Solving Linear DSGE Models with Bernoulli Iterations

Meye -Gohde, Alexande
A icle — Published Ve sion
Sol ing Linea DSGE Models wi h Be noulli I e a ions
Compu a ional Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Meye -Gohde, Alexande (2024) : Sol ing Linea DSGE Models wi h Be noulli
I e a ions, Compu a ional Economics, ISSN 1572-9974, Sp inge US, New Yo k, NY, Vol. 66, Iss. 1, pp.
593-643,
h ps://doi.o g/10.1007/s10614-024-10708-z
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/330775
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by/4.0/
Vol.:(0123456789)
Compu a ional Economics (2025) 66:593–643
h ps://doi.o g/10.1007/s10614-024-10708-z
Sol ing Linea DSGE Models wi hBe noulli I e a ions
Alexande Meye ‑Gohde1
Accep ed: 29 Augus 2024 / Published online: 26 Sep embe 2024
© The Au ho (s) 2024
Abs ac
This pape p esen s and compa es Be noulli i e a i e app oaches o sol ing linea
DSGE models. The me hods a e compa ed using 99 di e en models om he
mac oeconomic model da a base (MMB) and di e en pa ame e iza ions o he
mone a y policy ule in he medium-scale new Keynesian model o Sme s and
Wou e s (Am Econ Re 97(3):586–606, 2007. h ps:// doi. o g/ 10. 1257/ ae . 97.3. 586)
i e a i ely. I ind ha Be noulli me hods p o iding simila accu acy as measu ed
by he o wa d e o o he solu ion a a somewha highe compu a ion bu den o
he s anda d me hod o Dyna e when sol ing DSGE models. The me hod, howe e ,
has con e gence p ope ies use ul when a speci ic solu ion, e.g., unique s able,
is sough and can be combined wi h o he i e a i e me hods, such as he New on
me hod, lending hemsel es especially o e ining solu ions—ei he when s anda d
me hods ail o when one mo es h ough a pa ame e space i e a i ely—as I show in
applica ions o he me hods.
Keywo ds Func ional i e a ion· Nume ical accu acy· DSGE· Solu ion me hods
JEL Classi ica ion C61· C63· E17
1 In oduc ion
Sol ing linea DSGE models can be educed o sol ing a sys em o linea equa ions
and a ma ix quad a ic equa ion wi h he la e being he mo e challenging
compu a ionally. S anda d exis ing me hods p edominan ly ely on a gene alized
Schu o QZ decomposi ion (Mole & S ewa , 1973; Golub & an Loan, 2013)
o sol e his ma ix quad a ic equa ion. Al e na i e me hods om he applied
ma hema ics li e a u e o sol e his ma ix quad a ic equa ion ha e ye o be
sys ema ically s udied in a DSGE con ex . This pape ills pa o ha gap, using
* Alexande Meye -Gohde
meye [email p o ec ed]
1 Goe he-Uni e si ä F ank u andIns i u e o Mone a y andFinancial S abili y (IMFS),
Theodo -W.-Ado no-Pla z 3, 60629F ank u amMain, Ge many
594
A.Meye -Gohde
Be noulli-based solu ion me hods om he ma hema ics li e a u e, in oducing
line-sea ch and hyb id New on based me hods o ma ix quad a ic p oblems and
applying hem o he solu ion o linea DSGE models. The abili y o Be noulli
me hods o be o mula ed such ha hey con e ge o he solu ion wi h he smalles
eigen alues in magni ude (ideally he s able solu ion sough by he esea che ) allow
hem o eliably compu e he s able solu ion, gene ally wi h accu acy on he same
o de o magni ude bu highe compu a ional cos s han QZ-based me hods. When
combined wi h New on me hods and hei asymp o ic quad a ic a e o con e gence,
he i e a i e na u e o he algo i hms also enables he Be noulli based me hods I
in oduce o co ec inaccu a e solu ions and o gene a e solu ions mo e quickly han
QZ-based me hods when i e a i ely explo ing a pa ame e space.
Be noulli’s me hod is o iginally a me hod o compu ing he la ges oo o a
scala polynomial1 and Be noulli-based unc ional i e a ions, amilia o economis s
in such oo - inding se ings—see, e.g., Judd (1998), a e an al e na i e o QZ-based
me hods. The Be noulli algo i hm o ma ix polynomials, see Dennis e al. (1978)
and explici ly o mula ed o minimal sol en o a ma ix quad a ic in Higham and
Kim (2000), was in oduced by Binde and Pesa an (1999) o linea DSGE models
and Rendahl (2017) ex ended he heo e ical con e gence analysis o Higham and
Kim (2000) o po en ially singula (o non monic ma ix polynomials) models ha
a e pe asi e in he DSGE li e a u e. Ye his algo i hm and ecen ex ensions om
he applied ma hema ics li e a u e such as Bai and Gao (2007) ha e no ye been
sys ema ically s udied o hei nume ical p ope ies. This pape does p ecisely
his and in oduces a numbe o algo i hms o sol ing ma ix quad a ic p oblems
ha inco po a e insigh s om o he s udies and algo i hms, including exac line
sea ches om Higham and Kim (2001) and he solu ion o DSGE models using
New on me hods om Meye -Gohde and Saecke (2024). To build in ui ion I begin
Sec .3 wi h he scala case o he algo i hm o build in ui ion be o e in oducing he
mul idimensional algo i hms. I show hese ex ensions imp o e global con e gence
by making he Be noulli me hod as e and mo e eliable, imp o ing on he slow
con e gence (i.e., many i e a ions) o he baseline Be noulli algo i hm. Finally, I
assess he accu acy o he di e en me hods using he p ac ical o wa d e o bounds
o Meye -Gohde (2023), de i ed om a nume ical s abili y app oach inco po a ing
condi ioning numbe and backwa d e o bounds.
In his pape , I p esen ele en di e en Be noulli-based solu ion algo i hms
using a uni ied no a ion and o he applica ion o sol ing linea DSGE models
as an al e na i e o QZ-based me hods. I engage in a numbe o expe imen s
o compa e he algo i hms o QZ-based me hods2 and Meye -Gohde and
Saecke ’s (2024) New on algo i hms, ollowing exac ly hei expe imen s o
ensu e compa abili y. Fi s I apply he di e en me hods o he models in he
1 I can be cas as he powe me hod, see Golub and an Loan (2013,pp.366–367) ope a ing on he
polynomial’s companion ma ix.
2 I use Dyna e’s (Adjemian e al., 2011) implemen a ion o he QZ me hod, documen ed in Villemo
(2011), o compa ison. Va ying implemen a ions o he QZ o gene alized Schu decomposi ion o sol e
linea DSGE models can be ound in Uhlig (1999), Sims (2001), Klein (2000), Al-Sadoon (2018).
