Benigno, Gianluca; Foe s e , And ew; O ok, Ch is ophe M.; Rebucci, Alessand o
A icle
Es ima ing mac oeconomic models o inancial c ises: An
endogenous egime-swi ching app oach
Quan i a i e Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Benigno, Gianluca; Foe s e , And ew; O ok, Ch is ophe M.; Rebucci, Alessand o
(2025) : Es ima ing mac oeconomic models o inancial c ises: An endogenous egime-swi ching
app oach, Quan i a i e Economics, ISSN 1759-7331, The Econome ic Socie y, New Ha en, CT, Vol.
16, Iss. 1, pp. 1-47,
h ps://doi.o g/10.3982/QE2038
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Quan i a i e Economics 16 (2025), 1–47 1759-7331/20250001
Es ima ing mac oeconomic models o inancial c ises:
An endogenous egime-swi ching app oach
Gianluca Benigno
Depa men o Economics, Uni e si y o Lausanne and CEPR
And ew Foe s e
Economic Resea ch Depa men , Fede al Rese e Bank o San F ancisco
Ch is ophe O ok
Resea ch Depa men , Fede al Rese e Bank o Dallas
Alessand o Rebucci
Ca ey Business School, Johns Hopkins Uni e si y, ABFER, CEPR, and NBER
We de elop a new model o cycles and c ises in eme ging ma ke s, ea u ing an
occasionally binding bo owing cons ain and s ochas ic ola ili y, and es ima e
i wi h qua e ly da a o Mexico since 1981. We p opose an endogenous egime-
swi ching o mula ion o he occasionally binding bo owing cons ain , de elop
a gene al pe u ba ion me hod o sol e he model, and es ima e i using Bayesian
me hods. We ind ha he model i s he Mexican da a well wi hou sys ema ically
elying on la ge shocks, ma ching he ypical s ylized ac s o eme ging ma ke
business cycles and Mexico’s his o y o sudden s ops in capi al lows. We also ind
ha in e es a e shocks play a smalle ole in d i ing bo h cycles and c ises han
p e iously ound in he li e a u e.
Keywo ds. Business cycles, Bayesian es ima ion, endogenous egime-swi ching,
inancial c ises, Mexico, occasionally binding cons ain s, sudden s ops.
JEL classi ica ion. C11, E3, F41, G01.
1. In oduc ion
The global inancial c isis igge ed a s ong enewed in e es in unde s anding he
causes, consequences, and emedies o inancial c ises. In his con ex , dynamic
Gianluca Benigno: [email p o ec ed]
And ew Foe s e : [email p o ec ed]
Ch is ophe O ok: [email p o ec ed]
Alessand o Rebucci: [email p o ec ed]
We a e g a e ul o h ee anonymous e e ees, Yan Bai, Da io Calda a, Luca Gue ie i, Yoosoon Chang, Pablo
Gue on-Quin ana, Yasuo Hi ose, Gio gio P imice i, Felipe Sa ie, and F ank Scho heide o help ul com-
men s on p e ious d a s o his pape and discussions. We also hank pa icipan s a nume ous semina s
and con e ences. Sanha Noh p o ided ou s anding esea ch assis ance. The au ho s g a e ully acknowledge
inancial suppo om NSF G an SES1530707 and he Johns Hopkins Ca alys Awa d. The iews exp essed
a e solely hose o he au ho s and do no necessa ily e lec he iews o he Fede al Rese e Banks o Dal-
las, San F ancisco, o he Fede al Rese e Sys em.
©2025 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h p://qeconomics.o g.h ps://doi.o g/10.3982/QE2038
2Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
s ochas ic gene al equilib ium (DSGE) models wi h occasionally binding inancial ic-
ions o s ochas ic ola ili y p o ed success ul as labo a o ies o s udying he ana omy
o bo h business cycles and c ises and explo ing op imal policy esponses o hese dy-
namics. This success is because occasionally binding inancial ic ions a e mechanisms
ha c ea e an ampli ica ion o egula business cycle dynamics. A he same ime, a now
la ge li e a u e has shown ha unce ain y modeled as s ochas ic ola ili y also con-
ibu es o cycles and c ises, cap u ing a ails and skewness in he da a. The s uc u al
es ima ion o hese models is impo an o in e ence on key pa ame e s go e ning i-
nancial ic ions and exogenous p ocesses and he decomposi ion o impo an his o i-
cal episodes, ye i is e y challenging.
In his pape , we de elop a new model o cycles and c ises in eme ging ma ke s, ea-
u ing an occasionally binding bo owing cons ain and s ochas ic ola ili y, and es i-
ma e i wi h qua e ly da a o Mexico since 1981. The pape makes h ee con ibu ions.
Fi s , we p opose a new speci ica ion o he occasionally binding colla e al cons ain
ha pe mi s ma ching c ises o di e en du a ions and in ensi ies. Second, we de elop
a pe u ba ion solu ion me hod sui able o sol ing models like ou s in a way ha allows
o likelihood-based es ima ion. Thi d, we apply he p oposed amewo k o in es iga e
sou ces o business cycles and c ises in Mexico since 1981—a case s udied mos o en as
a ypical eme ging ma ke economy.
As a i s s ep, we p opose a new o mula ion o occasionally binding cons ain
models. As in he adi ional speci ica ion o such models, ou se up has wo s a es o
egimes: in he i s , limi ed le e age ampli ies egula shocks and gi es ise o i e sales
and deb -de la ion dynamics; in he second, access o inancing is uncons ained, and
he economy displays egula business cycles. Howe e , in ou speci ica ion, he ansi-
ions be ween he wo egimes depend on a ange a he han a unique le el o le e age,
wi h endogenous swi ching p obabili ies ha a e a unc ion o he bo owing capaci y
o he economy and he mul iplie associa ed wi h he le e age cons ain . This o mu-
la ion maps he model wi h an occasionally binding le e age cons ain in o an endoge-
nous egime-swi ching model. The pape ocuses on a pa icula cons ain and ype
o c isis, he so-called sudden s op in capi al lows, bu he p oposed speci ica ion has
b oade applicabili y o o he ypes o occasionally binding cons ain s and ic ions. Fo
example, ou p oposed app oach could be applied o he o mula ion and es ima ion o
models wi h housing cons ain s, downwa d wage igidi y, o he ze o lowe bound.
Nex , we de elop a pe u ba ion-based solu ion me hod o sol e he endogenous
egime-swi ching model. The pe u ba ion me hod is as enough o pe mi likelihood-
based es ima ion, is scalable o models la ge han he one we es ima e in his pape ,
and displays ypical le els o accu acy. We also analy ically show ha app oxima ing he
model solu ion o he second o de is necessa y and su icien o cap u e a leas some
o he e ec s o he endogenous ansi ion p obabili ies on he model’s policy unc ions,
including p ecau iona y beha io , and ha hese e ec s would be missed by linea ap-
p oxima ions o exogenous egime swi ching models. The second-o de solu ion also
allows us o cap u e he e ec s o egime shi s in he ola ili y o exogenous shocks.
Again, he solu ion me hod ha we de elop can be applied o a wide ange o models
wi h endogenous egime-swi ching.
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 3
Figu e 1. Cu en accoun and GDP in Mexico, 1981–2016. No e: Panel (a) plo s Mexico’s cu -
en accoun balance as a sha e o GDP. Panel (b) shows Mexico’s qua e ly log-change o eal
GDP. The ligh g ay egions deno e he pe iods o cu ency o ex e nal deb c isis acco ding o
Reinha and Rogo (2009), which we call he Ex e nal C isis Tally Index. See he Supplemen-
a y Appendix (Benigno, Foe s e , O ok, and Rebucci (2024)) o da a sou ces. Sample pe iod
1981:Q1-2016:Q4.
Finally, we apply he amewo k ha we de eloped o he Bayesian es ima ion and
analysis o Mexico’s his o y o cycles and sudden s op c ises since 1981. The Mexican
economy is a pa icula ly in e es ing labo a o y because many o he seminal con ibu-
ions o he li e a u e on business cycles and c ises in eme ging ma ke s p e iously s ud-
ied his case. Figu e 1plo s wo c i ical Mexican da a mac oeconomic se ies: he cu en
accoun balance as a sha e o GDP and he qua e ly eal GDP g ow h. The igu e also
indica es as (g ay) shaded a eas Mexico’s pe iods o cu ency and ex e nal deb c isis
iden i ied in Reinha and Rogo (2009). The igu e illus a es he egula luc ua ions in
he da a, as well as he mul iple episodes o la ge cu en accoun e e sals and ou pu
g ow h declines. La ge cu en accoun e e sals and ou pu d ops o he e ogeneous size
and pe sis ence a e he wo main empi ical ea u es commonly associa ed wi h sudden
s ops in capi al lows, no only in Mexico bu also in many o he eme ging ma ke s. In
his pape , we ocus on he challenge o i ing a s uc u al model o Mexico’s business
cycle and sudden s op his o y wi hou imposing ad hoc es ic ions on he magni ude o
pe sis ence o hese episodes. The igu e also displays he ma ked shi in he ola ili y
o he economy bo h be o e and a e he mid-1990s and du ing ce ain pe iods o ime,
which may no be cap u ed by an occasionally binding bo owing cons ain o inancial
shocks.
Despi e he econome ic challenges in cha ac e izing da a like hose shown in Fig-
u e 1, ou es ima ed model i s Mexico’s business cycles and sudden s op episodes
well, wi hou sys ema ically elying on la ge shocks o explain c ises, bu ins ead le ing
4Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
he economic s uc u e o he model explain hose e en s. The inclusion o s ochas ic
ola ili y helps us o dis inguish be ween he binding cons ain and pe iods o mo e
ola ile shocks as d i e s o luc ua ions. The model p oduces business cycle s a is ics
ha ma ch he second momen s o he da a and yields e idence ha nei he p oduc i -
i y no in e es a e shocks a e he mos impo an d i e s o Mexico’s business cycles.
