff/ dqg d kljk yrodwlolw| rqh zkhuh ' f=f1
Wkh frpsdulvrq ehwzhhq dojrulwkpv lv hvwdeolvkhg lq whupv ri wzr frpprqo| xvhg
phdvxuhv ri dffxudf|= dq hfrqrplf phdvxuh lq olqh zlwk Mxgg +4<<5, dqg d vwdwlvwlfdo
phdvxuh lq olqh zlwk Ghq Kddq dqg Pdufhw +4<<7,1 Wkh hfrqrplf phdvxuh hydoxdwhv
krz odujh duh +lq hfrqrplf whupv, wkh ghyldwlrqv iurp wkh sureohp*v htxloleulxp
frqglwlrqv jhqhudwhg e| wkh dssur{lpdwlrq surfhgxuh1 Lq wklv sduwlfxodu h{dpsoh/
wkh ghyldwlrq lv mxvw=
3
3
xw ' fw Hw f3
wn E wn nwn n  BANCO DE ESPAÑA
16
DOCUMENTO DE TRABAJO N.º 0838
+:,
H{suhvvlqj wkdw ghyldwlrq lq frqvxpswlrq whupv/ rqh pd| frqvlghu wzr glhuhqw
phdvxuhv ri dffxudf| diwhu hydoxdwlqj +:, iru d sduwlfxodu vhw ri srlqwv=;
xw +;,
Hphdq ' *L}f H fw
xw H4@ ' *L}f 4@ +<,
fw
Wkh lqwhusuhwdwlrq ri wkhvh phdvxuhv lv vwudljkwiruzdug1 Iru lqvwdqfh/ Hphdq ' D
+H4@ ' D, phdqv wkdw/ iru wkh vhw ri srlqwv frqvlghuhg/ lqglylgxdov xvlqj wkh
dssur{lpdwlrq uxoh pdnh/ rq dyhudjh +dw prvw,/ dq huuru ri '4 zkhq frqvxplqj
'433/3331
Uhjduglqj wkh vwdwlvwlfdo phdvxuh/ zh frqvlghu wkh whvw sursrvhg e| Ghq Kddq
dqg Pdufhw +4<<7,/ zklfk edvlfdoo| fkhfnv zkhwkhu vlpxodwhg vhulhv jhqhudwhg e| d
jlyhq qxphulfdo vroxwlrq phwkrg vdwlvi| wkh udwlrqdo h{shfwdwlrqv dvvxpswlrq ru qrw1
Dffruglqj wr wklv dvvxpswlrq/ huuruv pdgh e| lqglylgxdov dw d jlyhq shulrg zkhq
iruhfdvwlqj wkh ixwxuh pxvw eh ruwkrjrqdo wr wkdw shulrg lqirupdwlrq vhw/ lw 1 Pruh
vshflfdoo|/ lw pxvw kdsshq wkdw=
H dhw k E{w o ' f
iru dq| n0glphqvlrqdo yhfwru {w ehorqjlqj wr lw dqg dq| ixqfwlrq k G Un $ Ut /
zkhuh hw ghqrwhv wkh iruhfdvwlqj huuruv dw shulrg w1 Lq rxu sduwlfxodu h{dpsoh/<
3
3
hw ' Hw f3
wn E wn nwn n  fw
 nw nw3 nw32 w w3 w32
kw '
Lq sudfwlfh/ wklv whvw lv w|slfdoo| lpsohphqwhg lq wkh iroorzlqj zd|=
+l, Vlpxodwh wkh prgho d odujh qxpehu ri wlphv dqg/ iru hdfk ri wkhp/ frpsxwh
W EW D3
W EW / zkhuh
W
[
hw kw
W
W
[ 2 '
hw kw kw
W
EW '
DW
;
Qrwh wkdw/ e| frqvwuxfwlrq/ ghyldwlrqv lq +:, duh h{dfwo| }hur dw wkh froorfdwlrq srlqwv/ erwk lq
wkh Vpro|dn0SHD dqg lq wkh SHD froorfdwlrq dojrulwkpv1 Wkxv/ lw rqo| pdnhv vhqvh wr frpsxwh
wklv dffxudf| phdvxuh dw srlqwv glhuhqw iurp wkh froorfdwlrq srlqwv xvhg lq wkh hvwlpdwlrq ri wkh
dssur{lpdwlqj ixqfwlrq1
<
Zh xvh wkh vdph ixqfwlrq k dv Ghq Kddq dqg Pdufhw +4<<7, exw/ lq sulqflsoh/ d glhuhqw rqh
frxog eh xvhg1
BANCO DE ESPAÑA
17
DOCUMENTO DE TRABAJO N.º 0838
dqg W lv wkh qxpehu ri shulrgv frqvlghuhg lq hdfk vlpxodwlrq1
+ll, Ghq Kddq dqg Pdufhw +4<<7, vkrz wkdw/ xqghu vrph frqglwlrqv/ wkh vwdwlvwlf
2
W EW D3
W EW lv glvwulexwhg dffruglqj wr d "t 1 Wkhq/ revhuyh wkh shufhqwdjh ri wkh
vlpxodwlrqv lq zklfk wkh vwdwlvwlf olhv lq wkh orzhu +S=fD , dqg xsshu +S=bD , fulwlfdo 8(
ri wkdw glvwulexwlrq1 Wkh forvhu wkrvh wzr shufhqwdjhv duh wr 8(/ wkh ehwwhu1
Wdeohv 5 dqg 6 vkrz wkh shuirupdqfh ri wkh vwdqgdug SHD froorfdwlrq dqg wkh
Vpro|dn0SHD dojrulwkpv iru wkh orz yrodwlolw| dqg kljk0yrodwlolw| vfhqdulrv/ uhvshf0
wlyho|/ lq whupv ri wkh dffxudf| phdvxuhv ghvfulehg deryh1 Zh vwrs erwk dojrulwkpv
zkhq/ lq wkh fxuvh ri wkh lwhudwlyh surfhgxuh wkh| xvh wr qg wkh xqnqrzq frh!flhqwv
ri wkhlu dssur{lpdwlqj ixqfwlrqv/ lw kdsshqv wkdw n xl xln n? h S= Iru wkh SHD
froorfdwlrq h{huflvh zh frqvlghu qn ' q ' f= Zh frpsxwh wkh hfrqrplf dffxudf|
phdvxuhv iru wzr vhw ri srlqwv1 Rq wkh rqh kdqg/ zh vlpxodwh wkh prgho iru 4533
shulrgv dqg frpsxwh +;, dqg +<, xvlqj wkh odvw 4333 shulrgv +Vlpxodwlrq,1 Zh vwduw
wkh vlpxodwlrq lq wkh qrq0vwrfkdvwlf vwhdg| vwdwh dqg gurs wkh uvw 533 shulrgv vr dv
wr jhw ulg ri wkh hhfw ri lqlwldo frqglwlrqv1 Wkh vhtxhqfh ri vkrfnv lv wkh vdph iru
erwk dojrulwkpv/ zkdw pdnhv wkh dffxudf| phdvxuhv frpsxwhg lq wklv h{huflvh ixoo|
frpsdudeoh1 Rq wkh rwkhu kdqg/ zh dovr irup d qhz julg ri wkh vwdwh yduldeohv dqg
frpsxwh +;, dqg +<, xvlqj wkrvh julg srlqwv +Julg,1 Ilqdoo|/ wr lpsohphqw wkh Ghq
Kddq0Pdufhw whvw/ zh uxq 4333 vlpxodwlrqv ri wkh prgho ri 4333 shulrgv hdfk1
Wdeoh 51 Dffxudf| hydoxdwlrq= Orz yrodwlolw| vfhqdulr
Hfrqrplf
Vlpxodwlrq
Hphdq H4@
SHD froorfdwlrq
0:195 0:145
Vpro|dn0SHD + ' 2, 0:138 0918:
Vpro|dn0SHD + ' , 0:196 0:145
phdvxuh
Julg
Hphdq H4@
09178 09136
0913; 08197
09178 09137
Vwdwlvwlfdo phdvxuh
Ghq Kddq0Pdufhw
S=fD
S=bD
619(
815(
713(
61;(
71:(
71<(
Wdeoh 61 Dffxudf| hydoxdwlrq= Kljk yrodwlolw| vfhqdulr
Hfrqrplf
Vlpxodwlrq
Hphdq H4@
SHD froorfdwlrq
09157 09154
Vpro|dn0SHD + ' 2, 07155 061::
Vpro|dn0SHD + ' , 0915< 08187
phdvxuh
Julg
Hphdq H4@
09157 09136