595
Sol ing Linea DSGE Models wi hBe noulli I e a ions
Mac oeconomic Model Da a Base (MMB) (see Wieland e  al., 2012, 2016),
compa ing he pe o mance o he QZ-based me hod o Dyna e and he New on
me hod bo h uncondi ionally (i.e., eplacing he QZ me hod) and hen as a
e inemen (i.e., ini ializing he i e a i e me hods wi h he solu ion gene a ed
om QZ). I ind ha he baseline Be noulli me hod always con e ges o he
unique s able solu ion, bu p o ides a solu ion o he same o de o magni ude o
accu acy albei a an o de o magni ude highe compu a ional cos han New on
me hods. The di e en ex ended me hods gene ally pe o m compa ably— he
excep ion being Bai and Gao’s (2007) modi ied Be noulli i e a ion ha is se e al
o de s o magni ude mo e slow and equen ly aces con e gence issues— ading
he con e gence o he baseline algo i hm o agains mo e accu acy and/o less
compu a ional cos s.
The algo i hms a e i e a i e in na u e, enabling hem o e ine he solu ions
p o ided by he ano he me hod. Using he QZ solu ion om Dyna e o ini ialize,
he me hods imp o e he accu acy o he solu ion a an addi ional compu a ional
cos gene ally be ween 0.1 and 2 imes he o iginal cos o Dyna e’s QZ, wi h
con e gence o all o he algo i hms o he unique s able sol en o all o he
models in he MMB, excep Bai and Gao’s (2007) modi ied Be noulli and
columnwise New on–Be noulli combina ions. This i e a i e na u e also lends
i sel o i e a i e pa ame e expe imen s o es ima ions and I compa e he
algo i hms wi h Dyna e’s QZ me hod and New on algo i hm in sol ing o
di e en pa ame e iza ions o he mone a y policy ule in he celeb a ed Sme s
and Wou e s (2007) model o he US economy. Filling in a g id wi h di e en
alues o he eac ion o he nominal in e es a e ule o in la ion and eal
ac i i y, whe eas Dyna e’s QZ me hod s a s anew a each pa ame e iza ion,
i e a i e me hods can use he solu ion om he p e ious, nea by pa ame e iza ion
o ini ialize he algo i hm. As he densi y o he g id inc eases, all o he me hods
e en ually su pass QZ by oughly an o de o magni ude in e ms o compu a ion
cos . Finally, I show ha when Dyna e’s QZ p o ides a nume ically uns able
solu ion wi h high o wa d e o s ha leads he p edic ed momen s o he model’s
a iables o be w ong in all digi s, he Be noulli me hod ini ialized a his solu ion
p o ides a e ined solu ion wi h imp o ed p edic ed momen s and o wa d e o s.
Howe e , he Be noulli algo i hms a e a om being uni o m imp o emen s
o e he exis ing s anda d in he li e a u e and, in pa icula , ou side o si ua ions
whe e an in o ma i e ini ializa ion is a ailable, he s anda d QZ is o be p e e ed.
The pape is o ganized as ollows: Sec .2 lays ou he gene al DSGE model
and he unknown solu ion. In Sec .3, I begin by p esen ing he Be noulli me hod
o a scala quad a ic equa ion o build in ui ion be o e I hen p esen he se o
di e en Be noulli me hods om he applied ma hema ics li e a u e in a uni ied
no a ion as hey apply o he class o DSGE models. Sec ion4 examines p ac ical
and heo e ical conside a ions such as he choice o ini ial alue, accu acy, and
con e gence. In Sec .5, I compa e he di e en Be noulli me hods o he s anda d
QZ me hod and New on me hod o Meye -Gohde and Saecke (2024) in wo
applica ions, one using he MMB o 99 di e en models and he second o e a
ange o pa ame e iza ions wi hin he Sme s and Wou e s (2007) model. Finally,
Sec .6 concludes.
596
A.Meye -Gohde
2 P oblem S a emen
S anda d nume ical DSGE solu ion packages a ailable o economis s and policy
make s—e.g., Dyna e (Adjemian e al., 2011), Gensys (Sims, 2001), (Pe u ba ion)
AIM (Ande son & Moo e, 1985; Ande son e  al., 2006), Uhlig’s Toolki (Uhlig,
1999) and Solab (Klein, 2000)— analyze models ha , when (log)linea ized, can be
exp essed in he o m o he sys em o
ny
linea expec a ional di e ence equa ions
whe e A, B, and C a e
ny×ny
eal alued ma ices,
ℝn
y
→ℝny
, ha ope a e on
y ∈ℝny
he ec o o
ny
endogenous a iables; D is an
ny×ne
eal alued ma ix,
ℝn
e
→ℝny
, ha ope a e on
𝜀 ∈ℝne
he ec o o
ne
exogenous, se ially unco ela ed
shocks wi h a known dis ibu ion, assumed mean ze o3; whe e
ny
and
ne
a e posi i e
in ege s (
ny,ne∈ℕ
).
The solu ion o (1) is sough as he unknown linea solu ion in he o m
a ecu si e solu ion in he ime domain–solu ions ha posi
y
as a unc ion o i s
own pas ,
y −1
, and exogenous inno a ions,
𝜀
, which educes he p oblem o inding
wo eal alued ma ices, P and Q,
ny×ny
and
ny×ne
espec i ely.4
Inse ing (2) in o (1) and aking expec a ions (
E [
𝜀
+1]
=
0
), yields he es ic ions
Gene ally, a unique P wi h eigen alues inside he closed uni ci cle is sough . Lan
and Meye -Gohde (2014) p o e he la e can be uniquely sol ed o Q i such a P
can be ound. Hence, he hu dle is he o me , ma ix quad a ic equa ion.
Mos linea DSGE me hods use a gene alized Schu o QZ decomposi ion (Mole
& S ewa , 1973; Golub & an Loan, 2013) o he companion linea iza ion o (1) in
some o m o ano he . I will ake a di e en ou e and ins ead now sol e o P in (3)
using Be noulli i e a ions.
(1)
0
=AE
[
y
+1]
+By
+Cy
−1
+D𝜀
(2)
y =Py
−1+Q𝜀 ,
ℝ
n
y
+n
e→ℝ
ny
(3)
0=AP2+BP +C,0=(AP +B)Q+D
3 This assump ion ollows Dyna e o expediency a o acili a e he compa ison and implemen a ion
in he applica ions o Sec .5. Any pa e n o se ial co ela ion han can be cap u ed by a ini e o de
ARMA p ocess can b ough in o he abo e o m by expanding
y
app op ia ely. Meye -Gohde and Neu-
ho (2015) show ha he ma ix quad a ic and i s solu ion P is unchanged om he p esen a ion he e by
his assump ion—me ely he mapping om exogenous shocks o endogenous a iables becomes mo e
in ol ed.
4 As abo e his is a b ie s a emen o he linea p oblem, Binde and Pesa an (1997), Uhlig (1999),
Klein (2000), Sims (2001), and Ande son (2010) p o ide o e iews o he mul i a ia e app oaches ha
ha e become s anda d and, mo e ecen ly, Al-Sadoon (2018) and Al-Sadoon (2020) p o ide a much mo e
igo ous app oach ha go beyond he p oblem o sol ing o wo ma ices ha I add ess he e.