Mos impo an ly, ou speci ica ion o he colla e al cons ain iden i ies c isis episodes
and dynamics o a ying du a ion and in ensi y ha ma ch he c isis pe iods iden i ied
wi h a na a i e app oach in Reinha and Rogo (2009).
Rela ed li e a u e
Ou pape is connec ed o se e al s ands o li e a u e. The pape ela es o he la ge
li e a u e on he Bayesian es ima ion o DSGE models (e.g., Scho heide,2000,O ok,
2001,Sme s and Wou e s,2007,Iaco iello and Ne i,2010,Bianchi,2013). We ex end ha
success ul app oach o models wi h occasionally binding colla e al cons ain s, which
ha e become he benchma k o no ma i e analysis o mac o-p uden ial op imal policy.
Ou pape is closely ela ed o he empi ical wo k in Bocola (2016), whe e he model
is sol ed wi h global me hods and es ima ed. Howe e , ha es ima ion exe cise is made
possible by i s es ima ing he model ou side he c isis, and hen appending an es ima e
o he c isis in a second s ep. While his p ocedu e does no ma e o he speci ic ap-
plica ion in Bocola (2016), i is no necessa ily applicable mo e gene ally. Ou app oach
pe mi s join es ima ion o he model inside and ou side he c ises and is po en ially
scalable o la ge and mo e complex models, while main aining a sa is ac o y le el o
accu acy ela i e o global solu ion me hods.
The pape is also closely ela ed o he li e a u e on likelihood-based es ima ion o
Ma ko swi ching DSGE models ini ia ed by he seminal con ibu ion o Bianchi (2013),
andappliedinBianchi and Ilu (2017)andBianchi, Ilu , and Schneide (2018). The il e
we use in es ima ion di e s in wo key espec s. Fi s , ou egime-swi ching ansi ion
ma ix is endogenous. Second, condi ional on he egime, we sol e he model o he
second o de . So, we employ he Sigma Poin Fil e o e alua e he likelihood unc ion in
place o he modi ied Kalman il e in Bianchi (2013).
In he li e a u e on Ma ko -swi ching DSGE models, ou pape builds on he me hod
p oposed by Foe s e , Rubio-Rami ez, Waggone , and Zha (2016), who de eloped pe -
u ba ion me hods o he solu ion o exogenous egime-swi ching models. The pe -
u ba ion app oach ha we p opose allows o second- and highe -o de app oxima-
ions ha go beyond he linea models s udied by Da ig and Leepe (2007)andFa me ,
Daniel, Waggone , and Zha (2011). In ac , we show ha a leas a second-o de app ox-
ima ion is necessa y in o de o cap u e he e ec s o he endogenous swi ching.
The pape is na u ally ela ed o he li e a u e ha ocuses on endogenous egime-
swi ching models. Da ig and Leepe (2008), Da ig, Leepe , and Walke (2010), and Al-
panda and Uebe eld (2016) all conside endogenous egime-swi ching bu employ
global solu ion me hods ha hinde likelihood-based es ima ion. Lind (2014) de elops
a egime-swi ching pe u ba ion app oach o app oxima ing nonlinea models, bu i
equi es epea edly e ining he poin s o app oxima ion, and hence, i is no sui able o
es ima ion pu poses.
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 5
He e, ou pape ela es closely o OccBin, a se o p ocedu es o he solu ion o mod-
els wi h occasionally binding cons ain s, de eloped in Gue ie i and Iaco iello (2015).
OccBin is a ce ain y equi alen solu ion me hod ha cap u es nonlinea i ies bu no
p ecau iona y e ec s, which a e a c i ical ea u e o models wi h occasionally binding
colla e al cons ain s.1A key ea u e o ou app oach is o p ese e p ecau iona y sa ing
e ec s, as agen s in he model adjus hei beha io due o he p esence o he cons ain
e en when he cons ain does no bind, and ice e sa.
The speci ica ion o he inequali y cons ain and he accompanying solu ion
me hod ha we p opose can be applied o models wi h occasionally binding ze o-
lowe bound on in e es a es (e.g., Adam and Billi (2007), A uoba, Cuba-Bo da, and
Scho heide (2018), A kinson, Rich e , and Th ockmo on (2018)).2Exis ing me hods
o he es ima ion o such models may limi scalabili y due compu a ional cos s (Gus ,
He bs , Lopez-Salido, and Smi h (2017)). The app oach we p opose is scalable and ap-
plicable o la ge models.
The applica ion o he me hodology ha we p opose ela es o he li e a u e on
eme ging ma ke business cycles, which includes Aguia and Gopina h (2007), Men-
doza (2010), Ga cia-Cicco, Panc azi, and U ibe (2010), Fe nandez-Villa e de, Gue on-
Quin ana, Rubio-Rami ez, and U ibe (2011), among o he s. Encompassing mos shocks
p e iously conside ed, we conside ansi o y and pe manen echnology, p e e ence,
expendi u e, in e es a e, and e ms o ade shocks in ou analysis. We allow o egime
swi ching in he ola ili y o shocks as in (Liu, Waggone , and Zha (2011), Bianchi (2013))
since s ochas ic ola ili y has been shown o be an impo an ea u e o eme ging ma -
ke s da a (Fe nandez-Villa e de e al. (2011)). Rela i e o Mendoza (2010), we p o ide a
Bayesian es ima ion o he model and conside a wide se o s uc u al shocks. Rela i e
o Ga cia-Cicco, Panc azi, and U ibe (2010), we empi ically e alua e he ela i e impo -
ance o in e es a e shocks by modeling he ampli ica ion induced by inancial ic ions
wi h an occasionally binding bo owing cons ain .
The es o he pape is o ganized as ollows. Sec ion 2desc ibes he model and dis-
cusses he p oposed o mula ion o he colla e al cons ain and s ochas ic ola ili y.
Sec ion 3p esen s ou pe u ba ion solu ion me hod o endogenous egime-swi ching
models. Sec ion 4desc ibes he Bayesian es ima ion p ocedu e and epo s he es ima-
ion esul s. Sec ion 5discusses he main empi ical esul s on he analysis o Mexico’s
business cycle and his o y o sudden s op c ises. Sec ion 6concludes. Technical de ails
and addi ional esul s a e epo ed in he Appendix and in he Supplemen a y Appendix
(Benigno e al. (2024)).
2. The model
The amewo k is a medium-scale model o he analysis o business cycles and sudden-
s op c ises in eme ging ma ke economies. The model is a small, open p oduc ion econ-
1Cuba-Bo da, Gue ie i, Iaco iello, and Zhong (2019) s udy how he solu ion me hod and likelihood mis-
speci ica ion in e ac and possibly compound each o he .
2An occasionally binding ze o-lowe bound is no compa able o he cons ain wi h endogenous colla -
e al alue ha we es ima e in his pape . Indeed, endogenous colla e al alua ion ea u es di e en ampli-
ica ion mechanisms and en ails addi ional compu a ional complexi ies.
6Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
omy as in Mendoza (2010) wi h endogenous labo supply, in es men , and an occa-
sionally binding colla e al cons ain . We conside a la ge se o shocks as in Ga cia-
Cicco, Panc azi, and U ibe (2010), including pe manen and empo a y p oduc i i y
shocks (Aguia and Gopina h (2007)), in e empo al p e e ence, expendi u e, in e es
a e, and e ms o ade shocks. The model also ea u es s ochas ic ola ili y in all shocks
(Fe nandez-Villa e de e al. (2011)).
In he es o his sec ion, we b ie ly s a e he op imiza ion p oblem o he ep e-
sen a i e household- i m and hen discuss he speci ica ion o he bo owing cons ain
and he shock p ocesses, which a e he no el ea u es o ou model. The de i a ion o
he equilib ium condi ions and he o mal de ini ion o he compe i i e equilib ium o
he economy a e in Appendix A.
2.1 P e e ences, cons ain s, and shock p ocesses
The e is a ep esen a i e household- i m ha maximizes he ollowing u ili y unc ion:
U=E0
∞
=0d β C −Z −1
Hω
ω1−ρ
−1
1−ρ,(1)
whe e C deno es consump ion and H he supply o labo . The u ili y depends on an ex-
ogenous and s ochas ic p e e ence shock d , and he pe manen echnology le el Z −1,
which ollow he p ocesses speci ied below.3The household- i m chooses consump ion,
labo , capi al K , impo ed in e media e inpu s V gi en an exogenous s ochas ic o
hei ela i e p ice P also speci ied below, and holdings o eal one-pe iod in e na ional
bonds, B . Nega i e alues o B indica e bo owing om ab oad. The household- i m
can bo ow in in e na ional ma ke s by issuing one-pe iod bonds ha pay a ma ke o
coun y ne in e es a e .
The household- i m aces he ollowing budge cons ain :
C +I +E =Y −φ (W H +P V )−1
(1+ )B +B −1,(2)
whe e Y is g oss domes ic p oduc (GDP) gi en by
Y =A Kη
−1(Z H )αV1−α−η
−P V .(3)
He e, A deno es a s a iona y, exogenous, and s ochas ic le el o echnology, and Z is
a non-s a iona y, exogenous, and s ochas ic le el o echnology. E is an exogenous and
s ochas ic expendi u e p ocess possibly in e p e ed as a iscal o ne expo shock as in
Ga cia-Cicco, Panc azi, and U ibe (2010). The e m φ (W H +P V )desc ibes a wo king
capi al cons ain , s a ing ha a ac ion o he wage and in e media e goods bill mus
be paid in ad ance o p oduc ion wi h bo owed unds. The ela i e p ice o labo and
3Scaling hou s wo ked by Z −1pe mi s ob aining a balanced g ow h pa h wi h GHH p e e ences (see,
e.g., Ga cia-Cicco, Panc azi, and U ibe (2010)).