061<4 06165
08179 071:4
Vwdwlvwlfdo phdvxuh
Ghq Kddq0Pdufhw
S=fD
S=bD
716(
719(
61;(
919(
71<(
716(
Orrnlqj dw Wdeohv 5 dqg 6/ lw lv fohdu wkdw erwk wkh Vpro|dn0SHD dqg wkh SHD fro0
orfdwlrq dojrulwkpv uhdfk yhu| kljk ohyhov ri dffxudf|/ erwk lq whupv ri wkh hfrqrplf
dffxudf| phdvxuhv dqg lq whupv ri wkh Ghq Kddq0Pdufhw whvw1 Wkh idfw wkdw/ ryhudoo/
wkh Vpro|dn0SHD dojrulwkp h{klelwv d vlplodu ghjuhh ri dffxudf| dv wkh vwdqgdug
BANCO DE ESPAÑA
18
DOCUMENTO DE TRABAJO N.º 0838
froorfdwlrq dssurdfk lv uhpdundeoh dqg lw lv d yhu| surplvlqj uhvxow sulru wr dsso|lqj
wklv dojrulwkp wr sureohpv zlwk d odujh qxpehu ri vwdwh yduldeohv/ zkhuh phwkrgv olnh
SHD froorfdwlrq idfh wkh fxuvh ri glphqvlrqdolw| dqg fdq qrw eh xvhg143 Qrwh dovr
wkdw wklv uhvxow krogv ghvslwh wkh idfw wkdw wkh Vpro|dn0SHD dojrulwkp hpsor|v ihzhu
frh!flhqwv wkdq wkh SHD froorfdwlrq rqh1 Qdpho|/ lq wkh odwwhu zh duh hvwlpdwlqj
433 frh!flhqwv zkloh lq wkh iruphu rqo| 46 +5<, duh hvwlpdwhg zlwk ' 2 + ' ,1
Reylrxvo|/ zlwklq wkh Vpro|dn0SHD dojrulwkp/ wkh kljkhu wkh ehwwhu wkh ds0
sur{lpdwlrq1 Pruhryhu/ dv lq Pdolq/ Nuxhjhu dqg Nxeohu +533:,/ lw lv uhdvrqdeoh wr
h{shfw wkdw dffxudf| lq wkh kljk yrodwlolw| fdvh lv orzhu wkdq lq wkh orz yrodwlolw| rqh1
Ilqdoo|/ wkh hfrqrplf dffxudf| phdvxuh lv kljkhu zkhq frpsxwhg dw wkh vlpxodwhg
srlqwv udwkhu wkdq dw wkh julg srlqwv1 Wkh uhdvrq lv wkdw wkh julg srlqwv vhohfwhg iru
grlqj wkhvh dffxudf| whvwv duh hyhqo| vsdfhg lq wkh vwdwh vsdfh dqg/ frqvhtxhqwo|/
wkh| lqfoxgh vrph frpelqdwlrqv ri wkh vwdwh yduldeohv wkdw duh h{wuhph dqg zkhuh
qrq0olqhdulwlhv duh srwhqwldoo| lpsruwdqw1 Lq frqwudvw/ wkh vlpxodwhg srlqwv duh pruh
fhqwhuhg durxqg wkh uhdol}dwlrqv ri wkh vkrfnv wkdw duh pruh olnho| dqg zkhuh wkh
prgho ehkdyhv pruh vprrwko|1
6
Wzr0frxqwu| prgho zlwk sruwirolr ghflvlrqv dqg
lqfrpsohwh pdunhwv
Wklv vhfwlrq vkrzv krz wr dsso| wkh Vpro|dn0SHD dojrulwkp suhvhqwhg lq vhfwlrq 515
wr d sureohp zlwk sruwirolr ghflvlrqv dqg lqfrpsohwh pdunhwv1 Qdpho|/ frqvlghu d
rqh0vhfwru vwrfkdvwlf qhrfodvvlfdo jurzwk prgho zlwk wzr frxqwulhv1 D uhsuhvhqwdwlyh
djhqw lq hdfk frxqwu| vroyhv wkh iroorzlqj lqwhuwhpsrudo pd{lpl}dwlrq sureohp=
Pd{ Hf
"
[
w'f
vxemhfw wr
w
f3
l>w

2
2
2
[
[
[ 2
fl>w n
dl>m>wn n
d
'
E n um>w dl>m>w n zl>w Ol>w ;w
2 m' l>m>wn m'
m'
+43,
zkhuh fl>w ghqrwhv frqvxpswlrq lq frxqwu| l dw shulrg w/ l ' i> 2j/ dl>m>w lv wkh
fdslwdo ri frxqwu| m lq kdqgv ri frxqwu| l*v uhsuhvhqwdwlyh djhqw dw shulrg w/ um>w lv
wkh qhw udwh ri uhwxuq ri frxqwu| m*v fdslwdo dw shulrg w/ dqg z ghqrwhv zdjhv1 Hdfk
frxqwu|*v uhsuhvhqwdwlyh djhqw lv hqgrzhg rqh xqlw ri wlph lq hdfk shulrg/ zklfk
43
Wkh sxusrvh ri wklv h{huflvh lv qrw wr uhdfk wkh kljkhvw dffxudf| ohyho zlwk hdfk dojrulwkp/ exw
wr vhh li wkhvh phwkrgv* dffxudf| lv lq wkh vdph rughu ri pdjqlwxgh1 Lq wklv vhqvh/ lw lv fohdu wkdw
rqh frxog lqfuhdvh wkh dffxudf| ri wkh SHD froorfdwlrq dojrulwkp e| frqvlghulqj odujhu ydoxhv ri qn
dqg q / dqg ri wkh Vpro|dn0SHD e| lqfuhdvlqj 1
BANCO DE ESPAÑA
19
DOCUMENTO DE TRABAJO N.º 0838
kh lqhodvwlfdoo| vxssolhv wr klv grphvwlf oderu pdunhw +l1h1/ oderu lv lppreloh dfurvv
frxqwulhv,144 Wkh glvfrxqw sdudphwhu/ / lv dvvxphg wr eh wkh vdph iru doo djhqwv1
Lq wklv vhw xs/ wkh sruwirolr sureohp idfhg e| lqglylgxdov lq erwk frxqwulhv lv fohdu1
Lq rughu wr vprrwk wkhlu frqvxpswlrq ryhu wlph/ wr wudqvihu zhdowk dfurvv shulrgv/
lqglylgxdov fdq xvh glhuhqw pl{wxuhv ri wzr dvvhwv/ fdslwdo lq frxqwu| 4 dqg fdslwdo lq
frxqwu| 5/ ehlqj wkh uhwxuq ri erwk lqyhvwphqwv xqfhuwdlq145 Qrwh wkdw wkhvh sruwirolr
ghflvlrqv duh dhfwhg e| wkh suhvhqfh ri txdgudwlf sruwirolr frvwv rq hdfk w|sh ri
dvvhw kroglqjv/ wkh wklug whup rq wkh ohiw kdqg vlgh ri htxdwlrq +43,1 Wkh uhdvrq
wr lqfoxgh wkhvh frvwv lv wkh iroorzlqj1 Lw lv zhoo0nqrzq wkdw rshq hfrqrp| prghov
zlwk lqfrpsohwh pdunhwv idfh vwdwlrqdulw| sureohpv1 Xvlqj vpdoo txdgudwlf sruwirolr
frvwv lv rqh ri wkh urxwhv sxuvxhg lq wkh olwhudwxuh wr jhw ulg ri wkrvh sureohpv146
Lq sduwlfxodu/ zlwk yhu| vpdoo sruwirolr frvwv/ jryhuqhg e| wkh sdudphwhu / rqh fdq
lqgxfh vwdwlrqdulw| exw vwloo jhw sruwirolr ghflvlrqv forvh wr wkrvh uhvxowlqj li wkhvh
frvwv zhuh frpsohwho| devhqw1
Wkh wzr uvw rughu frqglwlrqv ri frxqwu| l*v uhsuhvhqwdwlyh djhqw sureohp duh=
Hw f3
E
n
u
>wn
l>wn
 n dl>>wn
'
Hw f3
E
n
u
2>wn
l>wn
 n dl>2>wn
f3
'
l>w
f3
l>w
D uhsuhvhqwdwlyh up lq hdfk frxqwu| kluhv oderu dqg fdslwdo lq hdfk shulrg lq d