597
Sol ing Linea DSGE Models wi hBe noulli I e a ions
3 Be noulli I e a ions o Linea DSGE Models
I will begin by analyzing a uni a ia e equa ion, see, e.g., Higham (2002,p.508) o
he unc ional i e a ion o ind he uns able solu ion and Judd (1992, pp. 152–153)
o unc ional i e a ions in gene al ix poin p oblems o economis s, o ix ideas and
illus a e how Be noulli unc ional i e a ions can be used o sol e quad a ic equa ions.
The p oblem gene a ed by (3) is a (ma ix) quad a ic p oblem. To ix ideas, conside
i s uni a ia e equi alen
whe e I conside (in acco dance wi h he DSGE model), a, b, and c
∈ℝ1
. A
unc ional i e a ion will e o mula e his as
gi ing an i e a i e p ocedu e o gene a e a solu ion
F om he quad a ic equa ion abo e, he e a e a numbe o possibili ies,
and so o h. I will ocus on he i s (x)=−
c
ax+b
as i is he uni a ia e coun e pa
o he mul i a ia e Be noulli algo i hm ha will be in oduced subsequen ly. Hence
Cha ac e ize he wo solu ions ia
whe e he solu ions a e—in analogy o he gene alized eigen alue o mula ion wi h
he gene alized Schu o Klein (2000)
(4)
0=ax2+bx +c
(5)
x= (x)
(6)
x
j+1
=
(
x
j)
, wi h some x
0
(7)
(x)=−
c
ax +b
(8)
(x)=−
b+
c
x
a
(9)
(x)=−
ax
2
+
c
b
(10)
x
k+1=
(
xk
)
=−
c
axk+b
,k=0, 1, 2..., x0
gi en
(11)
(
s1x− 1
)(
s2x− 2
)
=s1s2
⏟⏟⏟
a
x
2
−
(
s1 2+s2 1
)
⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟
+b
x+ 1 2
⏟⏟⏟
c
(12)
x
(i)=
{
i
s
i
, i si≠0; �, i si=0; ℝ, i si= i=0; i=1, 2
}
598
A.Meye -Gohde
Following, e.g., Judd (1998,pp.165–166), local con e gence equi es
| �(x)|<1
o
an x ha sol es
x= (x)
and as
o , say,
x(
1
)
=
1
s
1
and hence he i e a ion is con e gen i
o he e is local con e gence o he minimal (smalle in modulus) oo o he
quad a ic when he i e a ion is o mula ed ia (x)=−
c
ax+b
.
This highligh s a signi ican ad an age o e New on-based me hods, namely ha
he unc ional i e a ion can be ailo ed o deli e con e gence o he algo i hm o a
pa icula oo : o example, (x)=−
c
ax+b
o he minimal (as abo e) and (x)=−
b+
c
x
a
o he dominan (Higham & Kim, 2000) solu ion. As a pa icula solu ion o a
solu ion wi h pa icula p ope ies (namely he minimal solu ion in a saddle poin
s able p oblem) is sough in DSGE models, he Be noulli i e a ion
x
k+1=
(
xk
)
=−
c
axk+b
is po en ially e y use ul in sol ing DSGE models.
Tu ning now o he ma ix p oblem, I will o malize he ma ix quad a ic equa-
ion in (3). Fo A, B, and C
∈ℝn
y
×ny
, a ma ix quad a ic
M(P)∶
ℂ
n
y
×n
y→ℂ
n
y
×ny
is
de ined as
wi h i s solu ions, called sol en s, gi en by
P∈ℂn
y
×ny
i and only i
M(P)=0
. The
eigen alues o he sol en , called la en oo s o he associa ed lambda ma ix,5
M(𝜆)∶ℂ
→
ℂn×n
(he e o deg ee wo), a e gi en ia
The la en oo s a e (i) alues o
𝜆∈ℂ
such ha
de M(𝜆)=0
and (ii)
ny− ank(A)
in ini e oo s. An explici link be ween he quad a ic ma ix p oblem and he
quad a ic eigen alue p oblem is gi en ia
(13)
�(x)=
ac
(ax +b)2=
s
1
s
2
1
2
(
s
1
s
2
x
k
−
(
s
1
2
+s
2
1))
2
(14)
�
(
x(1)
)
=
s
1
s
2
1
2
(
s1s2
1
s1
−
(
s1 2+s2 1
))
2=
s
1
s
2
1
2
(
s1 2
)
2=
s
2
1
s1 2
=x
(1)
x(2)
(15)
|
|
|
�
(
x(1)
)|
|
|
=
|
|
|
|
x
(1)
x
(2)
|
|
|
|
<
1
(16)
M(P)≡AP
2+BP+C
(17)
M(𝜆)
≡
A𝜆2+B𝜆+C
(18)
𝜆
∈ℂ∶
(
A𝜆
2
+B𝜆+C
)
x=0 o some x≠
0
5 See, e.g., Dennis e al. (1976,p.835) o Gan mache (1959, ol.I,p.228).
599
Sol ing Linea DSGE Models wi hBe noulli I e a ions
which has been e iewed ex ensi ely by Tisseu and Mee be gen (2001) and o
which Hamma ling e al. (2013) p o ide a comp ehensi e me hod o imp o e he
accu acy o i s solu ions.
The ma ix quad a ic (16) can be expanded ollowing Higham and Kim (2001) as
whe e
DPM(ΔP)
is he F éche de i a i e o M a P in he di ec ion
ΔP
.
3.1 Baseline Be noulli I e a ion
The mul i a ia e coun e pa o he algo i hm
x
k+1=
(
xk
)
=−
c
ax
k
+
b
abo e is he
ollowing baseline Be noulli me hod o he minimal sol en . Beginning wi h (16)
and de ining he i e a ion ia
gi es he baseline Be noulli me hod
This i e a ion was no ed by Binde and Pesa an (1999) and is implemen ed in
hei RBCQDE Ma lab and Gauss code,6 bu wi hou discussion o i s con e gence
o compu a ional p ope ies. Rendahl (2017) s udies he Be noulli ecu sion in
(25) ollowing Higham and Kim (2000) by analyzing he heo e ical p ope ies
and explo ing a ew examples. The baseline Be noulli ecu sion is summa ized in
Algo i hm1.
(19)
M(P+ΔP)=A(P+ΔP)2+B(P+ΔP)+C
(20)
=AP
2+BP+C+AΔP2+A(PΔP+ΔPP)+BΔP
(21)
=M(P)+(AΔPP +(AP+B)ΔP)+AΔP2
(22)
=M(P)+
D
P(ΔP)+AΔP2
(23)
0=(AP+B)P+C
(24)
0
=
(
AP
j
+B
)
P
j+1
+C
(25)
Pj+1
=−
(
AP
j
+B
)−1C
6 See h ps:// dge. epec. o g/ codes/ binde / handb ook/.
600
A.Meye -Gohde
Algo i hm1: Baseline Be noulli me hod
Higham and Kim (2000) show ha he abo e ecu sion con e ges asymp o ically
a a linea a e o he minimal sol en , bu only unde he assump ion ha his sol-
en is in e ible (e.g., uling ou ze o la en oo s in (17)) and ha a dominan sol-
en also exis s (e.g., uling ou “in ini e” la en oo s in (17)), bo h o which abound
in he DSGE li e a u e. I e u n o his issue in Sec .4 and p o ide a p oo ha his
ecu sion con e ges a leas locally o he minimal sol en when his sol en is he
unique s able solu ion. Bo h a such a sol en and o he i s i e a ion i
P0=0
and
B is o ull ank, see Binde and Pesa an (1997) and Uhlig (1999), he coe icien
ma ix
AP
j+B
will be in e ible.7
3.2 Bai andGao’s (2007) Modi ied Be noulli Me hod
The modi ied Be noulli me hod o Bai and Gao (2007) is a s aigh o wa d
ex ension o he Be noulli me hod ha upda es he solu ion ma ix column by
column alongside he s anda d i e a ion abo e, inco po a ing he upda e o he
p e ious column when sol ing o he cu en column. Bai and Gao (2007) no e ha
he Be noulli ecu sion
sol es n linea equa ions a each i e a ion wi h
Pi,j+1
and
Ci
he i’ h columns o
Pj+1
and C espec i ely
(26)
Pj+1
=−
(
AP
j
+B
)−1C
7 In p ac ice, see he applica ions la e , singula i y is no a gene al issue, bu in he ew i e a ions whe e
i did occu , I se
Pj
+
1
o he minimum 2-no m solu ion o
(
AP
j
+B
)
P
j
+1=−
C
, see Higham and Kim
(2000, p. 512) de e mina ion ha he algo i hm can be e un o eini ialized i nume ical di icul ies
should be encoun e ed.
607
Sol ing Linea DSGE Models wi hBe noulli I e a ions
o he weigh ing abo e could be il ed owa ds one o he o he inc emen
wi h an exponen
p<1
shi ing he weigh in a o o he Be noulli inc emen (and
hence
sj(p)
eplacing
sj
in (54)), pe haps o educe he likelihood o he algo i hm
con e ging o a sol en o he han he unique s able one.
This gi es a combined Be noulli–New on p ocedu e as Algo i hm4.
Algo i hm4: Combined Be noulli–New on
While his p ocedu e will hope ully yield he bes o bo h wo lds: con e gence
o he desi ed sol en ia Be noulli and quad a ic con e gence ia New on, i
migh also do jus he opposi e—combining he linea con e gence o Be noulli o
he unp edic able sol en o New on. Til ing he weigh owa ds one o he o he
p ocedu e gi es he use lexibili y, bu i is s ill no clea a p io i how ha il migh
be chosen o deli e a p ocedu e wi h he desi ed p ope ies.
3.5 Be noulli andNew on Line Sea ch
As he weigh ing o he Be noulli and New on inc emen s elied on he same
in ui ion ha guided hei espec i e line sea ch algo i hms, he nex logical s ep
would be o combine bo h he line sea ches o he op imal magni udes o he
espec i e inc emen s and hen weigh he wo o p oduce a combined Be noulli
and New on inc emen wi h line sea ches. Tha is, i combining Be noulli and
New on migh b ing he slow con e gence o Be noulli oge he wi h he
(62)
cos
𝜃i,j=
Δ
NP
�
i,j
Δ
BPi,j
‖
‖
‖
ΔNPi,j
‖
‖
‖2
‖
‖
‖
ΔBPi,j
‖
‖
‖2
(63)