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 7
capi al a e gi en by W and q , espec i ely, bo h o which a e endogenous bu aken as
gi en by he indi idual household- i m. Capi al accumula ion depends on in es men
I and is subjec o adjus men cos s:
K =(1−δ)K −1+I −ι
2K −kK −1
K −12
K −1,(4)
whe e kdeno es he g ow h a e o capi al along he economy’s balanced g ow h pa h
(BGP).
All exogenous p ocesses ha e s ochas ic ola ili y, depending on a egime indica o
sσ
∈{H,L},whe eHand Lsigni y a high and low ola ili y egime, espec i ely, as in
Liu, Waggone , and Zha (2011)o Bianchi (2013), among o he s. The p e e ence p ocess
ollows
logd =ρdlogd −1+σdsσ
εd, .(5)
The s a iona y echnology p ocess ollows
logA =(1−ρA)logA∗+ρAlogA −1+σAsσ
εA, .(6)
The pe manen echnology p ocess ollows
logZ =(1−ρz)logZ∗+ρzlogZ −1+σzsσ
εz, .(7)
The in e es a e p ocess ollows
=(1−ρ ) ∗+ρ −1+σ sσ
ε , .(8)
The p ocess o he ela i e p ice o in e media e goods ollows
logP =(1−ρP)logP∗+ρPlogP −1+σPsσ
εP, .(9)
Finally, he p ocess o he exogenous componen o expendi u e ollows E =e /Z −1,
whe e
loge =(1−ρe)loge∗+ρeloge −1+σesσ
εe, . (10)
The s a ed a iables and he ρcoe icien s deno e he uncondi ional mean alues and
he pe sis ence pa ame e s o hese p ocesses, espec i ely. The εa e assumed i.i.d.
N(0, 1)inno a ions, and he σpa ame e s con ol he size o hei a iances. The an-
si ion ma ix o he ola ili y egimes, Pσ, is exogenous and gi en by
Pσ=Pσ
LL 1−Pσ
LL
1−Pσ
HH Pσ
HH ,
whe e Pσ
ij =P (sσ
+1=j|sσ
+1=i).
8Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
2.2 The occasionally binding bo owing cons ain : An endogenous egime-swi ching
speci ica ion
As in ypical models wi h occasionally binding inequali y cons ain s, he economy luc-
ua es be ween wo s a es o egimes. In one s a e, deno ed sc
=1 and called he binding
o cons ained egime, he ollowing cons ain on o al bo owing binds s ic ly:
1
(1+ )B −φ(1+ )(W H +P V )=−κq K , (11)
wi h λ deno ing he co esponding mul iplie . To al deb includes bo owing o con-
sump ion smoo hing plus wo king capi al o he pu chase o in e media e inpu s and
labo o p oduc ion. Cons ained wo king capi al limi s he supply esponse o he
economy o shocks in he binding egime. In he o he s a e, deno ed sc
=0 and called
he nonbinding o uncons ained egime, he bo owing limi is slack and λ =0, and
he only cons ain is he na u al deb limi .
Gi en hese wo egimes, which ep esen he occasionally binding na u e o he
cons ain , we cha ac e ize he ansi ion be ween hem s ochas ically in he sense ha ,
o gi en alues o capi al, bond holding, and exogenous p ocesses, he e is an endoge-
nous p obabili y o swi ching be ween he wo egimes. This o mula ion con as s wi h
he de e minis ic ela ionship be ween le e age and he egime o gi en alues o en-
dogenous and exogenous s a e a iables in a ypical occasionally binding speci ica ion.
In pa icula , we assume ha he p obabili ies o swi ching om one egime o he o he
ollow a logis ic unc ion o a es ic ed subse o he endogenous s a e a iables in he
model.4
De ine he “bo owing cushion,” B∗
, as he dis ance o ac ual bo owing om he
deb limi :
B∗
=1
(1+ )B −φ(1+ )(W H +P V )+κq K , (12)
so ha when B∗
is small, o al bo owing and le e age a e high ela i e o he alue o
he colla e al. We hen assume ha he ansi ion om he nonbinding o he binding
egime depends on ˜
B∗
=B∗
/Z −1acco ding o
P sc
+1=1|sc
=0, ˜
B∗
=exp−γ0˜
B∗
1+exp−γ0˜
B∗
. (13)
Thus, he likelihood ha he cons ain binds in he ollowing pe iod depends on
he size o he bo owing cushion in he cu en pe iod.5The pa ame e γ0con ols
4This is simila o he logis ic speci ica ion in Bocola (2016), whe e he economy swi ches be ween de-
aul and nonde aul s a es. Kumho , Ranciè e, and Winan (2015) also use a logis ic unc ion o model he
ansi ion o he de aul egime in he con ex o a posi i e analysis o he ela ionship be ween inancial
c ises and inequali y. Da ig, Leepe , and Walke (2010) and Bi and T aum (2014) use a simila logis ic o -
mula ion o s udy he mac oeconomic consequences o iscal limi s.
5Appendix Cshows ha his iming di e ence wi h espec o a model wi h a adi ional inequali y spec-
i ica ion o he occasionally bo owing cons ain does no a ec he i s and second momen s o he econ-
omy.
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 15
Ou solu ion me hod is as , and eadily scales o handle la ge models. In o al, we
ha e 20 equilib ium condi ions, wo endogenous and six exogenous s a e a iables, ou
egimes,andsixshocks.Themodelissol edinabou asecondonas anda dlap op.
13
As Appendix Cdiscusses in mo e de ail, he p oposed solu ion me hod is also ac-
cu a e. We in es iga e i s accu acy by applying i o he calib a ed model in Mendoza
and Villal azo (2020) and compa ing ou endogenous egime swi ching speci ica ion
sol ed by pe u ba ion wi h he adi ional inequali y cons ain speci ica ion sol ed
wi h global me hods. We compa e he i s and second momen s o all model a i-
ables and show ha ou solu ion me hod yields esul s p ac ically indis inguishable
om hose ob ained om a adi ional inequali y speci ica ion o he bo owing con-
s ain . Speci ically, we ind Eule equa ion e o s in line wi h he accu acy o pe u ba-
ion me hods applied o exogenous egime-swi ching models (Foe s e e al. (2016)) and
models wi hou egime-swi ching (A uoba, Fe nandez-Villa e de, and Rubio-Rami ez
(2006)). We also documen a solu ion speed mo e han 800 imes as e han he global
me hod. Indeed, his solu ion speed gain is wha makes likelihood-based es ima ion o
he model easible.
3.4 App oxima ion o de and endogenous swi ching
Ou endogenous egime-swi ching amewo k mus be sol ed a leas o he second o -
de o cap u e he e ec s o he endogenous swi ching on he policy ules, which include
p ecau iona y e ec s ha a y wi h he s a e o he economy. I we we e o app oxima e
only o he i s o de , we would no cap u e he p ecau iona y beha io s emming om
a ional expec a ions abou he dependency o he p obabili y o a egime change on
he bo owing cushion and he mul iplie . The ollowing p oposi ion s a es his esul
o mally.
P oposi ion 1 (P ope ies o he app oxima ed solu ion). The i s -o de app oxima-
ion o he endogenous egime-swi ching model is iden ical o he i s -o de app oxima e
solu ion o an exogenous egime-swi ching model in which he ansi ion p obabili ies
a e gi en by he s eady-s a e alue o he ime- a ying ansi ion ma ix.A second-o de
app oxima ion o he endogenous egime-swi ching model is necessa y and su icien o
cap u e p ecau iona y e ec s o he endogenous swi ching.
P oo . See Appendix B.
This esul is analogous o s a ing ha , in models wi hou egime-swi ching, a i s -
o de solu ion is in a ian o he size o he shocks, a second-o de solu ion cap u es
13A MATLAB code o he p oposed solu ion algo i hm is a ailable on he au ho s’ web pages. Fo his pa-
pe , ou compu a ional app oach is simila o ha in Fe nandez-Villa e de, Gue on-Quin ana, and Rubio-
Rami ez (2015). We use Ma hema ica o ake symbolic de i a i es and expo hese de i a i es o C++.
We hen in eg a e hem in Ma lab using mex iles o sol e he model o di e en pa ame e iza ions. In
p inciple, we could sol e and es ima e he model using only C++ o FORTRAN, bu he gains in e ms o
speed would be ela i ely mino ; o la ge models, hese languages migh yield mo e subs an ial e iciency
gains. Since we use a nonlinea il e , he il e ing in es ima ion, no he model solu ion, is he mos ime-
consuming compu a ional s ep.
16 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
p ecau iona y beha io , and a hi d-o de solu ion is needed o cap u e he e ec s o
s ochas ic ola ili y ha ollows an au o eg essi e p ocess as in Fe nandez-Villa e de,
Gue on-Quin ana, and Rubio-Rami ez (2015). In ou con ex , since we ha e exogenous
egime changes in ola ili y, a second-o de app oxima ion is su icien o cap u e p e-
cau iona y e ec s associa ed wi h ola ili y changes (Foe s e e al. (2016)). In ou se -
ing, he shocks o he ola ili y p ocesses o a s ochas ic ola ili y speci ica ion mani-
es hemsel es as disc e e changes in he ola ili ies and no as shocks o he p ocesses
hemsel es.
Un o una ely, howe e , using a second-o de app oxima ion wi h endogenous
egime-swi ching poses addi ional challenges o es ima ion pu poses. We now u n o
ou s a egy o add ess hese issues.