frpshwlwlyh idvklrq1 Qdpho|/ wkh uhsuhvhqwdwlyh up lq frxqwu| l vroyhv wkh iroorzlqj
sureohp dw shulrg w=
4@
3
l>w Nl>w
Ol>w Eul>w n Nl>w zl>w Ol>w
zkhuh hdfk up WIS +l1h1/ hdfk frxqwu| WIS, hyroyhv dffruglqj wr wkh iroorzlqj
yhfwru dxwruhjuhvvlyh surfhvv=
*L} >w
*L} >w3
%>w
 f
'
n
+44,
f 2
*L} 2>w
*L} 2>w3
%2>w
ru lq frpsdfw irup *L}Ew ' D *L}Ew3 n %w = Wkh lqqrydwlrqv %w duh vhuldoo|
lqghshqghqw dqg wkh| duh glvwulexwhg dffruglqj wr d elyduldwh qrupdo glvwulexwlrq
44
Wkh Vpro|dn0SHD dojrulwkp suhvhqwhg lq wklv sdshu pd| dovr vroyh prghov zlwk hqgrjhqrxv
oderu vxsso|1 Zh frqvlghu dq lqhodvwlf oderu vxssolhg khuh mxvw iru vlpsolflw|1
45
Dgglqj d ulvn0iuhh dvvhw wr wklv sureohp zrxog qrw lpso| dq| vljqlfdqw fkdqjh lq wkh vroxwlrq
phwkrg ghvfulehg khuh1
46
Wklv lv wkh dssurdfk iroorzhg/ iru lqvwdqfh/ e| Khdwkfrwh dqg Shuul +5335,1 Iru d glvfxvvlrq
rq glhuhqw dowhuqdwlyh zd|v ri lqgxflqj vwdwlrqdulw| lq wklv w|sh ri prghov vhh Vfkplww0Jurkh dqg
Xuleh +5336,1
BANCO DE ESPAÑA
20
DOCUMENTO DE TRABAJO N.º 0838
zlwk phdq }hur dqg yduldqfh0fryduldqfh pdwul{ P1 E| pdnlqj wkh r0gldjrqdo whupv
ri pdwul{ D htxdo wr }hur zh duh dvvxplqj/ dv prvw ri wkh olwhudwxuh/ wkdw wkhuh duh
qr vslooryhuv lq wkh wudqvplvvlrq ri vkrfnv +dowkrxjk vkrfnv pd| eh fruuhodwhg/ dv
ghwhuplqhg e| pdwul{ P,1 Krzhyhu/ hyhu|wklqj wkdw iroorzv zrxog shuihfwo| dsso| wr
dq| duelwudu| pdwul{ D1
Wkh uvw rughu frqglwlrqv ri upv* sureohpv ghqh wkh hfrqrp| sulfhv1 Qdpho|/
iru l ' i> 2j dqg ;w
ul>w
Nl>w
' l>w Ol>w
zl>w ' l>w E 3
Nl>w
Ol>w
Wkh prgho lv forvhg zlwk wkh pdunhw fohdulqj frqglwlrqv lq wkh oderu dqg fdslwdo
pdunhwv=
Ol>w '  ;l> w
2
[
Nl>w '
dm>l>w ;l> w
m'
614
Ixoo lqirupdwlrq yv1 sduwldo lqirupdwlrq
Wkh vwdwh yduldeohv ri wkh sureohp suhvhqwhg deryh duh wkh WIS ohyho lq hdfk frxqwu|
dqg wkh zkroh glvwulexwlrq ri dvvhwv lq wkh hfrqrp|1 Pruh vshflfdoo|/ wkh yhfwru ri
vwdwh yduldeohv lv=47
dd> d>2 d2> d2>2  2 o
+45,
Zh duh jrlqj wr frqvlghu wzr glhuhqw fdvhv= d ixoo lqirupdwlrq dqg d sduwldo
lqirupdwlrq fdvh1 Lq wkh ixoo lqirupdwlrq fdvh/ zh frqvlghu wkdw lqglylgxdov/ zkhq
vroylqj wkhlu frqvxpswlrq0sruwirolr sureohpv/ nqrz wkh zkroh vwdwh ri wkh hfrqrp|
jlyhq e| yhfwru +45,1 Lqvwhdg/ lq wkh sduwldo lqirupdwlrq fdvh/ zh frqvlghu wkdw
lqglylgxdov gr qrw nqrz wkh zkroh glvwulexwlrq ri dvvhwv lq wkh hfrqrp|/ dqg wkh|
rqo| nqrz wkhlu rzq dvvhw kroglqjv dqg wkh djjuhjdwh dprxqw ri fdslwdo lq hdfk
frxqwu|1 Pruh vshflfdoo|/ wkh uhsuhvhqwdwlyh djhqw lq frxqwu| l/ l ' i> 2j/ kdv wkh
iroorzlqj lqirupdwlrq=
ddl> dl>2 N N2  2 o
47
BANCO DE ESPAÑA
Zh gurs wlph vxevfulswv iru vlpsolflw|1
21
DOCUMENTO DE TRABAJO N.º 0838
+46,
Lq wklv wzr0frxqwu| vhw xs/ wkh ixoo dqg wkh sduwldo lqirupdwlrq dssurdfkhv kdyh
wkh vdph glphqvlrq/ 9148 Krzhyhu/ wr vhh wkh xvhixoqhvv ri wkh sduwldo lqirupdwlrq
dssurdfk/ qrwh wkdw lq d 90frxqwu| vhw xs/ iru lqvwdqfh/ wkh glphqvlrq ri wkh sureohp
lq wkh ixoo lqirupdwlrq fdvh zrxog eh 75/ zkloh lq wkh sduwldo lqirupdwlrq fdvh lw zrxog
rqo| eh 4;1 Wkxv/ hyhq li wkh Vpro|dn dojrulwkp lv deoh wr kdqgoh kljk0glphqvlrqdo
sureohpv/ li wkh glphqvlrq ri wkh sureohp lv wrr kljk/ lw pd| eh xvhixo wr frpelqh
wkh Vpro|dn dojrulwkp dqg wkh sduwldo lqirupdwlrq dssurdfk1 Zh dujxh wkdw wklv
lv vshfldoo| uhohydqw iru wkh vwxg| ri lqwhuqdwlrqdo sruwirolr sureohpv/ zklfk ehfrph
yhu| kljk0glphqvlrqdo sureohpv yhu| hdvlo|1
61414
Ixoo lqirupdwlrq1 Frpsxwdwlrq vwudwhj|
Wkh ixoo lqirupdwlrq sureohp fdq eh vroyhg zlwk wkh Vpro|dn0SHD dojrulwkp edvl0
fdoo| iroorzlqj wkh vwhsv ghvfulehg lq vhfwlrq 5 iru wkh fdvh ri d rqh0frxqwu| qhrfodvvl0
fdo jurzwk prgho1 Rqh mxvw qhhg wr uhdol}h wkdw/ lq wklv fdvh/ wkh prgho*v htxloleulxp
frqglwlrqv frqwdlq irxu frqglwlrqdo h{shfwdwlrqv wkdw qhhg wr eh sdudphwhul}hg1 Wkhvh
duh wkh vwhsv wr iroorz=
F410 Irup d julg ri wkh vwdwh yhfwru +45, dffruglqj wr wkh surfhgxuh E41
F510 Frqvwuxfw dq dssur{lpdwlqj ixqfwlrq iru hdfk ri wkh 7 frqglwlrqdo h{shfwd0
wlrqv ri wkh prgho dffruglqj wr E51 Qdpho|/
%
[ Ed> > d>2 > N > N2 >  > 2 ( x * Hw f3
>wn E >wn %
[2 Ed> > d>2 > N > N2 >  > 2 ( x2 * Hw f3
>wn E 2>wn %
[2 Ed2> > d2>2 > N > N2 >  > 2 ( x2 * Hw f3
2>wn E >wn [22 Ed2> > d2>2 > N > N2 >  > 2 ( x22 * Hw
%
f3
2>wn E 2>wn N>wn
O>wn
N2>wn
O2>wn
N>wn
O>wn
N2>wn
O2>wn
3
3
3
3
n  n  n  n  &
&
&
&
48
Wkh idfw wkdw lq wklv sduwldo lqirupdwlrq vhw xs frxqwu| l*v uhsuhvhqwdwlyh djhqw frxog/ lq sulq0
flsoh/ nqrz wkh dvvhw kroglqjv ri frxqwu| m*v uhsuhvhqwdwlyh djhqw vlpso| xvlqj wkh pdunhw fohdulqj