s
j(p)=s
p
j

608
A.Meye -Gohde
unp edic able con e gence des ina ion o New on, pe o ming line sea ches on
he s eps o bo h o hese migh a enua e his dange . F om abo e, he s ep size
o Be noulli can be inc eased pas 1 o imp o e he speed o con e gence, and he
s ep size o New on can be adjus ed op imally o balance he dange o aking oo
la ge o s eps.
This gi es a combined Be noulli–New on p ocedu e wi h line sea ches as
Algo i hm5.
Algo i hm5: Combined Be noulli–New on wi h line sea ches
As abo e, he unde lying s eps in he p ocedu e would end o limi he d awbacks
o he wo p ocedu es on hei own. Ye , i is no a p io i ce ain how hey will
pe o m when combined.
3.6 Op imal Be noulli andNew on
The inal se o algo i hms explici ly ake he op imali y app oach when de e mining
sj
, minimizing he same me i unc ion used in he line sea ches abo e. Fi s o he
ini ial inc emen s
and hen o he line-sea ch op imized inc emen s
whe e
(64)
s
=a gmin
0≤x≤1
‖
‖
M(xΔBP+(1−x)ΔNP+P)
‖
‖
2
F
(65)
s
=a gmin
0≤x≤1
‖
‖
M(x BΔBP+(1−x) NΔNP+P)
‖
‖
2
F
609
Sol ing Linea DSGE Models wi hBe noulli I e a ions
whe e
XB
es ic s x o be g ea e o equal o one, see abo e, and
XN
es ic s x o be
be ween ze o and wo (Higham & Kim, 2001).
This gi es an op imally (in he sense o he me i unc ion) combined
Be noulli–New on p ocedu e in algo i hm6.
Algo i hm6: Op imal Be noulli–New on
This algo i hm has he ad an age o weigh ing he Be noulli and New on
inc emen s in a non-a bi a y manne . Ye his comes a a cos , he e o an addi ional
op imiza ion p oblem o be sol ed, and is likely biased owa ds he New on
inc emen as i —in ui i ely ia quad a ic con e gence—gene ally akes la ge s eps,
making each inc emen mo e likely o be a o ed o e he imid Be noulli one. This
ca ies again he po en ial o losing he ad an age o Be noulli as o mula ed in he
baseline algo i hm ha gua an ees con e gence o a pa icula sol en .
4 Theo e ical andP ac ical Conside a ions
4.1 Ini ial Value
All Be noulli i e a ions need an ini ial alue,
P0
. In con as o New on me hods, he
baseline Be noulli me hod has s ong con e gence esul s, see he nex subsec ion,
and hence his ini ial alue is o lesse impo ance. Ye many o he algo i hms
p esen ed abo e combine Be noulli and New on me hods and a e subjec o his
conce n. As he goal is o ob ain he minimal sol en P, I choose he ini ial alue
P0=0
. In he absence o any o he guidance, his choice sa is ies he equi emen o
ha ing all eigen alues inside he uni ci cle.
(66)
i=a gmin
x∈Xi
‖
‖
M(xΔiP+P)
‖
‖
2
F, o i=B,
N
610
A.Meye -Gohde
Fu he mo e, Higham and Kim (2000,p.512) no e ha he Be noulli algo i hm
can be e un o es a ed wi h a di e en ini ializa ion should nume ical di icul ies
be encoun e ed. Fo he baseline Be noulli i e a ion which sol es
he (nea ) singula i y o
AP
j+B
would pose such a di icul y. In his case, he ank
de iciency o he leading coe icien ma ix admi s mul iple solu ions and I chose he
no m (
min ‖
‖
‖(
AP
j+B
)
Pj+1+C
‖
‖
‖F
) minimizing
Pj+1
=−
(
AP
j
+B
)+C
, whe e
+
indica es he Moo e-Pen ose in e se.
4.2 Con e gence
While Higham and Kim (2000) and Bai and Gao (2007) p o ide con e gence
esul s o Be noulli i e a ions, hei analyses assume ha A is nonsingula and
ha bo h a minimal and a dominan sol en exis . This is un enable in DSGE
models whe e singula A’s abound—associa ed wi h a iables ha a ise only a
ime and
−1
—and singula C’s—associa ed wi h a iables ha a ise only a
ime
+1
and —p e en he applica ion o hei esul s o he e e se quad a ic.
Rendahl (2017) adap s Higham and Kim’s (2000) app oach o DSGE models and
p o ides p oo s o he global con e gence o he Be noulli i e a ion o a bi a ily
small pe u ba ions o an o iginally singula model ha p oduce a nonsingula
sys em. Complemen ing his, I p o ide a local con e gence p oo o he o iginal
unpe u bed p oblem in he ollowing.
Recall he baseline Be noulli me hod,
he F éche de i a i e o F a
Pj
in he di ec ion
ΔPj
is
D
P
j
F
(
ΔPj
)
and be de ined
implici y by di e en ia ing
(
AP
j
+B
)
F
(
P
j)
=−
C
using he K onecke / ec o ized ep esen a ion (
ec (ABC)=(C�⊗A) ec (B)
)
he algo i hm con e ges (locally) o he minimal sol en .
(67)
(
AP
j
+B
)
P
j+1
=−
C
(68)
Pj+1
=−
(
AP
j
+B
)−1
C=F
(
P
j)
(69)
A
ΔPjF
(
Pj
)
+
(
AP
j+B
)
DP
j
F
(
ΔPj
)
=
0
(70)
D
P
j
F
(
ΔPj
)
=−
(
AP
j+B
)−1
AΔPjF
(
Pj
)
(71)
D
P
j
F
(
ΔPj
)
=
(
AP
j+B
)−1
AΔPj
(
AP
j+B
)−1
C
(72)
ec
(DPjF
(
ΔPj
)
)=
([(
AP
j+B
)
−1C
]
�
⊗
[(
AP
j+B
)
−1A
])
ec (ΔPj
)
611
Sol ing Linea DSGE Models wi hBe noulli I e a ions
Theo em 1 (Con e gence o he unique s able sol en P) Assume he e exis s a
unique sol en P o
M(P)≡AP
2+BP+C
in (16) such ha he eigen alues o P
comp ise all he la en oo s,
𝜆
o
M(𝜆)
—see 17—s able wi h espec o he closed
uni ci cle, i.e., less han o equal o one in absolu e alue. Then he Be noulli i e a-
ion
Pj+1
=−
(
AP
j
+B
)−1C
is s able in he neighbo hood o P.
P oo See he “Appendix”.
◻
The condi ions o he exis ence o he unique s able sol en P a e Blancha d
and Kahn’s (1980) celeb a ed ank and o de condi ions, see Lan and Meye -Gohde
(2014) and Meye -Gohde (2023) o he condi ions exp essed in e ms o he gene al
class o mul i a ia e singula leading A models pe asi e in he li e a u e oday. So
condi ional on i s exis ence, he Be noulli me hod abo e will con e ge asymp o ially
o P.
Equally in e es ing is he pe o mance in he absence o a unique s able sol en P,
ei he in he case o nonexis ence o , especially, inde e minacy. In he la e case, he
model pe mi s an en i e con inuum o solu ions and some s udies, such as Lubik and
Scho heide (2003), analyze he implica ions o he model in such a case. Bo h he
esul s he e, heo em1, and Higham and Kim’s (2000) app oach in he absence o
singula coe icien ma ices, show ha he Be noulli me hod con e ges o he mini-
mal sol en . Tha is, ha sol en P o
0=M(P)≡AP
2+BP+C
in such ha he
eigen alues o P he smalles la en oo s,
𝜆
o
M(𝜆)
, ha is, wi h he smalles pos-
sible spec al adius. In he case o inde e minacy, mo e la en oo s han he dimen-
sion o he p oblem (i.e., mo e han
ny
he leng h o he ec o
y
and he numbe o
equa ions in ), he Be noulli me hod will e u n ha s able solu ion wi h he small-
es oo s, lea ing he la ges s able oo (s) ou o he solu ion. Fo a non exis ence,
he Be noulli me hod will e u n ha solu ion wi h he smalles (in absolu e alue)
oo s again, bu some o hem wi h be ou side he uni ci cle and hence he solu-
ion e u ned will be uns able. This ollows locally om heo em1 as he c ossp od-
uc s o he
ny
smalles la es oo s wi h in e ses o he
ny
la ges la es oo s will be
bounded abo e by one gene ically.8
While he con e gence o a speci ic sol en (in his o mula ion, he unique
s able one) is an ad an age o e New on me hods, which canno gua an ee
con e gence o a pa icula sol en (see Higham & Kim, 2001), Be noulli me hods
con e ge only a a linea a e (gi en abo e by he a io o he la ges s able
and smalles uns able eigen alues) ins ead o New on me hods’ quad a ic a e.
In p ac ice, I ollow Higham and Kim (2001) and use he ela i e esidual
�
�
�
M(Pj)
�
�
�F
∕
�‖
A
‖
F
�
�
�
P2
j
�
�
�F
+
‖
B
‖
F
�
�
�
Pj
�
�
�F
+
‖
C
‖
F
�
<ny
𝜖
o assess whe he con e -
gence has occu ed.
8 The excep ion being i eigen alues o iden ical magni ude a e included in bo h se s. Limi ing he pe -
missible cases o
�𝜆
1
�
≤…≤
‖𝜆
n
y
�<�𝜆
n
y
+1≤
‖𝜆
2n
y
�
ules ou his excep ional case.
612
A.Meye -Gohde
4.3 Accu acy
The p ac ical o wa d e o bounds o Meye -Gohde (2023) can be used o assess he
accu acy o a compu ed solu ion