4. Es ima ing he endogenous swi ching model
We es ima e he model wi h a Bayesian ull in o ma ion p ocedu e. The pos e io dis-
ibu ion has no analy ical solu ion, and we use Ma ko -Chain Mon e Ca lo (MCMC)
me hods o sample om i . Since he Me opolis–Has ings algo i hm we use o sam-
pling is s anda d, we omi he discussion o his s ep in ou p ocedu e.
A c i ical challenge in pos e io sampling is he e alua ion o he likelihood unc-
ion. We ace h ee di icul ies ela i e o linea DSGE models (e.g., Sme s and Wou e s
(2007)). The i s is he nonlinea i y induced by he p esence o mul iple egimes. The
second is he need o app oxima e o he second o de . The hi d is he ac ha he
ansi ion p obabili ies a e endogenous. Bianchi (2013) de elops an algo i hm o ad-
d ess he i s di icul y. He e, we mus deal wi h he second-o de app oxima ion and
endogenous p obabili ies in a ac able manne . One al e na i e is he Pa icle Fil e
(Fe nandez-Villa e de and Rubio-Rami ez (2007)). Howe e , he egime swi ching leads
o he disca ding o a la ge numbe o simula ed pa icles, lowe ing he accu acy o a
gi en numbe o pa icles and g ea ly inc easing he compu a ional cos o eaching ac-
cu acy (Douce , Go don, and K ishnamu hy (2001)). To add ess hese challenges, we
use he Unscen ed Kalman Fil e (UKF) wi h Sigma Poin s (Julie and Uhlmann (1999)).
The Sigma Poin il e has been shown o be an e icien way o es ima ing egime-
swi ching models (Binning and Maih (2015)). De ails o he cons uc ion o he s a e
space ep esen a ion and he il e ing o he e alua ion o he likelihood a e epo ed in
he Supplemen a y Appendix (Benigno e al. (2024)).
4.1 Obse ables, da a, and measu emen e o s
We es ima e he model wi h qua e ly da a o GDP g ow h (g oss ou pu less in e me-
dia e inpu paymen s), consump ion g ow h, in es men g ow h, and in e media e im-
po p ice g ow h, as well as he cu en accoun - o-GDP a io, and a measu e o he
coun y eal in e es a e.14
14See he SA o de ails on a iable de ini ions and da a sou ces. The coun y in e es a e is cons uc ed
ollowing U ibe and Yue (2006), and i is he US 3-Mon h T easu y Bill minus ex pos US CPI in la ion a e
plus Mexico’s EMBI Sp ead.
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 17
As he e a e six shocks wi h six obse ables, we do no necessa ily need measu e-
men e o s. Howe e , measu emen e o s in he obse a ion equa ion imp o e he
pe o mance o he nonlinea il e and accoun o any ac ual measu emen e o in
he da a. As in Ga cia-Cicco, Panc azi, and U ibe (2010), we limi hei a iance o 5% o
he a iance o he obse able a iables. As a consequence, he model will i he da a
ela i ely closely on a e age, and model i o lack he eo will be assessed by checking
whe he i elies on la ge shocks o gene a e c isis episodes and by compa ing ou model
wi h al e na i e speci ica ions.
4.2 Calib a ed pa ame e s and p io dis ibu ions
Ou objec i e is o es ima e he c i ical pa ame e s go e ning dynamics in bo h he bind-
ing and nonbinding egimes. We calib a e a subse o model pa ame e s lis ed in Table 1
on which we ha e s ong p io in o ma ion om he exis ing li e a u e, including pa -
icula ly Mendoza (2010), who calib a ed hem based on he s ylized ac s o Mexico’s
Na ional Accoun s a an annual equency. Ou model is calib a ed and es ima ed a a
qua e ly equency. The SA p o ides de ails on he calib a ion o hese pa ame e s.
Fo he es ima ed pa ame e s, we se wo ypes o p io . The i s is di ec ly on he
pa ame e s. These p io s a e lis ed in Table 2. They a e ela i ely di use, only impos-
ing sign es ic ions o placing low p io p obabili y on pa ame e alues ha gene a e
implausible momen s in model simula ions. The second ype o p io is on a model-
implied objec . The model has an e godic mean condi ional on he nonbinding egime
ha depends on he model solu ion. This mean is he le el a which he economy s a-
bilizes wi hou any egime changes o shocks. The bo owing cushion associa ed wi h
his mean implies a ansi ion p obabili y o he binding egime in equa ion (13). We se
a p io on his model-objec ha is a Be a dis ibu ion wi h mean 0.01 and a iance o
0.05.15 This p io places e y low p obabili y mass on combina ions o pa ame e s ha
imply oo equen ansi ions o he binding egime. In ui i ely, i e lec s he belie ha ,
in he absence o shocks, he p obabili y ha he cons ain becomes binding is e y low.
Table 1. Calib a ed pa ame e s.
Pa ame e Desc ip ion Value
βDiscoun Fac o 0.99156
ρRisk A e sion 2
ωLabo Supply 1.846
ηCapi al Sha e 0.3053
αLabo Sha e 0.5927
δDep ecia ion Ra e 0.0228
A∗Mean Technology 1.7455
Z∗Mean G ow h 1.006
P∗Mean Impo P ice 1.028
e∗Mean Expendi u e 0.11
15P io s on model-implied objec s a e used and discussed, o example, in O ok (2001) and Del Neg o
and Scho heide (2008).
18 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
Table 2. Es ima ed pa ame e s.
Pos e io
Pa . Desc ip ion P io Mode 5% 50% 95%
¯
In . Ra e Mean N(0.0177, 0.005)0.006 0.005 0.006 0.007
ιCapi al Adj. N(10, 5)5.769 5.169 5.769 5.967
φWo king Cap. U(0, 1)0.769 0.737 0.769 0.799
κ∗Le e age U(0, 1)0.182 0.171 0.182 0.195
logγ0Logis ic, En e Binding U(−20, 20)2.065 2.033 2.065 2.090
logγ1Logis ic, Exi Binding U(−20, 20)4.925 4.900 4.925 4.961
ρaAu oco , TFP B(0.6, 0.2)0.982 0.961 0.982 0.988
ρzAu oco , TFP G ow h B(0.6, 0.2)0.811 0.751 0.811 0.873
ρpAu oco , Imp P ice B(0.6, 0.2)0.978 0.970 0.978 0.986
ρ Au oco , In Ra e B(0.6, 0.2)0.956 0.951 0.956 0.957
ρeAu oco , Expend B(0.6, 0.2)0.879 0.848 0.878 0.907
ρdAu oco , P e B(0.6, 0.2)0.882 0.861 0.882 0.904
σa(L)Low SD, TFP IG(0.005, 0.01)0.005 0.004 0.005 0.005
σa(H)High SD, TFP IG(0.005, 0.01)0.012 0.011 0.012 0.012
σz(L)Low SD, TFP G ow h IG(0.005, 0.01)0.003 0.001 0.003 0.003
σz(H)High SD, TFP G ow h IG(0.005, 0.01)0.010 0.009 0.010 0.010
σp(L)Low SD, Imp P ice IG(0.05, 0.01)0.027 0.025 0.027 0.029
σp(H)High SD, Imp P ice IG(0.05, 0.01)0.063 0.061 0.063 0.065
σ (L)Low SD, In Ra e IG(0.01, 0.025)0.002 0.001 0.002 0.002
σ (H)High SD, In Ra e IG(0.01, 0.025)0.007 0.006 0.007 0.008
σe(L)Low SD, Exp IG(0.5, 0.5)0.160 0.117 0.160 0.194
σe(H)High SD, Exp IG(0.5, 0.5)0.384 0.322 0.384 0.438
σd(L)Low SD, P e IG(0.05, 0.01)0.048 0.039 0.049 0.057
σd(H)High SD, P e IG(0.05, 0.01)0.060 0.053 0.060 0.066
PσlP ob, S ay Low Vol B(0.975, 0.025)0.958 0.937 0.958 0.976
PσhP ob, S ay High Vol B(0.975, 0.025)0.949 0.941 0.949 0.955
No e: This able epo s he p io dis ibu ion and he pos e io momen s o he es ima ed pa ame e s. P io s a e No mal
(N), Uni o m (U), Be a (B), o In e se Gamma (IG) and show mean and a iance, excep o he uni o m dis ibu ion showing
he lowe and uppe bounds. Pos e io dis ibu ions show mode, along wi h 5 h, 50 h, and 95 h pe cen iles o he MCMC
pos e io d aws.
4.3 Es ima ed pa ame e s and model i
We now discuss he es ima ed pa ame e s and he model’s i o he da a. Table 2 epo s
he mode, he median, he 5 h, and he 95 h pe cen ile o he pos e io dis ibu ion. The
pa ame e s ha e igh ly es ima ed pos e io s, so we ocus on he pos e io modes.
The es ima ed mean in e es a e, wi h a mode o 0.6% pe qua e , is highe han he
calib a ed alue in Mendoza (2010). Since we use da a on he in e es a e as an obse -
able, his pa ame e es ima e is di ec ly linked o ha obse able. The obse ed se ies
exceeds ou es ima e o 0.6% o he ea ly pa o he sample, be o e g adually declining
o a alue close o ou es ima e. The es ima ed model in e p e s he sample in e es a e
as a pe sis en p ocess slowly con e ging o om a highe o a lowe mean a e.