5
S
dm>l>w / frxog eh plvohdglqj1 Lq sduwlfxodu/ lw frxog lqgxfh
frqglwlrq lq fdslwdo pdunhwv/ Nl>w @
m@4
vrphrqh wr wklqn wkdw wkh lqirupdwlrq lq kdqgv ri lqglylgxdov lv wkh vdph lq wkh ixoo dqg lq wkh
sduwldo lqirupdwlrq vhw xsv1 Wklv lv qrw fruuhfw1 Dv lw zloo eh fohduhu ehorz/ lq wkh sduwldo lqirupdwlrq
dssurdfk rqh frxqwu|*v uhsuhvhqwdwlyh djhqw ehkdyhv dv li kh frxog qrw uhdoo| nqrz wkh rwkhu djhqw*v
fdslwdo kroglqjv +lq idfw/ lq d pxowl0frxqwu| vhwwlqj zlwk pruh wkdq 5 frxqwulhv/ kh zrxogq*w eh deoh
wr nqrz wkdw hyhq diwhu xvlqj wkh pdunhw fohdulqj frqglwlrqv,1
BANCO DE ESPAÑA
22
DOCUMENTO DE TRABAJO N.º 0838
F610 Iru hdfk frqglwlrqdo h{shfwdwlrq/ iroorz wkh lwhudwlyh surfhgxuh D6l0D6y1
Mxvw qrwh wkdw rqh qhhgv wr vroyh vlpxowdqhrxvo| iru fl>w / dl>>wn dqg dl>2>wn / l ' i> 2j1
2
S
Lq wkh frpsxwdwlrqv/ Nl>wn ' dm>l>wn ;l1
m'
61415
Sduwldo lqirupdwlrq1 Frpsxwdwlrq vwudwhj|
Wkh sduwldo lqirupdwlrq sureohp fdq eh vroyhg lq wkh iroorzlqj idvklrq=
G410 Dffruglqj wr wkh surfhgxuh E4/ irup wzr julgv/ rqh iru wkh vwdwh yhfwru ri
frxqwu| 4*v uhsuhvhqwdwlyh djhqw +l '  lq +46,, dqg dqrwkhu rqh iru wkh vwdwh yhfwru
ri frxqwu| 5*v uhsuhvhqwdwlyh djhqw +l ' 2 lq +46,,1
G510 Frqvwuxfw dq dssur{lpdwlqj ixqfwlrq iru hdfk ri wkh 7 frqglwlrqdo h{shfwd0
wlrqv ri wkh prgho dv lq vwhs F51
G610 Frqvlghu wkdw lqglylgxdov pdnh iruhfdvwv derxw qh{w shulrg ydoxhv ri frxqwu|
4 dqg frxqwu| 5 djjuhjdwh fdslwdo/ ghqrwhg e| N dqg N2 / frpelqlqj fxuuhqw dqg
sdvw lqirupdwlrq lq d sduwlfxodu idvklrq1 Qdpho|/ zh dvvxph wkdw doo djhqwv xvh wkh
iroorzlqj iruhfdvwlqj uxohv=
*L} N ' f n  *L} N n 2 *L} N2 n  *L}  n e *L} 2
+47,
*L} N2 ' 2f n 2 *L} N n 22 *L} N2 n 2 *L}  n 2e *L} 2
+48,
ru lq frpsdfw irup/ Nl ' xEN > N2 >  > 2 ( l > l ' i> 2j > zkhuh wkh *v duh sdud0
phwhuv wkdw qhhg wr eh irxqg rswlpdoo| zlwklq wkh qxphulfdo surfhgxuh1 Fhuwdlqo|/
wkhuh duh pdq| dowhuqdwlyh iruhfdvwlqj uxohv wkdw frxog eh xvhg1 Wkh| frxog lqfoxgh/
iru lqvwdqfh/ pruh odjv ri wkh djjuhjdwh vwdwh yduldeohv1 Krzhyhu/ dv zh zloo vkrz
ehorz/ wkh iruhfdvwlqj uxohv +47, dqg +48,/ wkdw duh vlplodu wr rwkhuv xvhg lq wkh olw0
hudwxuh wkdw frqvlghuv sduwldo lqirupdwlrq vhw xsv/ zrun uhdoo| zhoo lq rxu sureohp lq
whupv ri dffxudf|1
Iruhfdvwlqj uxohv duh qhhghg iru dq reylrxv uhdvrq1 Iru h{dfwo| wkh vdph uhdvrq
lqglylgxdov qhhg wr nqrz wkh odz ri prwlrq ri hdfk frxqwu| WIS/ htxdwlrq +44,/ lq
rughu wr wdnh wkhlu ghflvlrqv/ wkh| dovr qhhg wr nqrz krz wkh rwkhu djjuhjdwh vwdwh
yduldeohv lq wkhlu lqirupdwlrq vhwv/ N dqg N2 / duh h{shfwhg wr ehkdyh lq wkh ixwxuh1
Wkh iruhfdvwlqj uxohv surylgh wkdw lqirupdwlrq1
G710 Pdnh dq lqlwldo dvvxpswlrq/ f dqg f2 / derxw wkh ydoxhv ri wkh *v sdudphwhuv
lq wkh iruhfdvwlqj uxohv dqg lqlwldwh wkh lwhudwlyh surfhgxuh D6l0D6y1 Wkh glhuhqfh
zlwk uhvshfw wr wkh vwudwhj| ghvfulehg lq vhfwlrq 61414 iru wkh ixoo lqirupdwlrq fdvh
lv wkdw/ lq wklv fdvh/ wkh Nl>wn *v wkdw hqwhu lqwr wkh frqglwlrqdo h{shfwdwlrqv duh
BANCO DE ESPAÑA
23
DOCUMENTO DE TRABAJO N.º 0838
qrw frqvwuxfwhg dv Nl>wn '
2
S
dm>l>wn ;l/ exw dv Nl>wn ' xEN>w > N2>w > >w > 2>w ( fl m'
;l1 Lq wklv vhqvh/ qrwh wkdw lqglylgxdov gr qrw kdyh hqrxjk lqirupdwlrq wr lqihu rwkhu
lqglylgxdov* qh{w shulrg dvvhw kroglqjv dqg/ wkhuhiruh/ fdq qrw xvh wkh iruphu irupxod1
G810 Dw wkh hqg ri hdfk lwhudwlrq ri wkh surfhgxuh D6l0D6y xsgdwh/ qrw rqo|
wkh frh!flhqwv lq wkh dssur{lpdwlqj ixqfwlrq/ x/ exw dovr wkh frh!flhqwv lq wkh
iruhfdvwlqj uxohv/ 1 Wr xsgdwh wkh *v/ dw wkh hqg ri hdfk lwhudwlrq/ vlpxodwh wkh prgho
iru d jlyhq vhtxhqfh ri whfkqrorj| vkrfnv dssur{lpdwlqj wkh frqglwlrqdo h{shfwdwlrqv
zlwk wkh odvw x frpsxwhg1 Wkhq/ zlwk wkh vlpxodwhg gdwd pdnh wkh uhjuhvvlrqv
lpsolflw lq +47, dqg +48, dqg rewdlq qhz *v1
G910 Vwrs wkh lwhudwlyh surfhgxuh zkhq wkhuh lv frqyhujhqfh erwk lq x dqg lq 1
61416
Txdqwlwdwlyh hydoxdwlrq
Wklv vhfwlrq hydoxdwhv krz dffxudwho| wkh Vpro|dn0SHD dojrulwkp lv deoh wr vroyh wkh
ixoo lqirupdwlrq dqg wkh sduwldo lqirupdwlrq vfhqdulrv ghvfulehg deryh1 Iru vlpsolflw|/
zh dvvxph wkdw wkh wzr frxqwulhv duh lghqwlfdo dqg xvh wkh vdph sdudphwhu ydoxhv dv lq
wkh rqh0frxqwu| fdvh +Wdeoh 4,1 Lq dgglwlrq/ zh vhw ' f=fff/ zklfk pdnhv txdgudwlf
sruwirolr frvwv vr vpdoo wkdw wkh| gr qrw lqwhuihuh vljqlfdqwo| lq wkh lqglylgxdov*
sruwirolr ghflvlrqv1 Dv lq vhfwlrq 516/ wkh
hydoxdwlrq
lv shuiruphg erwk
txdqwlwdwlyh
f=ff2
f
lq d orz yrodwlolw| vfhqdulr/ zlwk P '
/ dqg lq d kljk yrodwlolw|
f
f=ff2
f=f2
f
rqh zlwk P '