P
whe e
R
P=
A

P2
+
B

P
+C
is he esidual o he ma ix quad a ic and
H

P=In
y
⊗
(
A

P+B
)
+

P�⊗
A
. S ewa ’s (1971) sepa a ion unc ion, see also Kåg-
s öm (1994), Kågs öm and Po omaa (1996), and Chen and L (2018), is
whe e
𝜆(
A,A

P+B
)
is he spec um o se o (gene alized) eigen alues o he pencil
(
A,A

P+B
)
(and, acco dingly,
𝜆(
P
)
he se o eigen alues o

P
) and he las line
holds wi h equali y o
A=I
and

P
and

P+B
egula —hence, he sepa a ion
be ween he wo pencils— he smalles singula alue o
H
P
—is gene ically smalle
han he minimal sepa a ion be ween hei spec a. Analogously o he gene alized
Syl es e and algeb aic Ricca i equa ions, he sepa a ion unc ion p o ides he
na u al ex ension o he condi ioning numbe om s anda d linea equa ions o hese
s uc u ed p oblems, and he a pos e io i condi ion numbe o he ma ix quad a ic
is gi en by
sep
−1
[(
A,A
P+B
)
,
(
I,−
P
)]
=
‖
‖
‖
H−1

P‖
‖
‖2
=𝜎min
(
H
P
)−1
, which— om
abo e—can be a bi a ily la ge han he in e se o he minimal dis ance be ween
he spec a o he pencils
(
A,A

P+B
)
,
(
I,−

P
)
. This in e se o he sepa a ion ela es
an uppe bound o he o wa d e o di ec ly o he esidual, like he condi ion
numbe o a s anda d linea sys em, and a igh e bound akes in o accoun he
s uc u e mo e ca e ully and conside s he linea ope a o
H
P
and he esidual
R
P
join ly.
5 Applica ions
I conduc wo se s o expe imen s o assess he pe o mance o he Be noulli
algo i hms p esen ed abo e. These wo se s a e chosen o assess he di e en
me hods in a speci ic, policy ele an model bu also in as non-model speci ic an
(73)
�
�
�P−
P
�
�
�F
‖
P
‖
F
�������
Fo wa d E o
≤
�
�
�H−1

P
ec
�
R
P
��
�
�2
�
�
�

P
�
�
�
F
�������������������
Fo wa d E o Bound 1
≤�
�
�
H−1

P�
�
�
2
�
�R
P�
�F
�
�
�

P
�
�
�
F
���������������
Fo wa d E o Bound 2
(74)
sep��
A,A

P+B
�
,
�
I,−

P
��
=min
‖X‖F=1�
�
�
AX

P+
�
A

P+B
�
X
�
�
�
F
(75)
=
min
‖ ec(X)‖2=1
�
�
H
P ec(X)
�
�2
(76)
=
𝜎min
(
H
P
)
≤min
|
|
|
𝜆
(
A,A
P+B
)
−𝜆
(