The mode o he es ima ed in es men adjus men cos pa ame e (ι) is 5.769. This
pa ame e c i ically in e ac s wi h he inancial ic ion pa ame e s (wo king capi al,
le e age, and logis ic unc ion) o de e mine he dynamics o he model. I s es ima ed
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 19
magni ude canno be compa ed wi h he da a ou side he model. The es ima ed wo k-
ing capi al cons ain pa ame e (φ) is plausible, indica ing ha 77% o he wage and
in e media e goods bill mus be paid in ad ance wi h bo owed unds. This alue is sub-
s an ially highe han he 25.79% calib a ed in Mendoza (2010) bu lowe han he 100%
assumed by Neumeye and Pe i (2005) o he 125% used by U ibe and Yue (2006). The
es ima e is close o he 60% alue es ima ed by A es and Sa ie (2016) using in e es pay-
men s and p oduc ion cos s om Chilean mic oeconomic da a. The es ima ed alue o
he le e age pa ame e in he bo owing cons ain (κ) is 0.18, sugges ing ha less han
a i h o he alue o capi al se es as colla e al. This alue is sligh ly igh e han he
baseline alue o 0.20 calib a ed in Mendoza (2010), and is a he low end o he 0.15–
0.30 ange conside ed in ha s udy.
The pos e io modes o he logis ic pa ame e s in equa ions (13)and(14) a e 2.065
and 4.925 (in log poin s), espec i ely. They a e es ima ed o be in a igh ange ela i e
o he e y loose p io s. Figu e 2illus a es hei implica ions o model dynamics. The
igu e plo s he implied p obabili ies om equa ion (13)and(14), e alua ed a he pos-
e io mode alue o γ0and γ1, oge he wi h he es ima ed e godic dis ibu ions o hei
a gumen s, he bo owing cushion ˜
B∗and he cons ain mul iplie ˜
λ.
Figu e 2. The logis ic unc ions and he dis ibu ions o hei a gumen s. No e: The op panel
shows he model-implied dis ibu ion o he bo owing cushion ˜
B∗in he nonbinding egime,
and he logis ic ansi ion unc ion o he binding egime implied by ou es ima es in o equa ion
(13). The bo om panel shows he model-implied dis ibu ion o he mul iplie λin he binding
egime, and he ansi ion unc ion o he nonbinding egime as implied by ou es ima es in
equa ion (14).
20 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
The op panel o Figu e 2shows ha he bulk o he p obabili y mass is loca ed on
he posi i e side o he e godic suppo o his a iable, as he economy spends mos o
i s ime in he nonbinding egime, abo e he bo owing limi . As he bo owing cushion
declines, he p obabili y o swi ching o he binding egime inc eases e y apidly o
1 o small nega i e alues o ˜
B∗, wi h a negligible p obabili y mass on la ge nega i e
ealiza ions. This implies a ela i ely quick ansi ion in o he binding egime once he
bo owing cushion is exhaus ed as a esul o shocks and agen s’ decisions.
The bo om panel o Figu e 2shows ha once he economy is in he binding egime,
he e godic dis ibu ion o he mul iplie is cen e ed on ze o wi h app oxima ely equal
p obabili y on bo h ails o he suppo . As ˜
λapp oaches 0 om he posi i e side o he
suppo , he p obabili y o swi ching o he nonbinding egime inc eases only g adually,
eaching 1 o λsmalle han −0.02.16 As we no ed ea lie , nega i e alues o λ e lec in-
s ances in which had he economy been in he nonbinding egime, he bo owing cush-
ion would ha e been posi i e, bu a swi ch o he nonbinding egime has no been d awn
ye . The es ima ed alues o he logis ic unc ion pa ame e s imply ha he economy
suddenly en e s a binding s a e bu exi s i only g adually, consis en wi h he e idence
in Ce a and Saxena (2008)andBoissay, Colla d, and Sme s (2016), among o he s.
Tu ning o he exogenous p ocess es ima es, all pe sis en pa ame e s a e es ima ed
p ecisely, wi h hei 95 h pe cen ile alue below one, excep o he empo a y p oduc-
i i y shock, whose au oco ela ion is e y close o one. The es ima ed ola ili ies show
ha he high- and low- ola ili y egimes a e well iden i ied. Howe e , egime change is
a e, pe haps e lec ing he one-o change in he obse able a iable beha io a e he
1994 Tequila c isis, which is dis inc ly isible in Figu e 3.
The obse ables used in he es ima ion a e shown in Figu e 3, oge he wi h he
model-implied smoo hed se ies based on he ull sample pe iod. The igu e also high-
ligh s pe iods o cu ency o ex e nal deb c isis as iden i ied in Reinha and Rogo
(2009), shown in ligh g ay a eas. Since we ha e assumed a small a iance o he mea-
su emen e o s as a sha e o he obse ables’, he model acks he da a closely by con-
s uc ion. Howe e , he acking is consis en h oughou he sample pe iod, du ing
bo h he egula business cycle and he highligh ed c isis pe iods. Fo example, a he
beginning o he 1980s deb c isis and du ing he 1994–1995 equila c isis, he da a show
huge swings in he cu en accoun and la ge d ops and ebounds in ou pu , consump-
ion, and in es men g ow h wi hou losing i . I ins ead one we e o obse e a loss o
i du ing c isis episodes, i would sugges ha he model inds i di icul o ma ch he
da a dynamics du ing hese episodes o c i ical in e es in ou empi ical analysis.17
Likelihood-based es ima ion pe mi s us o eco e he his o ical shock se ies ha
d i e he obse able a iables, which is no possible in a calib a ed model. In o he
wo ds, he es ima ion no only pe mi s o assess which shocks d i e he business cycle
in he model, bu also which shocks we e his o ically mo e impo an in d i ing speci ic
episodes o c ises in Mexico’s his o y.
16Values o λapp oxima ely below −0.2, p oduce a nea ly de e minis ic swi ch back o he binding
egime. The e godic dis ibu ion o λin he binding egime (Figu e 2b) implies ha he p obabili y o exi -
ing ha egime exceeds 99% abou a qua e o he ime.
17See he SA o mo e de ails.
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 21
Figu e 3. Da a and model es ima es. No e: The igu e plo s obse able a iables used in es ima-
ion (solid black lines) and i ed alues (i.e., model-implied smoo hed es ima ed se ies based on
he ull sample, dashed ed lines). Ligh g ay a eas indica e pe iods o cu ency o ex e nal deb
c isis as iden i ied in Reinha and Rogo (2009).
Figu e 4plo s he shocks implied by he es ima ed model in s anda d de ia ion uni s
oge he wi h a wo-s anda d de ia ion band. Because he model acks he obse ed se-
ies closely and e enly o e ime, he eco e ed s uc u al shocks a e in o ma i e abou
22 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
Figu e 4. Model es ima ed shocks. No es: The igu e plo s he es ima ed model implied shocks,
in s anda d de ia ion uni s, oge he wi h a wo-s anda d de ia ion band (black dashed lines).
Ligh g ay a eas indica e pe iods o cu ency o ex e nal deb c isis as iden i ied in Reinha and
Rogo (2009).
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 23
he model mechanisms’ abili y o d i e he ou size mo emen s in he da a du ing c isis
pe iods. I only no mally sized shocks a e needed, hen he model’s in e nal p opaga-
ion mechanisms mus accoun o he abno mal luc ua ions in he da a; al e na i ely,
eliance on la ge, low-p obabili y shocks would cas doub on he model’s abili y o cap-
u e he c isis dynamics. Well-beha ed ealized shocks also p o ide e idence ha sup-
po s ou choice o using he Sigma Poin il e .
As we can see om Figu e 4, he es ima ed model i s he da a wi hou sys ema i-
cally elying on la ge shocks, al hough he e a e wo ins ances in which la ge shocks a e
needed. Fi s , a he beginning o he sample pe iod in 1982:Q3 and Q4, a he peak o
he deb c isis, when Mexico de alued he Peso, decla ed de aul on i s ex e nal deb ,
and na ionalized he banking sys em, e y la ge expendi u e and especially p e e ence
shocks help o ma ch he cu en accoun , which e e ed by abou 12 pe cen age poin s
o GDP by he end o ha yea . Second, an ou size in e es a e shock is needed in
1995:Q1, when ou pu d opped by mo e han 5% in a qua e ( he la ges change in
he sample pe iod), a he end o he Fede al Rese e’s igh ening cycle ha s a ed in
1994:Q1, and a e he peg was abandoned in Decembe 1994. Two la ge empo a y p o-
duc i i y shocks also ma ch wo high g ow h qua e s a e he equila c isis. All o he
shock ealiza ions a e wi hin he wo-s anda d de ia ion band.18
4.4 Model compa isons
As a u he s ep in alida ing ou model, Table 3 epo s he esul s o a model com-
pa ison exe cise using he Schwa z In o ma ion C i e ion (SIC). We compa e ou model
wi h a speci ica ion wi h exogenous egime swi ching and ano he one wi hou he bo -
owing cons ain . Fo each o hese h ee models, we also conside a e sion wi hou
s ochas ic ola ili y. The SA p o ides mo e de ail on hei es ima ion and epo s model-
speci ic pa ame e es ima es. The s a is ic epo ed is he model’s pos e io densi y a
he pos e io mode, adjus ed by he Schwa z In o ma ion C i e ia (SIC) o penalize o
addi ional pa ame e s.19 A di e ence o 10 indica es “s ong” e idence in a o o he
Table 3. Model compa ison—Schwa z in o ma ion c i e ia.
Model Wi h S och Vol No S och Vol
Endogenous Swi ching −4074 −3663
Exogenous Swi ching −4039 −3651
No Cons ain −3842 −3465
18Th ee mo e impo p ice shocks a e ou side he wo-s anda d de ia ion e o bands: in 2008:Q3, when
he oil p ice eached i s his o ical eco d high le el o $150 pe ba el; in 1986:Q1 du ing he I an–I aq Wa
be o e he oil p ice collapse la e in 1986, and in 1990:Q4 and 1991:Q1 due o he 1991 I aq Wa . Howe e ,
impo p ice shocks a e di ec ly implied om he obse able se ies based on commodi y e ms o ade
da a.