=
f
f=f2
Wkh xqnqrzq frh!flhqwv ri wkh dssur{lpdwlqj ixqfwlrqv dqg/ lq wkh fdvh ri wkh
sduwldo lqirupdwlrq vfhqdulr/ ri wkh iruhfdvwlqj uxohv duh ghwhuplqhg iroorzlqj wkh
vwudwhjlhv ghvfulehg lq vhfwlrqv 61414 dqg 614151 Zlwk wkhp/ zh uxq 433 vlpxodwlrqv ri
wkh prgho/ ri 4533 shulrgv hdfk/ erwk lq wkh orz yrodwlolw| dqg lq wkh kljk0yrodwlolw|
vfhqdulrv1 Lq hdfk vlpxodwlrq/ +;, dqg +<, duh frpsxwhg iru wkh odvw 4333 shulrgv +zh
gurs wkh uvw 533 shulrgv wr holplqdwh wkh hhfw ri lqlwldo frqglwlrqv, dqg hdfk ri wkh
7 uvw rughu frqglwlrqv ri wkh sureohp1 Iljxuhv 407 vkrz wkhvh huuruv iru wkh ixoo dqg
wkh sduwldo lqirupdwlrq fdvhv/ lq doo vlpxodwlrqv dqg lq wkh wzr yrodwlolw| vfhqdulrv1
Wdeoh 7 vxppdul}hv wkh lqirupdwlrq lq wkh jxuhv e| wdnlqj wkh dyhudjh dqg wkh pd{
dfurvv vlpxodwlrqv1
BANCO DE ESPAÑA
24
DOCUMENTO DE TRABAJO N.º 0838
Figure 1. E mean - Low volatility
Full Inf.
Figure 2. E max - Low volatility
Full Inf.
Partial Inf.
-6.2
-5.1
Emax
Emean
-6.35
-6.5
-5.5
-5.9
-6.3
-6.65
0
50
Simulation
0
100
Figure 3. E mean - High volatility
Full Inf.
Partial Inf.
Full Inf.
-4.1
-3
-4.6
-3.6
-5.1
50
Simulation
100
Figure 4. E max - High volatility
Emax
Emean
Partial Inf.
Partial Inf.
-4.2
-4.8
-5.6
0
50
Simulation
100
0
50
Simulation
100
Dv rqh frxog h{shfw/ wkh dssur{lpdwlrq lv pruh dffxudwh lq wkh ixoo lqirupdwlrq
fdvh wkdq lq wkh sduwldo lqirupdwlrq fdvh/ dqg lq wkh orz yrodwlolw| fdvh wkdq lq wkh
kljk yrodwlolw| rqh1 Pruh lpsruwdqwo|/ +l, wkh dojrulwkp*v dffxudf| lq doo vfhqdulrv
phhwv xvxdo vwdqgdugv lq wkh olwhudwxuh dqg +ll, lw vhhpv wkdw wkh orvv lq dffxudf|
dvvrfldwhg wr wkh xvh ri wkh sduwldo lqirupdwlrq dssurdfk lv qhjoljleoh1 Wkh odwwhu
uhvxow lv hvshfldoo| lqwhuhvwlqj1 Uhvhdufkhuv kdyh h{whqvlyho| xvhg wkh Nuxvhoo0Vplwk
sduwldo lqirupdwlrq dssurdfk zkhq vroylqj wkh ixoo lqirupdwlrq yhuvlrq ri wkhlu prghov
zdv gl!fxow1 Wkhq/ wkh| xvxdoo| uhsruw wkh U2 rewdlqhg lq wkh hvwlpdwlrq ri wkh
iruhfdvwlqj uxohv/ exw gr qrw frpsduh wkhlu dffxudf| uhvxowv zlwk wkrvh ri wkh ixoo
lqirupdwlrq vfhqdulr1 Zh gr wklv khuh dqg qg wkdw wkh dffxudf| orvv ri xvlqj d
sduwldo lqirupdwlrq dssurdfk lv yhu| vpdoo1 Rqh pxvw qrwh/ krzhyhu/ wkdw wklv lv
reylrxvo| d prgho0vshflf qglqj dqg lw frxog qrw krog lq pruh frpsoh{ prghov1 Iru
frpsohwhqhvv/ Wdeoh 8 uhsruwv wkh hvwlpdwhg iruhfdvwlqj uxohv dqg wkhlu U2 =
BANCO DE ESPAÑA
25
DOCUMENTO DE TRABAJO N.º 0838
Wdeoh 71 Ixoo lqirupdwlrq yv1 sduwldo lqirupdwlrq= Dffxudf|
Orz yrodwlolw|
Kljk yrodwlolw|
phdqEHphdq 4@ EH4@ phdqEHphdq 4@ EH4@ Ixoo lqirupdwlrq
0917;43
081<389
071<976
061458<
Sduwldo lqirupdwlrq
09167:8
0814<97
071643;
0613;74
Wdeoh 81 Sduwldo lqirupdwlrq= Hvwlpdwhg iruhfdvwlqj uxohv
Lqghs= ydu= q
Ghs= ydu
Frqvwdqw
*L} N
*L} N2
*L} 
*L} 2
U2
615
Orz yrodwlolw|
Kljk yrodwlolw|
*L} N
*L} N2
*L} N
*L} N2
314575
314574
314593
314598
317:56< 317:56< 317:44; 317:436
317:56: 317:56; 317:5:< 317:5:4
31:6948 0318<68; 31:6:4: 0318<56<
0318<68< 31:6947 0318<766 31:6866
31<<<<<8 31<<<<<8 31<<<7<: 31<<<997
Sruwirolr frqvwudlqwv
Erwk lq qdqfldo pdunhwv duh lq wkhruhwlfdo prghov/ sruwirolr ghflvlrqv duh riwhq frq0
glwlrqhg e| rffdvlrqdoo| elqglqj frqvwudlqwv1 Wklv vhfwlrq vkrzv krz hdvlo| wkh
Vpro|dn0SHD dojrulwkp fdq dffrpprgdwh wklv w|sh ri frqvwudlqwv1 Lq idfw/ dv
srlqwhg rxw e| Fkulvwldqr dqg Ilvkhu +5333,/ wkh sdudphwhul}hg h{shfwdwlrq dssurdfk
kdv d uhodwlyh dgydqwdjh ryhu rwkhu dssurdfkhv irfxvlqj rq srolf| ixqfwlrqv suhflvho|
lq wkh suhvhqfh ri wkhvh frqvwudlqwv1 Wkxv/ zkloh wkh iruphu phwkrg rqo| qhhgv wr
xqghuwdnh voljkw prglfdwlrqv lq wkh dojrulwkp wr ghdo zlwk wkhvh frqvwudlqwv +dv zh
zloo vkrz ehorz,/ wkh odwwhu rqhv qhhg wr vlpxowdqhrxvo| sdudphwhul}h wkh srolf| dqg
wkh pxowlsolhu ixqfwlrqv/ zkdw vxevwdqwldoo| frpsolfdwhv wkh vhdufk surfhvv1 Wkhuh0
iruh/ zh vhh wklv sdshu/ wkdw dssolhv wkh Vpro|dn dojrulwkp wr wkh sdudphwhul}dwlrq
ri frqglwlrqdo h{shfwdwlrqv/ dv frpsohphqwdu| wr wkh zrun lq Nuxhjhu dqg Nxeohu
+5337, dqg lq Pdolq/ Nuxhjhu dqg Nxeohu +533:,/ zklfk dsso| wkh Vpro|dn dojrulwkp
wr wkh sdudphwhul}dwlrq ri srolf| ixqfwlrqv/ ehlqj sduwlfxoduo| xvhixo lq wkh suhvhqfh
ri rffdvlrqdoo| elqglqj frqvwudlqwv1
Wr looxvwudwh krz wkh Vpro|dn0SHD dojrulwkp zrunv lq wkh suhvhqfh ri rffd0
vlrqdoo| elqglqj frqvwudlqwv/ zh frqvlghu wkh wzr0frxqwu| prgho ghvfulehg deryh dqg
lpsrvh wkh iroorzlqj uhvwulfwlrq rq wkh sruwirolr ghflvlrqv ri frxqwu| 4*v uhsuhvhqwdwlyh
djhqw=
d>2>wn 9 dd>>wn n d>2>wn o ;w