P
)|
|
|

613
Sol ing Linea DSGE Models wi hBe noulli I e a ions
en i onmen as possible. This is in con as o much o he exis ing li e a u e ha
compa es al e na e solu ion in a single model, equen ly a s ochas ic g ow h o eal
business cycle model, such as Taylo and Uhlig (1990), A uoba e  al. (2006), o
Calda a e al. (2012) o in a ew s ylis ic models and in Rendahl (2017). To mo e
sys ema ically asses he di e en Be noulli based algo i hms om abo e, I compa e
hese algo i hms wi h Dyna e’s QZ implemen a ion9 and he baseline New on
algo i hm om Meye -Gohde and Saecke (2024).10 bo h in he model o Sme s
and Wou e s (2007) and on he sui e o models in he Mac oeconomic Model Da a
Base (MMB) (see Wieland e  al., 2012, 2016), a model compa ison ini ia i e a
he Ins i u e o Mone a y and Financial S abili y (IMFS),.11 The pe o mance is
measu ed in e ms o accu acy, compu a ional ime, and con e gence o he s able
sol en , ini ializing bo h om ze o ma ix (an unin o med ini ializa ion o a s able
sol en ) and he ou pu om he QZ algo i hm.
5.1 Sme s andWou e s’s (2007) Model
I begin wi h he medium scale, es ima ed model o Sme s and Wou e s (2007) ha
is a guably he benchma k o policy analysis. In hei model hey analyze and
es ima e a New Keynesian DSGE model wi h US da a ea u ing he usual ic ions,
s icky p ices and wages, in la ion indexa ion, consump ion habi o ma ion as well
as p oduc ion ic ions conce ning in es men , capi al and ixed cos s. Among he
equa ions is he ollowing log-linea ized mone a y policy ule ha will be he ocus
o he inal expe imen , assessing he accu acy o he me hods he e when sol ing
unde al e na e, bu nea by pa ame e iza ions.
This Taylo ule se s he in e es a e
acco ding o in la ion
𝜋
, he cu en ou pu
gap
(
y
−y
p
)
and he change in he ou pu gap, wi h he pa ame e s
𝜋
,
Y
and
Δy
desc ibing he s eng h o each o hese eac ions and
𝜌
con olling he deg ee o
in e es a e smoo hing. The mone a y policy shock,
𝜀
, ollows an AR(1)-p ocess
wi h an iid no mal e o . The Bayesian es ima ion o he model employs se en
mac oeconomic ime se ies om he US economy o es ima e he model pa ame e s
and he au ho s show ha he model ma ches he US mac oeconomic da a e y
closely and ha ou -o -sample o ecas ing pe o mance is a o able compa ed o (B)
VAR models.
(77)
=
𝜌
−1
+(1−
𝜌
)(
𝜋𝜋
+
Y
(y
−y
p
)) +
Δy
((y
−y
p
)−(y
−1
−y
p
−1
)) +
𝜀
,
9 A u he de elopmen o Klein (2000), see Villemo (2011). No e ha Dyna e uses he eal Schu
decomposi ion as p o ided by LAPACK’s ou ine DGGES, see h ps:// gi . dyna e. o g/ Dyna e/ dyna e/-/
ee/ mas e / mex/ sou c es/ mjdgg es.
10 Addi ionally, no e ha I ollow Dyna e and educe he dimensionali y o he p oblem by g ouping a -
iables and s uc u ing he ma ix quad a ic acco ding o he classi ica ion o “s a ic”, “pu ely o wa d”,
“pu ely backwa d looking”, and “mixed” a iables. The de ails a e in he appendix and Meye -Gohde
and Saecke (2024).
11 See h p:// www. mac o model base. com
614
A.Meye -Gohde
Table1 summa izes he esul s a he pos e io mode calib a ion o he model o
Sme s and Wou e s (2007). The baseline Be noulli me hod akes 12 imes longe
han Dyna e’s QZ, which would appea o pu i a a disad an age compa ed o he
baseline New on, howe e he la ge maximal absolu e di e ence o he QZ solu-
ion o he New on algo i hm shows ha i has con e ged o a di e en solu ion
han he unique s able solu ion ound by Dyna e’s QZ. Indeed, his dange looms
wi h New on ela ed algo i hms as can be seen he e o he New on–Be noulli and
New on–Be noulli op imal algo i hms, bo h o which also con e ged o a di e en
sol en . The baseline Be noulli equi ed 440 i e a ions and line sea ches educed
his numbe o 420, bu he educ ion in i e a ions was ou weighed by he cos liness
o he line sea ch algo i hm, esul ing al oge he in a longe compu a ion ime. The
columnwise New on–Be noulli and he modi ied Be noulli bo h ook ex ao dina ily
long imes o sol e he model, wi h he New on–Be noulli wi h line sea ches and
op imal New on–Be noulli wi h line sea ches p o iding solu ions wi hin an o de o
magni ude o compu a ion ime ela i e o Dyna e’s QZ, 33 and 19 i e a ions espec-
i ely o do so, and p o iding solu ions ha a e an o de o magni ude mo e accu a e
han QZ.
Table 2 assesses he di e en me hods as solu ion e inemen echniques,
by pa ame e izing he model o Sme s and Wou e s (2007) wi hin he p io o
Table 1 Resul s: model o Sme s and Wou e s (2007), pos e io mode
Fo Dyna e, e e o Adjemian e al. (2011)
Run Time o Dyna e in seconds, o all o he s, un ime ela i e o Dyna e
Max Abs. Di . measu es he la ges absolu e di e ence in he compu ed P o each me hod om he P
p oduced by Dyna e
Fo wa d e o 1 and 2 a e he uppe bounds o he ue o wa d e o , see (73)
Me hod Rela i e pe o mance Fo wa d e o s I e a ions
Run ime Max Abs. Di Bound 1 Bound 2
Dyna e (QZ) 0.00063 5.5e−14 2.4e−11 1
Baseline New on 2.4 108 4.1e−14 1.6e−09 11
Baseline Be noulli 12 7.7e−13 3.8e−14 2.6e−11 436
Modi ied Be noulli (MBI) 132 8.8e−13 3.5e−14 2.5e−11 423
Be noulli wi h Line Sea ches 20 6.9e−13 3.7e−14 2.4e−11 423
New on–Be noulli 8.9 108 2.3e−14 1.5e−09 39
New on–Be noulli Column 611 6.9e−13 0.49 3.8e+06 2385
New on–Be noulli 1/3 11 6.8e−13 4.7e−15 4.1e−12 47
New on–Be noulli LS 8.5 6.7e−13 1.1e−14 2.2e−12 33
New on–Be noulli Column LS 9.3 7.2e−13 5.1e−15 9.6e−12 32
New on–Be noulli LS 1/3 15 5.9e−13 1.1e−14 5.1e−12 57
New on–Be noulli Op 5.6 108 2.3e−14 1e-09 18
New on–Be noulli Op LS 6.5 7.7e−13 1.5e−15 2.2e−12 19
615
Sol ing Linea DSGE Models wi hBe noulli I e a ions
demons a e a nume ical ins abili y whose consequences a e economically ele an ,
wi h momen s o a iables p edic ed by Dyna e’s QZ di e ing in all digi s. The sec-
ond column now displays he a iance o in la ion as p edic ed by he di e en solu-
ion me hods,12 A his pa ame e iza ion, Dyna e’s QZ solu ion p edic s an in la ion
a iance o 0.28. Howe e , e en he lowe o he wo uppe bounds on he o wa d
e o is consis en wi h a nume ical ins abili y being se e al o de s o magni ude
beyond machine p ecision.
In he inal expe imen wi h he model o Sme s and Wou e s (2007), I use algo-
i hms o sol e i e a i ely o di e en pa ame e iza ions o he Taylo ule. The goal
he e is o explo e whe he solu ions om p e ious, nea by pa ame e iza ions can be
used o e icien ly ini ialize he Be noulli me hods simila ly o he expe imen abo e
wi h he QZ solu ion as he ini ial guess. Fo he pa ame e s de e mining he Taylo
ule eac ion o in la ion and he long un eac ion o he ou pu gap, he expe imen
i e a es h ough a g id o
10 ×10
pa ame e alues a ying he size o he in e al
conside ed—se ing
𝜋∈[1.5, 1.5 (1+10−x)]
and
Y∈[0.125, 0.125 (1+10−x)]
,
whe e
x∈ [−1, 8]
(Sme s & Wou e s (2007) calib a e hem o
𝜋
=2.0443
and
Table 2 Resul s: model o Sme s and Wou e s (2007), nume ically p oblema ic pa ame e iza ion
Fo Dyna e, e e o Adjemian e al. (2011)
Run ime o Dyna e in seconds, o all o he s, un ime ela i e o Dyna e
Va iance
𝜋
gi es he associa ed alue o he popula ion o heo e ical a iance o in la ion—no e ha
wo algo i hms did no con e ge o a s able P and hence he a iance could no be calcula ed o hem
Fo wa d e o 1 and 2 a e he uppe bounds o he ue o wa d e o , see (73)
Me hod Rela i e pe o mance Fo wa d e o s I e a ions
Run ime Va iance
𝜋
Bound 1 Bound 2
Dyna e (QZ) 0.0046 0.28 1e-11 4.6 1
Baseline New on 4.3 0.45 2.4e−14 0.00058 4
Baseline Be noulli 3 0.39 3.8e−13 0.018 90
Modi ied Be noulli (MBI) 2814 – 2.9 6.6e+06 5e+04
Be noulli wi h Line Sea ches 504 – 0.99 1.9e+11 5e+04
New on–Be noulli 5.1 0.39 3.3e−15 0.00081 8
New on–Be noulli Column 3.7 0.4 3.7e−14 0.019 5
New on–Be noulli 1/3 1.2 0.46 1.4e−14 0.0027 8
New on–Be noulli LS 7.7 0.39 7.9e−15 0.0023 6
New on–Be noulli Column LS 5.7 0.44 1.7e−14 0.00079 6
New on–Be noulli LS 1/3 1 0.38 3.3e−14 0.0022 8
New on–Be noulli Op 3.7 0.45 2.4e−14 0.00058 4
New on–Be noulli Op LS 4.2 0.5 1.3e−15 0.0011 4
12 See Meye -Gohde (2023) o mo e de ails on he pa ame e iza ion. Sme s and Wou e s (2007) epo a
a iance o in la ion in he en i e sample o 0.62 and 0.55 and 0.25 in wo subsamples.
616
A.Meye -Gohde
Y=0.0882
). The algo i hm i e a es h ough he wo-dimensional g id aking he
solu ion unde he p e ious pa ame e iza ion as he ini ializa ion o he nex i e a-
ion. A dec ease in he spacing be ween he 100 g id poin s hus inc eases he p eci-
sion o he s a ing guess, he solu ion om he p e ious pa ame e iza ion.13
Figu e 1 summa izes he expe imen g aphically. Figu e 1c con i ms a
dec ease in un ime pe g id poin wi h a na owe g id o he i e a i e algo-
i hms he e and an i ele ance o he g id spacing o QZ. As he g id becomes
na owe , he i e a i e Be noulli and New on p ocedu es inc easingly bene i
om s a ing om he solu ion o he p e ious i e a ion as i becomes close o
he unknown solu ion o he cu en i e a ion. The QZ algo i hm does no ope a e
Fig. 1 Fo wa d e o s and compu a ion ime pe g id poin o di e en pa ame e iza ions o he model
by Sme s and Wou e s (2007). Figu e1a, b plo he uppe o wa d e o bounds 1 and 2 agains he g id
size, log10 scale on bo h axes. Figu e1c plo s he compu a ion pe g id poin agains he numbe o g id
poin s, log10 scale on bo h axes
13 Al e na i ely, one could ix he end poin s o he g id and inc ease he numbe o g id poin s— his
would p o ide he same message bu would possess a compu a ionally p ohibi i e numbe o g id poin s
a he na owes spacing I examine he e.
623
Sol ing Linea DSGE Models wi hBe noulli I e a ions
endogenous a iables—and o wa d e o s (bound 1 and bound 2) ela i e o QZ
a e plo ed in log10. The mos s iking esul is he di e ence o accu acy, wi h
algo i hms in ol ing a New on s ep mo e o en below he x axis han he emain-
ing me hods, indica ing ha hey a e gene ally associa ed wi h mo e accu acy—an
obse a ion I e u n o sho ly in a densi y compa ison. The e appea s o be a down-
wa d end, indica ing ha he me hods become mo e accu a e ela i e o Dyna e’s
QZ o la ge models, which is con i med by looking a Fig.4d, which shows he
clea downwa d end o he op imal Be noulli–New on me hod wi h line sea ches
and is exempla y o many o he me hods. Finally, Fig.4c shows ha he e appea s
o be a ela ionship be ween he size o he model and he compu a ion ime ela i e
o Dyna e o a leas he e y la ge models owa ds he igh o he igu e, indica ing
ha he me hods he e a e likely o be pa icula ly compe i i e al e na i es o la ge
scale applica ions, wi h Meye -Gohde and Saecke ’s (2024) New on algo i hm p e-
sen ing he mos con incing e idence in his ega d.
Fig. 6 Dis ibu ion o o wa d e o bounds ela i e o dyna e o he mac oeconomic model da a base
(MMB). Figu e6a, b plo he dis ibu ion o model solu ions agains he uppe bounds o he o wa d
e o 1 and 2 o all algo i hms, log10 scale on he x axis, 99 MMB models (s a ing guess: solu ion
Dyna e(QZ))