19See Liu, Waggone , and Zha (2011) o a discussion o he SIC as a goodness-o - i measu e and he
challenges o compu ing ma ginal da a densi ies in he con ex o egime swi ching models, e en wi hou
conside ing endogenous swi ching as in ou model.
24 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
model wi h he lowe SIC, and a di e ence o 100 indica es ‘decisi e’ e idence (Kass and
Ra e y (1995)).
The SIC esul s in Table 3ha e h ee impo an implica ions abou he i o ou
model. Fi s , o all h ee model speci ica ions, he da a p e e he e sion wi h s ochas-
ic ola ili y. This esul unde sco es he impo ance o conside ing changes in ola ili y
as a d i ing o ce in eme ging ma ke s, as s essed in Fe nandez-Villa e de e al. (2011).
Second, he da a p e e swi ching models wi h a colla e al cons ain ela i e o a e sion
wi hou he colla e al cons ain , whe e inancial ic ions ake he o m o a deb -elas ic
componen on he in e es a e. Finally, he da a p e e he model e sion wi h endoge-
nous swi ching ha we p opose o a adi ional one in which egime swi ches a e exoge-
nous e en s wi h cons an ansi ion p obabili ies. In o he wo ds, he model ha i s
he da a bes is one in which he e is s ochas ic ola ili y, egimes based on he s a e o
he colla e al cons ain (and hence a colla e al cons ain ), and swi ching be ween he
s a es o his cons ain ha is endogenously d i en by le e age a he han exogenous
p obabili ies.
5. The ana omy o Mexico’s business cycles and inancial c ises
In his sec ion, we s udy Mexico’s his o y o business cycles and sudden-s op c ises
h ough he lens o ou es ima ed model. We i s compa e momen s simula ed by he
model and in he da a and assess he ela i e impo ance o di e en shocks in he busi-
ness cycle by means o a a iance decomposi ion. We hen ocus on he model’s i and
he d i e s o Mexico’s his o y o sodden s op c ises.
5.1 Business cycles
Table 4compa es he da a and he simula ed second momen s o he model, epo ing
esul s o ou pu g ow h, consump ion g ow h, in es men g ow h, he in e es a e,
ade balance, and he cu en accoun .20
Table 4. Simula ed second momen s: da a and model.
S d. De . Rela i e S d. De . Co ela ions
Da a Se ies Da a Model Da a Model Da a Model
GDP G ow h 1.41 3.34 1 1 1 1
Cons G ow h 1.76 6.44 1.25 1.93 0.73 0.75
In G ow h 7.57 54.69 5.37 16.38 0.53 0.35
In e es Ra e 1.92 1.35 1.36 0.41 −0.11 −0.04
TB/GDP 2.9 9.5 2.0 2.8 0.14 −0.31
CA/GDP 2.2 9.2 1.56 2.76 −0.1 −0.3
No e: The able compa es he second momen s o he da a ela i e o he momen s simula ed om he model.
20 All model-based s a is ics a e based on simula ed da a om he pos e io mode es ima es. We gene -
a e 100,000 samples o 144 qua e s leng h ( he same as ou da a sample), a e a bu n-in pe iod o 1000
qua e s. We hen compu e he median alues o hese 100,000 uns. We use he p uning me hod in An-
d easen, Fe nandez-Villa e de, and Rubio-Rami ez (2018) o a oid explosi e simula ion pa hs. All epo ed
simula ed momen s a e uncondi ional a he han condi ional on a pa icula egime.
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 31
subjec o
C +I +E =A Kη
−1(Z H )αV1−α−η
−P V −φ (W H +P V )
−1
(1+ )B +B −1. (A.2)
The p e e ence shock ollows
logd =ρdlogd −1+σdεd, . (A.3)
T ansi o y echnology ollows
logA =(1−ρa)log ¯
A+ρalogA −1+σaεa, , (A.4)
while pe manen echnology ollows
logZ =(1−ρz)log ¯
Z+ρzlogZ −1+σzεz, . (A.5)
To al alue added o GDP is gi en by
Y =A Kη
−1(Z H )αV1−α−η
−P V . (A.6)
Capi al accumula es acco ding o
K =(1−δ)K −1+I −ι
2K −kK −1
K −12
K −1, (A.7)
whe e I deno es in es men and k he g ow h a e o capi al along he balanced
g ow h pa h. The expendi u e p ocess E ollows
loge =(1−ρe)log ¯
e+ρeloge −1+σeεe, , (A.8)
whe e
e =E /Z −1. (A.9)
In he binding egime, he colla e al cons ain is gi en by
1
(1+ )B −φ(1+ )(W H +P V )=−κq K , (A.10)
wi h he co esponding mul iplie deno ed λ . In he nonbinding egime, he colla e al
cons ain disappea s, and he mul iplie is λ =0. The cons ain is implemen ed by
de ining
B∗
=1
(1+ )B −φ(1+ )(W H +P V )+κq K (A.11)
and using he egime-swi ching slackness condi ion
ϕ(s )B∗
ss +ν(s )B∗
−B∗
ss=1−ϕ(s )λss +1−ν(s )(λ −λss )(A.12)
32 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
The household- i m maximizes he ollowing Lag angian:
L=E0
∞
=0
d β C −Z −1
Hω
ω1−ρ
−1
1−ρ
+
∞
=0
β μ
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A Kη
−1(Z H )αV1−α−η
−P V −φ (W H +P V )
−1
(1+ )B +B −1
−C −K +(1−δ)K −1−ι
2K −kK −1
K −12
K −1−E
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+
∞
=0
β λ 1
(1+ )B −φ(1+ )(W H +P V )+κq K .
The i s -o de condi ions a e
C :d C −Z −1
Hω
ω−ρ
−μ =0; (A.13)
P :μ (1−α−η)A Kη
−1(Z H )αV−α−η
−P −φ P −λ φ(1+ )P =0; (A.14)
H :⎡
⎢
⎣
−d C −Z −1
Hω
ω−ρ
Z −1Hω−1
+μ αA Kη
−1Zα
Hα−1
V1−α−η
−φ W −λ φ(1+ )W
⎤
⎥
⎦=0; (A.15)
B :−μ
1+
+βμ +1+λ
1+
=0; (A.16)
K :⎡
⎢
⎢
⎢
⎢
⎢
⎣
μ −1−ιK −kK −1
K −1+λ κq
+βμ +1⎛
⎜
⎝
ηA +1Kη−1
(Z +1H +1)αV1−α−η
+1+1−δ
+ιkK +1−kK
K +ι
2K +1−kK
K 2⎞
⎟
⎠
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=0. (A.17)
Combining and simpli ying hese exp essions p oduces
d C −Z −1
Hω
ω−ρ
=μ ; (A.18)
(1−α−η)A Kη
−1(Z H )αV−α−η
=P 1+φ +λ
μ
φ(1+ ); (A.19)
αA Kη
−1Zα
Hα−1
V1−α−η
=φW +λ
μ
(1+ )
+Z −1Hω−1
; (A.20)
μ =λ +β(1+ )E μ +1; (A.21)
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 33
βE μ +1
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ηA +1Kη−1
(Z +1H +1)αV1−α−η
+1
+1−δ
+ιkK +1−kK
K
+ι
2K +1−kK
K 2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=μ 1+ιK −kK −1
K −1
−λ κq . (A.22)
Fac o p ices mus sa is y W =−∂U/∂H
∂U/∂C and q =∂I /∂K , which means
W =Z −1Hω−1
, (A.23)
q =1+ιK −kK −1
K −1. (A.24)
The in e es a e is an exogenous p ocess gi en by
∗
=(1−ρ )¯
∗+ρ ∗
−1+σ ε , , (A.25)
wi h he possibili y o a deb -p emium (only in an al e na i e model speci ica ion, in he
baseline ψ=0) speci ied as ollows:
= ∗
+ψexp(¯
b−B /Z −1)−1. (A.26)
The ex e nal inancing p emium on deb is
EFPD =λ
βE μ +1
. (A.27)
A.2 Compe i i e equilib ium
A compe i i e equilib ium o ou economy is a sequence o quan i ies {K ,B ,C ,H ,V ,
I ,A ,Z ,d ,E ,B∗
}and p ices {P , ∗
, ,q ,w ,μ ,λ } ha , gi en he 5 exogenous p o-
cesses (A.3), (A.4), (A.5), (A.8), (A.25), sa is ies he i s -o de condi ions o he ep esen-
a i e household- i m (A.18)–(A.22), he ma ke p ice equa ions (A.23)–(A.24), he ma -
ke clea ing condi ions (A.2)–(A.7), he deb cushion de ini ion (A.11), egime-swi ching
slackness condi ion (A.12), and he equa ion o he in e es a e (A.26).
Appendix B: Pe u ba ion solu ion me hod
This Appendix p o ides de ails abou wo aspec s o he solu ion me hod: (1) he de ini-
ion o , and solu ion o , he s eady s a e o he endogenous egime-swi ching economy;
and (2) he pe u ba ion me hod ha gene a es second-o de Taylo expansions o he
solu ion o he economy a ound he s eady s a e. Fo no a ional simplici y, we ocus on
a e sion o he model wi hou ola ili y swi ching.