BANCO DE ESPAÑA
26
DOCUMENTO DE TRABAJO N.º 0838
+49,
Lq zrugv/ wklv djhqw fdq qrw kdyh pruh wkdq d iudfwlrq 5 df> o ri klv zhdowk
lqyhvwhg deurdg149 Li ' f/ iruhljq lqyhvwphqw lv qrw doorzhg/ zkloh wkh frqvwudlqw
+49, qhyhu elqgv li ' 1 Wkh htxloleulxp frqglwlrqv dw shulrg w ri wklv djhqw*v
pd{lpl}dwlrq sureohp duh qrz=4:
3
f3
>w E n d>>wn ' Hw f>wn E n u>wn n w

3
3
f>w E n d>2>wn ' Hw f>wn E n u2>wn w
w
d>>wn d>2>wn ' f

2
2
2
[
[
[ 2
f>w n
d>m>wn n
d>m>wn '
E n um>w d>m>w n z>w
2
m'
m'
m'
zkhuh lv wkh Odjudqjh pxowlsolhu dvvrfldwhg wr +49,1
Xvlqj wkh Vpro|dn0SHD dojrulwkp wr vroyh wklv qhz sureohp lv yhu| hdv|1 Wkhuh
lv rqo| rqh glhuhqfh zlwk uhvshfw wr wkh surfhgxuh ghvfulehg lq vhfwlrqv 61414 dqg
614151 Lq wkh fxuvh ri wkh lwhudwlyh surfhgxuh D6l0D6y/ zkhq vroylqj vlpxowdqhrxvo|
iru fl>w / dl>>wn dqg dl>2>wn / l ' i> 2j/ dvvxph uvw wkdw wkh sruwirolr frqvwudlqwv duh
qrw elqglqj +lq wklv fdvh/ w ' f, dqg vroyh wkh sureohp1 Wkhq/ fkhfn li rswlpdo
ghflvlrqv irxqg xqghu wkdw dvvxpswlrq dfwxdoo| vdwlvi| wkh frqvwudlqwv1 Li wkh| gr/
rqh fdq surfhhg1 Li wkh| grq*w/ vroyh wkh sureohp djdlq exw/ dw wklv wlph/ orrnlqj
dovr iru d juhdwhu wkdq }hur Odjudqjh pxowlsolhu1 Wklv srvlwlyh pxowlsolhu pxvw dovr
eh wdnhq lqwr dffrxqw zkhq fkhfnlqj wkh dffxudf| ri dssur{lpdwlrqv1
61514
Srolf| ixqfwlrqv
Wklv vhfwlrq looxvwudwhv wkh hhfwv ri +49, rq wkh srolf| ixqfwlrqv ri erwk frxqwulhv*
uhsuhvhqwdwlyh djhqwv1 Lq sduwlfxodu/ zh vkrz krz erwk lqglylgxdov* rswlpdo dvvhw
kroglqjv/ dv d ixqfwlrq ri frxqwu| 5*v fxuuhqw whfkqrorj| ohyho/ fkdqjh ghshqglqj ri
wkh vhyhulw| ri frxqwu| 4*v sruwirolr frqvwudlqw1 Zh frqvlghu wkuhh ydoxhv ri 1 Wkh
uvw/ ' f=S/ edvlfdoo| lpsolhv wkdw +49, qhyhu elqgv1 Lq wkh vhfrqg fdvh/ ' f=D/
wkh frqvwudlqw lv rffdvlrqdoo| elqglqj1 Ilqdoo|/ zlwk ' f=e/ wkh frqvwudlqw lv elqglqj
doprvw dozd|v1 Wkh h{huflvhv duh shuiruphg lq d ixoo lqirupdwlrq vfhqdulr dqg xvh
wkh vdph sdudphwhu ydoxhv frqvlghuhg lq vhfwlrq 614161 Ixuwkhupruh/ frxqwu| 4*v WIS
ohyho lv dozd|v dvvxphg wr eh dw lwv phdq ydoxh1 Iljxuh 8 vkrzv wkh uhvxowv1
49
Vlplodu uhvwulfwlrqv rq iruhljq lqyhvwphqw duh riwhq hqfrxqwhuhg/ iru lqvwdqfh/ e| pdqdjhuv ri
shqvlrq ixqgv lq pdq| frxqwulhv1
4:
Wkh sureohp ri frxqwu| 5*v uhsuhvhqwdwlyh djhqw lv lghqwlfdo wr wkh rqh suhvhqwhg lq wkh suhylrxv
vhfwlrqv dqg zh rplw lw khuh1
BANCO DE ESPAÑA
27
DOCUMENTO DE TRABAJO N.º 0838
Iljxuh 8= Srolf| ixqfwlrqv
6
ȝ=0.6
ȝ=0.5
ȝ=0.4
5.6
5
4.6
4.5
ȝ=0.6
ȝ=0.5
ȝ=0.4
3.6
-0.25 -0.15 -0.05 0.05 0.15 0.25
ș2
7.5
ȝ=0.5
ȝ=0.4
a22
a21
5.5
3.5
4.5
2.5
-0.25 -0.15 -0.05 0.05 0.15 0.25
ș2
ȝ=0.6
ȝ=0.5
ȝ=0.4
3.5
-0.25 -0.15 -0.05 0.05 0.15 0.25
ș2
1.4
1
0.8
0.6
-0.25 -0.15 -0.05 0.05 0.15 0.25
ș2
a21/a22
1.2
a12/a11
ȝ=0.6
6.5
4.5
1.4
ȝ=0.4
4.1
4
-0.25 -0.15 -0.05 0.05 0.15 0.25
ș2
5.5
ȝ=0.5
a12
5.1
a11
5.5
ȝ=0.6
ȝ=0.6
ȝ=0.5
ȝ=0.4
1
0.6
0.2
-0.25 -0.15 -0.05 0.05 0.15 0.25
ș2
Dv h{shfwhg/ doo djhqwv zdqw wr lqyhvw pruh lq frxqwu| 5*v fdslwdo wkh kljkhu lv
frxqwu| 5*v WIS ohyho wrgd|/ vlqfh wklv vljqdov d kljkhu uhwxuq wr fdslwdo lq wkdw frxq0
wu| lq wkh iroorzlqj shulrg gxh wr wkh shuvlvwhqfh lq 1 Krzhyhu/ wkh lqyhvwphqw lq
frxqwu| 5*v fdslwdo ri frxqwu| 4*v uhsuhvhqwdwlyh djhqw lv uhvwulfwhg1 Zkhq ' f=S
wklv uhvwulfwlrq grhv qrw elqg1 Lqvwhdg/ iru ' f=D wkh uhvwulfwlrq ehjlqv wr elqg
BANCO DE ESPAÑA
28
DOCUMENTO DE TRABAJO N.º 0838
dv 2 ghsduwv vljqlfdqwo| iurp  = Ilqdoo|/ iru ' f=e wkh frqvwudlqw dozd|v elqgv1
Wzr uhvxowv duh zruwk0phqwlrqlqj1 Iluvw/ qrwh wkdw grphvwlf lqyhvwphqw ri frxqwu|
4*v uhsuhvhqwdwlyh djhqw lqfuhdvhv zlwk 2 zkhq kh lv frqvwudlqhg1 Wkh uhdvrq ehklqg
wklv dssduhqwo| frxqwhulqwxlwlyh uhvxow lv reylrxv1 Jlyhq wkh sduwlfxodu sruwirolr frq0
vwudlqw zh kdyh frqvlghuhg/ +49,/ frqvwudlqhg lqglylgxdov pd| lqfuhdvh wkhlu iruhljq
lqyhvwphqw e| lqfuhdvlqj wkhlu grphvwlf lqyhvwphqw1 Wkdw lv zk| zh qg wkdw grphv0
wlf lqyhvwphqw ri frqvwudlqhg lqglylgxdov lqfuhdvhv hyhq zkhq wkdw lqyhvwphqw jhwv
uhodwlyho| ohvv surwdeoh1 Dqg vhfrqg/ qrwh wkdw frxqwu| 5*v uhsuhvhqwdwlyh djhqw/ zkr
grhv qrw idfh dq| sruwirolr frqvwudlqw/ vwloo prglhv klv rswlpdo ghflvlrqv ghshqglqj
rq wkh ghjuhh ri frxqwu| 4*v sruwirolr frqvwudlqw1 Wklv lv vlpso| gxh wr wkh jhqhudo
htxloleulxp qdwxuh ri rxu h{huflvh1
7
Frqfoxglqj uhpdunv
Lq wklv sdshu zh kdyh sursrvhg d joredo vroxwlrq phwkrg wkdw frpelqhv wkh sdudph0
whul}hg h{shfwdwlrqv +SHD, dqg wkh Vpro|dn dojrulwkpv/ dqg lw lv zhoo0vxlwhg wr ghdo