624
A.Meye -Gohde
Figu e 5 p o ides an o e iew o he en i e dis ibu ion o o wa d e o s, he
uppe ow ela i e o hose om Dyna e’s QZ me hod and he lowe in absolu e
e ms. Fo wa d e o s le o he e ical line a e hus smalle han Dyna e o
bo h igu es in he uppe ow. Fo bo h he i s , Fig.5a, and second, Fig.5b, uppe
bounds on he o wa d e o , he e is an ob ious shi o he le o abou one o de
o magni ude o all he me hods in ol ing a New on s ep and less isually compel-
ling e idence o Be noulli algo i hms no combined wi h New on (again, he modi-
ied Be noulli algo i hm pe o ms wo s ). F om he lowe ow, his en ails igh ening
he dis ibu ions as well as shi ing hem close o machine p ecision—a lowe con-
e gence c i e ion would allow mo e i e a ions and likely b ing ye mo e solu ions
below machine p ecision.
To assess he po en ial o imp o ing on solu ions, I epea he exe cise, bu now
ini ialize wi h he solu ion p o ided by QZ, see able4. He e he baseline Be noulli
me hod is he op pe o me —wi h i s low pe i e a ion cos , i uns one i e a ion a
a small ac ion o he o iginal Dyna e QZ cos and p o ides a signi ican imp o e-
men in accu acy. The modi ied Be noulli algo i hm again pe o ms unsa is ac o-
ily and, in e es ingly, he combined Be noulli and New on me hods all equi e mo e
han one i e a ion o con e ge, which would seem o imply ha he New on and Be -
noulli s eps indi idually we e gene ally pulling in di e en di ec ions in he icini y
o he solu ion p o ided by Dyna e’s QZ. No e ha he modi ied Be noulli algo i hm
along wi h he New on Be noulli Column algo i hms did no always con e ge—all
h ee o hese algo i hms ope a e column-wise on he p oblem which appa en ly can
in e e e wi h he con e gence, e en when ini ialized close o he solu ion.
Figu e6, like Fig.5 bu now ini ialized a he Dyna e’s QZsolu ion, p o ides an
o e iew o he en i e dis ibu ion o o wa d e o s, he uppe ow ela i e o hose
om Dyna e’s QZ me hod and he lowe in absolu e e ms. Fo wa d e o s le o
he e ical line a e hus smalle han Dyna e’s QZ o bo h igu es in he uppe
ow. Fo bo h he i s , Fig.6a, and second, Fig.6b, uppe bounds on he o wa d
e o , he e is again an ob ious shi o he le o abou one o de o magni ude o
all he me hods in ol ing a New on s ep and a ma ginal a bes imp o emen using
algo i hms wi hou a New on s ep. This is consis en wi h he quad a ic con e gence
p ope ies o New on me hods, see Meye -Gohde and Saecke (2024), when close o
a solu ion. The modi ied Be noulli algo i hm now pe o ms compa ably o he o he
me hods wi hou a New on s ep, highligh ing some applicabili y o he conclusions
o Bai and Gao (2007) o he DSGE con ex .
6 Conclusion
I ha e applied and ex ended Be noulli-based me hods o sol ing he ma ix
quad a ic equa ion unde lying he solu ion o linea DSGE models. This adds a se o
al e na i es alongside Meye -Gohde and Saecke ’s (2024) New on-based and Hube
e al. (2023) doubling algo i hms o he cu en s anda d o a gene alized Schu o
QZ decomposi ion (Mole & S ewa , 1973; Golub & an Loan, 2013). Applying he
625
Sol ing Linea DSGE Models wi hBe noulli I e a ions
me hods o he sui e o models in he Mac oeconomic Model Da a Base (MMB), I
demons a e ha Be noulli-based me hods a e a po en ial al e na i e, wi h a adeo
be ween con e gence o a pa icula solu ion (he e he unique, s able sol en ) and
pe o mance in e ms o compu a ional cos s.
Pa icula ly in i e a i e en i onmen s o when a solu ion e inemen is sough do
hese algo i hms show po en ial o u u e applica ion. In illing in an inc easingly
dense g id o pa ame e iza ions o he Taylo ule in he model o Sme s and Wou -
e s (2007), i e a i e me hods like he Be noulli-based me hods he e can ini ialize
wi h he solu ion om he p e ious pa ame e iza ion and signi ican ly ou pe o m
he cu en gene alized Schu o QZ me hod bo h in e ms o compu a ional cos s
and o wa d e o . Taking he solu ion om QZ as he ini ializa ion, he me hods
p o ide oughly an o de o magni ude imp o emen in he accu acy o he solu-
ion a a ac ion o he o iginal compu a ional cos . This ini ializa ion and i e a-
ion makes applying he se o Be noulli me hods o imp o ing he accu acy o solu-
ions o linea DSGE models a po en ially use ul di ec ion o applica ion, which is
demons a ed he e wi h a p oblema ic pa ame e iza ion o he Sme s and Wou e s
(2007) model as p esen ed by Meye -Gohde (2023). S a ing wi h he QZ solu ion
ha unde p edic s he a iance o in la ion by abou 25% and demons a es la ge
o wa d e o s, he Be noulli me hod e ines his solu ion, p o iding o wa d e o s
se e al o de s o magni ude smalle and a p edic ed a iance o in la ion in line wi h
New on me hods.
Tha being said, he baseline Be noulli algo i hm is no a uni o m imp o emen
o e he s anda d QZ me hod and, in pa icula , o mos models when an in o ma i e
ini ializa ion is no a ailable, he s anda d QZ is o be p e e ed. Fu u e esea ch
migh explo e he applica ion o he me hods he e o educe he compu a ional
bu den associa ed wi h sol ing he model o i e a i e es ima ion p ocedu es and
migh be adap ed o mo e quickly and/o accu a ely pe o m likelihood calcula ions
o sol e he e ogenous agen models.
626
A.Meye -Gohde
Appendix
O e iew
The ollowing low cha isualizes he decision p ocess ha leads o each algo i hm.
The code o eplica e he analysis can be ound a h ps:// gi hub. com/ AlexM
eye - Gohde/ Linea - DSGE- wi h- Be no ulli The di ec o y algo i hm con ains
he algo i hms used he e (wi h he excep ion o Dyna e wi h can be downloaded
a h ps:// www. dyna e. o g/). The ile be noulli_ma ix_quad a ic.m
con ains he Be noulli algo i hms in oduced he e and ou lined by he lowcha
abo e. The di ec o y mmb_expe imen con ains he MMB expe imen s a ing
a he ze o ma ix in Table 3 and he Sme s and Wou e s (2007) compa ison o
Table1 ( he model o Sme s and Wou e s (2007) is among he 99 MMB models
used in he compa ison, hemsel es con ained in he di ec o y mmb_expe imen ).
627
Sol ing Linea DSGE Models wi hBe noulli I e a ions
The di ec o y mmb_expe imen _2 con ains he MMB expe imen s a ing
a he QZ solu ion o Table 4. The nume ically p oblema ic pa ame e iza ion
o Sme s and Wou e s (2007) displayed in Table2 can be ound in he di ec o y
imp o emen _expe imen . Finally he g id size expe imen o Sme s and
Wou e s (2007) o Fig.1 is in he di ec o y policy_expe imen _2.
De ailed Dyna e Va iable Classi ica ion
He e I summa ize he de ails in he ma ix quad a ic ha ollows om he clas-
si ica ion o a iables om Dyna e as laid ou in Villemo (2011). See Meye -
Gohde and Saecke (2024) o de ails.
Subdi iding he sys em o equa ions in acco dance wi h he QR decomposi ion
yields
whe e
nd
is he numbe o dynamic a iables, he sum o numbe o pu ely back-
wa d-looking,
n
−−
, mixed
nm
, and pu ely o wa d-looking a iables,
n++
. The num-
be o o wa d-looking a iables,
n+
, is he sum o he numbe o mixed,
nm
, and
pu ely o wa d-looking a iables,
n++
, and he numbe o backwa d-looking a i-
ables,
n
−
, is he sum o he numbe o pu ely backwa d-looking,
n
−−
and mixed a i-
ables
nm
. Hence, he numbe o endogenous a iables is he sum o he numbe o
s a ic,
ns
, and dynamic a iables,
nd
, o he sum o he numbe o s a ic,
ns
, pu ely
backwa d-looking,
n
−−
, mixed
nm
, and pu ely o wa d-looking a iables,
n++
. The
dimensions sa is y he ollowing
The ansi ion ma ix, P, om (2) ha sol es he ma ix equa ion (16) can be subdi-
ided in acco dance o Dyna e’s ypology as
n
d
=n−− +n
m
+n++
,n+=n
m
+n++
,
n
−=n−− +nm
,
n=ns+nd=ns+n−− +nm+n
++
628
A.Meye -Gohde
The ma ix quad a ic can be exp essed as
Fo a sol en P o he ma ix quad a ic, aking he s uc u e o C om he Dyna e
ypology abo e in o accoun yields
Following Meye -Gohde and Saecke (2024), who apply co olla y 4.5 o Lan and
Meye -Gohde (2014), i P is he unique sol en o M(P) s able wi h espec o he
closed uni ci cle,
𝐆
has ull ank and hence he columns o P associa ed wi h
nonze o columns in C, he s a ic and o wa d-looking a iables a e ze o
→𝐏
∙,s
=𝟎
n×n
s
,𝐏
∙,++
=𝟎
n×n++
, whence
𝐏
is and
𝐌
(𝐏)=
[
𝟎
n×ns
&𝐌(𝐏)−−
n×n
−−
𝐌(𝐏)m
n×n
m
𝟎
n×n++
]
. Consequen ially, he i s
ns
ows o he
ma ix quad a ic, aking as gi en, yield as
𝐌
(𝐏
n×n
)= 𝐀
n×n
𝐏
2
+𝐁
n×n
𝐏+𝐂
n×
n
=
(
𝐀𝐏 +𝐁
)
⏟⏞⏞⏟⏞⏞⏟
≡𝐆
𝐏+𝐂