34 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
B.1 Regime swi ching equilib ium
W i e he equilib ium condi ions as
E (y +1,y ,x ,x −1,χε +1,ε ,θ +1,θ )=0. (B.1)
He e, y deno es he non-p ede e mined a iables, x p ede e mined a iables, ε he
exogenous shocks, θ he egime-swi ching pa ame e s, and χ he pe u ba ion pa am-
e e . In gene al, he egime-swi ching pa ame e s a e pa i ioned in o hose ha a ec
he s eady s a e, θ1, ,and hose ha dono ,θ2, .23 In he case o ou speci ic applica ion,
he pa i ion is
θ1, =!ϕ(s )"θ2, =!ν(s )".(B.2)
In o de o sol e he model, we assume he unc ional o ms
θ1, +1=
¯
θ1+χˆ
θ1(s +1),θ1, =
¯
θ1+χˆ
θ1(s ),(B.3)
θ2, +1=θ2(s +1),θ2, =θ2(s ),(B.4)
x =hs (x −1,ε ,χ),(B.5)
y =gs (x −1,ε ,χ),y +1=gs +1(x ,χε +1,χ)(B.6)
and
Ps ,s +1, =πs ,s +1(y ).(B.7)
Now, subs i u ing hese unc ional o ms in he equilib ium condi ions and being mo e
explici abou he expec a ion ope a o , gi en (x −1,ε ,χ)and s ,weha e
Fs (x −1,ε ,χ)=#1
s=0
πs ,sgs (x −1,ε ,χ)
× ⎛
⎜
⎜
⎜
⎜
⎜
⎝
gs +1hs (x −1,ε ,χ),χε,χ,
gs (x −1,ε ,χ),
hs (x −1,ε ,χ),
x −1,χε,ε ,
¯
θ+χˆ
θs,
¯
θ+χˆ
θ(s )
⎞
⎟
⎟
⎟
⎟
⎟
⎠
dμε,(B.8)
whe e dμεdeno es he join pd o he shocks.
Finally, s acking all condi ions by egime yields
F(x −1,ε ,χ)=Fs =0(x −1,ε ,χ)
Fs =1(x −1,ε ,χ)=0. (B.9)
23In he e sion wi h ola ili y swi ches, hese pa ame e s belong o he second se .
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 35
B.2 S eady-s a e de ini ion and solu ion
The model has wo ea u es ha make de ining a s eady s a e challenging. Fi s , as i is
common in a egime-swi ching amewo k, some s uc u al pa ame e s may be swi ch-
ing. In he case o ou applica ion, he e is only one swi ching pa ame e ha a ec s he
s eady s a e, ϕ(s ). None heless, in p inciple, one could allow o egime swi ching also
in he pa ame e s o he exogenous p ocesses, A∗(s )and P∗(s ), o he s uc u al pa-
ame e κ∗(s ), which would a ec he le el o he economy and he s eady-s a e calcu-
la ions. Following Foe s e e al. (2016), we de ine he s eady s a e in e ms o he e godic
means o hese pa ame e s ac oss egimes. To de ine he s eady s a e, we se ε =0and
χ=0, which implies ha he s eady s a e is gi en by
yss,yss,xss,xss,0,0,
¯
θ1,θ2s,
¯
θ1,θ2(s)=0 (B.10)
o all s,s.
In ou case, he ansi ion ma ix e alua ed a s eady-s a e Pss is endogenous, since
i depends on a iables ha in u n depend on he s eady-s a e alue o he ansi ion
ma ix. To ind a solu ion o he s eady s a e (balanced g ow h pa h), we p oceed in wo
s eps. Fi s , we assume he s eady-s a e ansi ion ma ix is known and sol e o all he
s eady- s a e p ices and quan i ies. Second, we use he s eady-s a e alues o he bo -
owing cushion ˜
B∗
ss and mul iplie λss om s ep 1 o upda e he s eady-s a e ansi ion
ma ix. We hen i e a e o con e gence.
S ep 1: Sol e s eady s a e using a gi en s eady-s a e ansi ion ma ix. Fi s , as-
sume ha he s eady-s a e ansi ion ma ix a i e a ion i,P(i)
ss , is known. Nex , le ξ=
[ξ0,ξ1]deno e he e godic ec o o P(i)
ss . Then, as no ed in he pape , de ine he e godic
means o he swi ching pa ame e s as
¯ϕ=ξ0ϕ(0)+ξ1ϕ(1).
The s eady s a e o he egime-swi ching economy depends on hese e godic means, and
we can now sol e o he s eady s a es o all a iables.
S ep 2: Upda ing he ansi ion ma ix. S ep 1 yields he a iables ˜
B∗
ss and λss,and
hence, p o ides a new alue o he ansi ion ma ix o i e a ion i+1:
P(i+1)
ss =p00,ss p01,ss
p10,ss p11,ss=⎡
⎢
⎢
⎢
⎣
1−exp−γ0˜
B∗
ss
1+exp−γ0˜
B∗
ss
exp−γ0˜
B∗
ss
1+exp−γ0˜
B∗
ss
exp(−γ1λss )
1+exp(−γ1λss )1−exp(−γ1λss )
1+exp(−γ1λss)
⎤
⎥
⎥
⎥
⎦
, (B.11)
which can be checked agains he guess in S ep 1. We hen i e a e o con e gence un il
$
$P(i+1)
ss −P(i)
ss $
$< ole ance,
whe e in ou applica ion we use a ole ance o 10−10.
36 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
B.3 Gene a ing app oxima ions
To compu e a second-o de app oxima ion o he endogenous egime-swi ching model
solu ion, we la gely ollow Foe s e e al. (2016), adap ing o he case wi h endogenous
p obabili ies.
We ake he s acked equilib ium condi ions F(x −1,ε ,χ), and di e en ia e wi h e-
spec o (x −1,ε ,χ). The i s -o de de i a i e wi h espec o x −1p oduces a polyno-
mial sys em deno ed
Fx(xss,0,0
)=0. (B.12)
In Foe s e e al. (2016), when he ansi ion p obabili ies a e exogenous and ixed, his
sys em needs o be sol ed ia G öbne bases, which inds all possible solu ions in o -
de o check hem o s abili y. The ele an s abili y concep is mean squa e s abili y
(MSS), which equi es he expec a ion o i s and second momen s o be ini e (see
Cos a, F agoso, and Ma ques (2005)). In ou case wi h endogenous p obabili ies, he
check o MSS is no applicable, so we ocus on inding a single solu ion and igno e
he possibili y o mul iple solu ions o inde e minacy, a common simpli ica ion in he
egime-swi ching li e a u e wi h and wi hou endogenous swi ching (e.g., Fa me , Wag-
gone , and Zha (2011), Foe s e (2015), Maih (2015), Lind (2014)). This simpli ica ion
is also common o global solu ion me hods o models wi h occasionally binding con-
s ain s, whe e nume ical me hods con e ge o a gi en solu ion bu do no gua an ee
uniqueness o ha solu ion. Ins ead, he ocus ypically is on checking obus ness o he
solu ion o ini ial condi ions. While in some simple models wi h colla e al cons ain s i
is possible o impose pa ame ic es ic ions ha ule ou mul iple equilib ia (Schmi -
G ohe and U ibe (2020), Benigno e al. (2016)), in he case o ou model, as in Mendoza
(2010)andBianchi and Mendoza (2018), he e a e no such es ic ions and uniqueness
mus be e i ied nume ically.
To ind a model solu ion, we guess a se o policy unc ions o egime s =1, which
educes he equilib ium condi ions Fx(xss,0,0;s =0) o a ixed- egime eigen alue
p oblem, and sol e o he policy unc ions o s =0. Then, using his ini ial solu ion
as a guess, we sol e o egime s =0 unde he ixed- egime eigen alue p oblem, and
i e a e o con e gence. A e sol ing he i e a i e eigen alue p oblem, he emaining sys-
ems o sol e a e
Fε(xss,0,0
)=0, (B.13)
Fχ(xss,0,0
)=0, (B.14)
and he second-o de sys ems o he o m
Fi,j(xss,0,0
)=0, i,j∈{x,ε,χ}. (B.15)
Recalling now ha he decision ules ha e he o m
x =hs (x −1,ε ,χ), (B.16)
y =gs (x −1,ε ,χ), (B.17)
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 37
he second-o de app oxima ion a e
x ≈xss +H(1)
s S +1
2H(2)
s (S ⊗S ), (B.18)
y ≈yss +G(1)
s S +1
2G(2)
s (S ⊗S )(B.19)
whe e S =[(x −1−xss )ε
1],wi hxss deno ing he alue o he s eady-s a e a iables.
B.4 P oo o P oposi ion 1(p ope ies o he app oxima ed solu ion)
To p o e P oposi ion 1, ake he i s -o de de i a i es o (B.9) wi h espec o i s a gu-
men s, e alua ed a he s eady s a e. This yields
Fx,s (xss,0,0
)=
s
πs ,s,y(yss )gx,s sss,s
+
s
πs ,s(yss ) y +1s,s gx,shx,s + y s,s gx,s
+ x s,s hx,s + x −1s,s , (B.20)
Fε,s (xss,0,0
)=
s
πs ,s,y(yss )gε,s sss,s
+
s
πs ,s(yss ) y +1s,s gx,shε,s + y s,s gε,s
+ x s,s hε,s + ε s,s (B.21)
and
Fχ,s (xss,0,0
)=
s
πs ,s,y(yss )gχ,s sss,s
+
s
πs ,s(yss )⎡
⎢
⎣
y +1s,s gx,shχ,s + y s,s gχ,s
+ x s,s hχ,s
+ θ +1s,s ˆ
θ(s +1)+ θ s,s ˆ
θ(s )
⎤
⎥
⎦. (B.22)
No e now ha , by de ini ion o a s eady s a e, ss(s,s )=0, and so he i s e m o each
o hese exp essions equals ze o. Hence, we a e le wi h he exp essions o he exoge-
nous ansi ion p obabili ies as in Foe s e e al. (2016), gi en by Pss =πs ,s(yss ).