zlwk kljk0glphqvlrqdo vwrfkdvwlf sureohpv1 Zh kdyh vkrzq krz wr dsso| wklv dojr0
ulwkp wr pxowl0frxqwu| prghov zlwk sruwirolr ghflvlrqv dqg lqfrpsohwh pdunhwv/ zlwk
dqg zlwkrxw rffdvlrqdoo| elqglqj frqvwudlqwv/ dqg lq d ixoo dqg sduwldo lqirupdwlrq
vfkhphv1 Lq doo wkhvh lqvwdqfhv/ wkh dojrulwkp pd| eh lpsohphqwhg dw d uhodwlyho|
orz frvw dqg dssur{lpdwhv wkh sureohpv* vroxwlrqv zlwk d kljk ghjuhh ri dffxudf|1
Zh wkxv fodlp wkdw wkh Vpro|dn0SHD dojrulwkp lv d srzhuixo vroxwlrq phwkrg wkdw
pd| eh dssolhg wr d zlgh ydulhw| ri uhohydqw sureohpv1
BANCO DE ESPAÑA
29
DOCUMENTO DE TRABAJO N.º 0838
References
[1] CHRISTIANO, L. J., and J. FISHER (2000). "Algorithms for Solving Dynamic
Models with Occasionally Binding Constraints", Journal of Economic Dynamics
and Control, vol. 24, pp. 1179-1232.
[2] DEN HAAN, W., and A. MARCET (1990). "Solving the Stochastic Growth
Model by Parameterizing Expectations", Journal of Business and Economic Statistics, vol. 8(1), pp. 31-34.
[3] — (1994). "Accuracy in Simulations", Review of Economic Studies, vol. 61, pp.
3-17.
[4] DEVEREUX, M. B., and A. SUTHERLAND (2007). "Solving for Country Portfolios in Open Economy Macro Models", IMF Working Papers, No. 07/284.
[5] DOMEIJ, D., and M. FLODEN (2006). "Population Aging and International
Capital Flows", International Economic Review, vol. 47(3), pp. 1013-1032.
[6] EVANS, M. D., and V. HNATKOVSKA (2005). Solving General Equilibrium
Models with Incomplete Markets and Many Assets, NBER Technical Working
Papers, No. 0318.
[7] HEATHCOTE, J., and F. PERRI (2002). "Financial Autarky and International
Business Cycles", Journal of Monetary Economics, vol. 49, pp. 601-627.
[8] JUDD, K. L. (1992). "Projection Methods for Solving Aggregate Growth Models", Journal of Economic Theory, vol. 58(2), pp. 410-452.
[9] KRUEGER, D., and F. KUBLER (2004). "Computing Equilibrium in OLG Models with Stochastic Production", Journal of Economic Dynamics and Control,
vol. 28(7), pp. 1411-1436.
[10] KRUEGER, D., and A. LUDWIG (2007). "On the Consequences of Demographic
Change for Rates of Returns to Capital, and the Distribution of Wealth and
Welfare", Journal of Monetary Economics, vol. 54(1), pp. 49-87.
[11] KRUSELL, P., and A. A. SMITH (1998). "Income and Wealth Heterogeneity in
the Macroeconomy", Journal of Political Economy, vol. 106(5), pp. 867-896.
[12] MALIAR, L., and S. MALIAR (2003). "Parameterized Expectations Algorithm
and the Moving Bounds", Journal of Business and Economic Statistics, vol.
21(1), pp. 88-92.
[13] MALIN, B., D. KRUEGER and F. KUBLER (2007). Computing Stochastic Dynamic Economic Models with a Large Number of State Variables: A Description
BANCO DE ESPAÑA
30
DOCUMENTO DE TRABAJO N.º 0838
and Application of a Smolyak-Collocation Method, NBER Working Papers, No.
13517.
[14] MARCET, A., and G. LORENZONI (1999). "The Parameterized Expectation
Approach: Some Practical Issues", in R. Marimon and A. Scott (eds.), Computational Methods for Study of Dynamic Economies, New York, Oxford University
Press, pp. 143-171.
[15] MARCET, A., and K. SINGLETON (1999). "Equilibrium Asset Prices and Savings of Heterogeneous Agents in the Presence of Incomplete Markets and Portfolio Constraints", Macroeconomic Dynamics, vol. 3, pp. 243-277.
[16] SCHMITT-GROHE, S., and M. URIBE (2003). "Closing Small Open Economy
Models", Journal of International Economics, vol. 61(1), pp. 163-185.
[17] VAN WINCOOP, E., and C. TILLE (2007). International Capital Flows, NBER
Working Papers, No.12856.
BANCO DE ESPAÑA
31
DOCUMENTO DE TRABAJO N.º 0838
BANCO DE ESPAÑA PUBLICATIONS
WORKING PAPERS1
0721
0722
0723
0724
0725
0726
0727
0728
0729
0730
0731
0732
0733
0734
0736
0737
0738
0739
0740
0741
0801
0802
0803
0804
0805
0806
0807
0808
0809
0810
0811
0812
0813
0814
0815
CLAUDIA CANALS, XAVIER GABAIX, JOSEP M. VILARRUBIA AND DAVID WEINSTEIN: Trade patterns, trade
balances and idiosyncratic shocks.
MARTÍN VALLCORBA AND JAVIER DELGADO: Determinantes de la morosidad bancaria en una economía
dolarizada. El caso uruguayo.
ANTÓN NÁKOV AND ANDREA PESCATORI: Inflation-output gap trade-off with a dominant oil supplier.
JUAN AYUSO, JUAN F. JIMENO AND ERNESTO VILLANUEVA: The effects of the introduction of tax incentives on
retirement savings.
DONATO MASCIANDARO, MARÍA J. NIETO AND HENRIETTE PRAST: Financial governance of banking
supervision.
LUIS GUTIÉRREZ DE ROZAS: Testing for competition in the Spanish banking industry: The Panzar-Rosse
approach revisited.
LUCÍA CUADRO SÁEZ, MARCEL FRATZSCHER AND CHRISTIAN THIMANN: The transmission of emerging
market shocks to global equity markets.
AGUSTÍN MARAVALL AND ANA DEL RÍO: Temporal aggregation, systematic sampling, and the Hodrick Prescott
filter.
LUIS J. ÁLVAREZ: What do micro price data tell us on the validity of the New Keynesian Phillips Curve?
ALFREDO MARTÍN-OLIVER AND VICENTE SALAS-FUMÁS: How do intangible assets create economic value? An
application to banks.
REBECA JIMÉNEZ-RODRÍGUEZ: The industrial impact of oil price shocks: Evidence from the industries of six
OECD countries.
PILAR CUADRADO, AITOR LACUESTA, JOSÉ MARÍA MARTÍNEZ AND EDUARDO PÉREZ: El futuro de la tasa de
actividad española: un enfoque generacional.
PALOMA ACEVEDO, ENRIQUE ALBEROLA AND CARMEN BROTO: Local debt expansion… vulnerability
reduction? An assessment for six crises-prone countries.
PEDRO ALBARRÁN, RAQUEL CARRASCO AND MAITE MARTÍNEZ-GRANADO: Inequality for wage earners and
self-employed: Evidence from panel data.0735 ANTÓN NÁKOV AND ANDREA PESCATORI: Oil and the Great
Moderation.
MICHIEL VAN LEUVENSTEIJN, JACOB A. BIKKER, ADRIAN VAN RIXTEL AND CHRISTOFFER KOK-SØRENSEN:
A new approach to measuring competition in the loan markets of the euro area.
MARIO GARCÍA-FERREIRA AND ERNESTO VILLANUEVA: Employment risk and household formation: Evidence
from differences in firing costs.
LAURA HOSPIDO: Modelling heterogeneity and dynamics in the volatility of individual wages.
PALOMA LÓPEZ-GARCÍA, SERGIO PUENTE AND ÁNGEL LUIS GÓMEZ: Firm productivity dynamics in Spain.
ALFREDO MARTÍN-OLIVER AND VICENTE SALAS-FUMÁS: The output and profit contribution of information
technology and advertising investments in banks.
ÓSCAR ARCE: Price determinacy under non-Ricardian fiscal strategies.
ENRIQUE BENITO: Size, growth and bank dynamics.
RICARDO GIMENO AND JOSÉ MANUEL MARQUÉS: Uncertainty and the price of risk in a nominal convergence
process.
ISABEL ARGIMÓN AND PABLO HERNÁNDEZ DE COS: Los determinantes de los saldos presupuestarios de las
Comunidades Autónomas.
OLYMPIA BOVER: Wealth inequality and household structure: US vs. Spain.
JAVIER ANDRÉS, J. DAVID LÓPEZ-SALIDO AND EDWARD NELSON: Money and the natural rate of interest:
structural estimates for the United States and the euro area.
CARLOS THOMAS: Search frictions, real rigidities and inflation dynamics.
MAXIMO CAMACHO AND GABRIEL PEREZ-QUIROS: Introducing the EURO-STING: Short Term INdicator of Euro
Area Growth.
RUBÉN SEGURA-CAYUELA AND JOSEP M. VILARRUBIA: The effect of foreign service on trade volumes and
trade partners.
AITOR ERCE: A structural model of sovereign debt issuance: assessing the role of financial factors.
ALICIA GARCÍA-HERRERO AND JUAN M. RUIZ: Do trade and financial linkages foster business cycle
synchronization in a small economy?
RUBÉN SEGURA-CAYUELA AND JOSEP M. VILARRUBIA: Uncertainty and entry into export markets.
CARMEN BROTO AND ESTHER RUIZ: Testing for conditional heteroscedasticity in the components of inflation.
JUAN J. DOLADO, MARCEL JANSEN AND JUAN F. JIMENO: On the job search in a model with heterogeneous
jobs and workers.
SAMUEL BENTOLILA, JUAN J. DOLADO AND JUAN F. JIMENO: Does immigration affect the Phillips curve? Some
evidence for Spain.
ÓSCAR J. ARCE AND J. DAVID LÓPEZ-SALIDO: Housing bubbles.
1. Previously published Working Papers are listed in the Banco de España publications catalogue.
0816
0817
0818
0819
0820
0821
0822
0823
0824
0825
0826
0827
0828
0829
0830
0831
0832
0833
0834
0835
0836
0837
0838
GABRIEL JIMÉNEZ, VICENTE SALAS-FUMÁS AND JESÚS SAURINA: Organizational distance and use of collateral
for business loans.
CARMEN BROTO, JAVIER DÍAZ-CASSOU AND AITOR ERCE-DOMÍNGUEZ: Measuring and explaining the volatility
of capital flows towards emerging countries.
CARLOS THOMAS AND FRANCESCO ZANETTI: Labor market reform and price stability: an application to the Euro
Area.
DAVID G. MAYES, MARÍA J. NIETO AND LARRY WALL: Multiple safety net regulators and agency problems in the
EU: Is Prompt Corrective Action partly the solution?
CARMEN MARTÍNEZ-CARRASCAL AND ANNALISA FERRANDO: The impact of financial position on investment:
an analysis for non-financial corporations in the euro area.
GABRIEL JIMÉNEZ, JOSÉ A. LÓPEZ AND JESÚS SAURINA: Empirical analysis of corporate credit lines.
RAMÓN MARÍA-DOLORES: Exchange rate pass through in New Member States and candidate countries of the EU.
IGNACIO HERNANDO, MARÍA J. NIETO AND LARRY D. WALL: Determinants of domestic and cross-border bank
acquisitions in the European Union.
JAMES COSTAIN AND ANTÓN NÁKOV: Price adjustments in a general model of state-dependent pricing.
ALFREDO MARTÍN-OLIVER, VICENTE SALAS-FUMÁS AND JESÚS SAURINA: Search cost and price dispersion in
vertically related markets: the case of bank loans and deposits.
CARMEN BROTO: Inflation targeting in Latin America: Empirical analysis using GARCH models.
RAMÓN MARÍA-DOLORES AND JESÚS VAZQUEZ: Term structure and the estimated monetary policy rule in the
eurozone.
MICHIEL VAN LEUVENSTEIJN, CHRISTOFFER KOK SØRENSEN, JACOB A. BIKKER AND ADRIAN VAN RIXTEL:
Impact of bank competition on the interest rate pass-through in the euro area.
CRISTINA BARCELÓ: The impact of alternative imputation methods on the measurement of income and wealth:
Evidence from the Spanish survey of household finances.
JAVIER ANDRÉS AND ÓSCAR ARCE: Banking competition, housing prices and macroeconomic stability.
JAMES COSTAIN AND ANTÓN NÁKOV: Dynamics of the price distribution in a general model of state-dependent
pricing.
JUAN A. ROJAS: Social Security reform with imperfect substitution between less and more experienced workers.
GABRIEL JIMÉNEZ, STEVEN ONGENA, JOSÉ LUIS PEYDRÓ AND JESÚS SAURINA: Hazardous times for
monetary policy: What do twenty-three million bank loans say about the effects of monetary policy on credit risktaking?
ENRIQUE ALBEROLA AND JOSÉ MARÍA SERENA: Sovereign external assets and the resilience of global
imbalances.
AITOR LACUESTA, SERGIO PUENTE AND PILAR CUADRADO: Omitted variables in the measure of a labour
quality index: the case of Spain.
CHIARA COLUZZI, ANNALISA FERRANDO AND CARMEN MARTÍNEZ-CARRASCAL: Financing obstacles and
growth: An analysis for euro area non-financial corporations.
ÓSCAR ARCE, JOSÉ MANUEL CAMPA AND ÁNGEL GAVILÁN: Asymmetric collateral requirements and output
composition.
ÁNGEL GAVILÁN AND JUAN A. ROJAS: Solving Portfolio Problems with the Smolyak-Parameterized Expectations
Algorithm.
Unidad de Publicaciones
Alcalá, 522; 28027 Madrid
Telephone +34 91 338 6363. Fax +34 91 338 6488
e-mail: [email protected]
www.bde.es