To p o e he second pa o he p oposi ion, namely ha endogenous egime-
swi ching shows up a second o de , i su ices o show ha he second de i a i es o
he sys em wi h espec o x −1a e dependen on he de i a i es o he p obabili y unc-
ions. We include he ull se o he second-o de de i a i es in he SA. Now no e ha he
de i a i es ha e he ollowing o m:
[Fs,xx]a
b,c=A+
s
j
πs,s,yj, gj
s,xbB+
s
j
πs,s,yj, gj
s,xcC, (B.23)
whe e A,B,Ca e exp essions include scala s and unknowns. I is now e iden ha
his equa ion is a unc ion o πs,s,yj, , meaning ha second-o de coe icien s o he
decision- ules a e unc ions o he change in he p obabili ies. QED.
38 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
Appendix C: Solu ion accu acy and compa ison wi h adi ional inequali y
speci ica ion
To gauge he accu acy and speed o ou solu ion me hod and o compa e he endoge-
nous egime swi ching speci ica ion o he bo owing cons ain ela i e o he adi-
ional inequali y one, we compa e a sui ably modi ied and calib a ed e sion o ou
model o Mendoza (2010)sol edwi h heFiPI me hodo Mendoza and Villal azo
(2020).24 To do so, we calib a e ou model a an annual equency and e ain he in e -
es a e, p oduc i i y, and in e media e inpu p ice shocks, as in Mendoza and Villal azo
(2020), and d op he expendi u e and p e e ence shocks and he pe manen p oduc i -
i y shock. We hen compa e Eule equa ion e o s and solu ion speed, simula ed model
i s and second momen s, e godic dis ibu ions, and decision ules o bond holding
and capi al.
In he endogenous egime swi ching model, we calib a e all common model pa am-
e e s as in Mendoza and Villal azo (2020). The logis ic unc ion pa ame e s a e speci ic
o ou model. We se γ0=γ1=30 o ma ch he p obabili y o a posi i e mul iplie on
he bo owing cons ain . Bo h Mendoza (2010)andMendoza and Villal azo (2020)use
ini e-s a e Ma ko p ocesses, while we use disc e e- ime au o eg essi e p ocesses o e
a con inuous suppo . We se hese pa ame e s so ha he uncondi ional momen s o
he wo se s o p ocesses coincide.
Table C.1 epo s he s a is ics on he e o s in he Eule equa ion and he compu a-
ion ime. The able illus a es a s a k ade-o be ween speed and accu acy. Ou me hod
is abou 800 imes as e han he FiPI , wi h log-10 absolu e Eule equa ion e o s ha
a e 1–3 imes la ge han FiPI . The size o ou Eule equa ion e o s is in line wi h he
alues ypically ound when sol ing exogenous egime-swi ching models wi h pe u -
ba ion me hods (Foe s e e al. (2016)) and models wi hou egime swi ching (A uoba,
Fe nandez-Villa e de, and Rubio-Rami ez (2006)). To pu hese numbe s in pe spec i e,
he implied accu acy di e ences ep esen only a dolla e o pe 1000 dolla s o con-
sump ion, which is e y small in absolu e e ms by he s anda ds in he li e a u e.
Table C.2 compa es he i s and second momen s and he p obabili y o a posi i e
mul iplie . To size he di e ences be ween FiPI and ou model, as a e e ence, we also
Table C.1. Solu ion accu acy and speed.
FiPI Mendoza (2010)End.Swi ch.
Eule Equa ion E o s (log10 uni s)
Bond – Mean −6.27 na −2.92
Bond – Max −1.56 na −1.61
Capi al – Mean −7.04 na −3.61
Capi al – Max −6.68 na −2.41
Compu ing Time (seconds) 810 na 1.00
24See Binning and Maih (2017) o an analysis o he p ope ies o ou solu ion me hod applied o o he
s uc u al models, such as he ze o lowe bound, in which hey ound a high deg ee o accu acy.
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 39
Table C.2. Fi s and second momen s.
FiPI Mendoza (2010)End.Swi ch.
Means
gdp 393.619 388.339 393.168
c 274.123 267.857 268.826
in 67.481 65.802 67.136
nx/gdp (%) 1.5 2.4 −0.3
k 765.171 747.709 763.154
b/gdp (%) 1.3 −10.4 −17.9
q111
le (%) −10.3 −15.9 −19.4
42.617 41.949 42.600
wc 76.658 75.455 76.598
S anda d de ia ions (%)
gdp 3.94 3.85 3.81
c 4.03 3.69 3.628
in 13.33 13.45 10.884
nx/gdp 2.94 2.58 1.307
k 4.49 4.31 4.185
b/gdp 19.62 8.9 2.154
q 3.2 3.23 2.568
le 9.22 4.07 8.560
5.89 5.84 5.87
wc 4.35 4.26 4.21
Co ela ions wi h wi h gdp
gdp 1 1 1
c 0.842 0.931 0.984
in 0.641 0.641 0.748
nx/gdp −0.117 −0.184 −0.047
k 0.761 0.744 0.849
b/gdp −0.12 −0.298 −0.172
q 0.387 0.406 0.478
le −0.111 0.258 −0.076
0.832 0.823 0.832
wc 0.994 0.987 0.994
Au oco ela ions
gdp 0.825 0.815 0.815
c 0.83 0.766 0.804
in 0.501 0.483 0.441
nx/gdp 0.601 0.447 0.246
k 0.962 0.963 0.975
b/gdp 0.99 0.087 0.779
q 0.447 0.428 0.350
le 0.992 0.04 0.864
0.777 0.764 0.774
wc 0.801 0.777 0.784
Sudden-s op s a is ics (%)
P ob. posi i e mul iplie 2.6 na 3.615
40 Benigno, Foe s e , O ok, and Rebucci Quan i a i e Economics 16 (2025)
epo he simula ed second momen s om he o iginal Mendoza (2010). The compa -
ison shows ha ou speci ica ion o he occasionally binding bo owing cons ain es-
sen ially p oduces he same i s and second momen s as he FiPI , wi h di e ences ha
a e e y small ela i e o he gaps be ween he FiPI and Mendoza (2010). Two excep ions
a e he ne o eign asse posi ion as a sha e o GDP (b/GDP) and he ne expo - o-GDP
a io (nx/GDP). The e godic mean o b/GDP is abou −17.9% in ou model, while i is
small and posi i e in he FiPI model (1.3%).25 In Mendoza (2010), his simula ed mo-
men is abou −10%. Mexico’s ne o eign asse posi ion a e aged −37% o GDP and was
ne e posi i e om 1970 o 2015. The e o e, ou model gene a es an e godic mean o
deb signi ican ly close o he a e age in he da a compa ed o he FiPI .
The second disc epancy is nx/GDP, which is nega i e −0.3% in ou model, while i
is posi i e 1.5% in he FiPI , and 2.4% in Mendoza (2010). The FiPI gene a es a coun e -
ac ual ne o eign asse posi ion wi h a posi i e ne expo - o-GDP a io ha should be
nega i e in he e godic dis ibu ion i he bond posi ion is posi i e. Ou model gene a es
a da a-consis en deb posi ion wi h a small nega i e e godic ade su plus. In con as ,
in Mendoza (2010), he e godic ade balance and deb p opo ions ha e he opposi e
sign as one would expec . Ou model also gene a es a less ola ile bond posi ion and
ade balance ela i e o FiIP , which in u n gene a es mo e ola ili y han he o iginal
Mendoza (2010). A esul ha could be due o he mechanics o ou speci ica ion o he
bo owing cons ain .
Figu e C.1 plo s he e godic dis ibu ions o deb (panel a) and capi al (panel b), o-
ge he wi h hei join dis ibu ion as in Figu e 1 o Mendoza and Villal azo (2020). The
deb dis ibu ion o ou model has li le o no densi y ou side he [−100, 0] in e al o
he suppo . In con as , he FiPI dis ibu ion has a signi ican p obabili y mass on posi-
i e and la ge alues, which is coun e ac ual as we discussed abo e. Ins ead, he e godic
dis ibu ion o capi al has essen ially he same suppo in he wo models. The join dis-
ibu ion e lec s hese di e ences.
The policy unc ions o ou model sol ed wi h pe u ba ion me hods illus a e i s
abili y o cap u e he occasionally binding na u e o he bo owing cons ain consis en
wi h P oposi ion 1. Figu e C.2 plo s he decision ules o capi al, bonds, and he bo ow-
ing cushion in ou model, in bo h he nonbinding and he binding egimes. These ules
a e e alua ed a he s eady-s a e alues o he echnology, in e es a e, and in e media e
inpu p ice p ocesses, co e ing a ec angula suppo associa ed wi h he join e godic
dis ibu ion o capi al and deb . The e o e, he decision ules p esen ed he e a e condi-
ional on no shocks. Impo an ly, no e he e ha he ac ual beha io o he model also
depends on he speci ic alues o he exogenous s a es. In ou amewo k, endogenous
s a es, exogenous p ocesses, and shocks all ha e con inuous suppo ; while FiPI ’s use
o a disc e ized s a e space means ha a ini e se o s a es a e epea edly isi ed. This
ac is c ucial because he e godic se s a e in luenced by he unde lying p ocesses. In
o he wo ds, he po ions o he dis ibu ion’s suppo ha he economy isi s depend
25All model and solu ion a ian s epo ed in Mendoza and Villal azo (2020) also ha e a posi i e le el o
e godic deb .
Quan i a i e Economics 16 (2025) Es ima ing models o inancial c ises 47
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Co-edi o Mo en O. Ra n handled his manusc ip .
Manusc ip ecei ed 22 No embe , 2021; inal e sion accep ed 8 No embe , 2024; a ailable on-
line 14 No embe , 2024.
The eplica ion package o his pape is a ailable a h ps://doi.o g/10.5281/zenodo.14026657.
The Jou nal checked he da a and codes included in he package o hei abili y o ep oduce
he esul s in he pape and app o ed online appendices.