629
Sol ing Linea DSGE Models wi hBe noulli I e a ions
and he i s
ns
ows o P a e .
The las
nd
columns and ows o P sol e he educed ma ix quad a ic equa ion
Recalling ha
𝐏
∙,++
=𝟎
n×n++
,

𝐏
can be educed and wo subma ices
𝐏
and

𝐏
de ined
ia
630
A.Meye -Gohde
whe e and allow
he ma ix quad a ic o be w i en as
which can be educed o
This leads o he Be noulli i e a ion
631
Sol ing Linea DSGE Models wi hBe noulli I e a ions
De ailed Dyna e Topology‑Line Sea ch
The line sea ch me hods in he ex equi e inding he minimum o he me i unc ion
The i s o de condi ion o an in e io solu ion equi es inding he ze os o he
polynomial
whe e
𝛼
=
‖
M(P)
‖2
F
,
𝛽
=
‖
Ad𝐏⋅P
‖2
F
,
𝛾
=
‖
‖
‖
A(d𝐏)2
‖
‖
‖
2
F
,
𝜉
=
(
(Ad𝐏⋅𝐏)
∗
A(d𝐏)
2+
(
A(d𝐏)2
)
∗Ad𝐏⋅𝐏
)
,
𝜎
=
(
M(P)∗A(d𝐏)2+
(
A(d𝐏)2
)∗
M(P)
)
, and
𝛿=
(M(P)∗
Ad
𝐏⋅𝐏+(Ad𝐏⋅𝐏)
∗
M(P)
)
, which ollows om
‖
M(P+xΔP)
‖2
F=(1−x)
2‖
M(P)
‖2
F+x
2‖
AΔP(xΔP+P)
‖2
F
+(1−x)x∗ (M(P)*AΔP(xΔP+P)
+(xΔP*+xP*)ΔP*A*M(P))
=(1−x)2‖M(P)‖2
F+x2‖AΔP(xΔP+P)‖2
F
+(1−x)x2∗ (M(P)*AΔP2+ΔP*2A*M(P))
+(1−x)x∗ (M(P)*AΔP⋅P+P*ΔP*A*M(P
))
=(1−x)2‖M(P)‖2
F+x4�
�
�AΔP2�
�
�
2
F+x2‖AΔP⋅P‖2
F
+x3∗ (ΔP*2A*AΔP⋅P+P*ΔP*A*AΔP2)
+(1−x)x2∗ (M(P)*AΔP2+ΔP*2A*M(P))
+(1−x)x∗ (M(P)*A𝛿P⋅P+P*ΔP*A*M(P))
≡
g(x)
(A1)
g�(x)=4𝛾x3+3(𝜉−𝜎)x2+2(𝛼+𝛽−𝛿+𝜎)x+𝛿−2𝛼
632
A.Meye -Gohde
Using he ypology om Dyna e and he esul s abo e
g�
(x)=2(1−x)(−1)‖M(P)‖
2
F
+2x‖AΔP⋅P‖2
F+4x3�
�
�AΔP2�
�
�
2
F
+3x2∗ (ΔP*2A*AΔP⋅P+P*ΔP*A*AΔP2)
+ [(1−x)2x+ (−1)x2]∗ (M(P)*AΔP2+ΔP*2A*M(P))
+ [(1−x)x+ (−1)x]∗ (M(P)*AΔP⋅P+P*ΔP*A*M(P))
=−2‖M(P)‖2
F+2‖M(P)‖2
Fx+2‖AΔP⋅P‖2
Fx+4�
�
�AΔP2�
�
�
2
Fx3
+3∗ (ΔP*2A*AΔP⋅P+P*ΔP*A*AΔP2)x2
+ (M(P)*AΔP⋅P+P*ΔP*A*M(P)) − 2∗ (M(P)*AΔP⋅P+P*ΔP*A*M(P))
x
+2∗ (M(P)*AΔP2+ΔP*2A*M(P))x
−3∗ (M(P)*AΔP2+ΔP*2A*M(P))x2
=4�
�
�AΔP2�
�
�
2
Fx3+(3∗ (ΔP*2A*AΔP⋅P+P*ΔP*A*AΔP2)
−3 (M(P)*AΔP2+ΔP*2A*M(P)))x2
+2‖M(P)‖2
Fx+2‖AΔP⋅P‖2
Fx
−2∗ (M(P)*AΔP⋅P+P*ΔP*A*M(P))x
+2∗ (M(P)*AΔP2+ΔP2*A*M(P))x
+ (M(P)*AΔP⋅P+P*ΔP*A*M(P)) − 2‖M(P)‖2
F
639
Sol ing Linea DSGE Models wi hBe noulli I e a ions
Unsing he Dyna e ypology abo e
‖
𝐌(⋅)‖
2
F=‖𝐚‖
2
F+s( (𝐚
∗
⋅𝐛)+ (𝐛
∗
⋅𝐚))+s
2
⋅
�
‖𝐛‖
2
F+ (𝐚
∗
𝐜)+ (𝐜
∗
𝐚)
�
+s3( (𝐛∗⋅𝐜)+ (𝐜∗⋅𝐛))+s4
⋅‖𝐜‖2
F
=‖𝐚‖2
F+s⋅2⋅ (𝐚∗𝐛)+s2
⋅�‖𝐛‖2
F+2 (𝐚∗𝐜)�+s3
⋅2 (𝐛∗⋅𝐜)+s4
⋅‖𝐜‖2
F
≡𝐭(s)
𝐭�(s)=2⋅ (𝐚∗𝐛)+2⋅(
‖
𝐛
‖
2
F
+2⋅ (𝐚∗𝐜)⋅s+6⋅ (𝐛∗⋅𝐜)⋅s2+4⋅
‖
𝐜
‖
2
F
⋅s3

640
A.Meye -Gohde
P oo o Theo em1
Following Higham (2008,Sec ion4.9.4.), local s abili y equi es he F éche de i a i e
o F a P o ha e bounded powe s, which holds i i s spec al adius is less han one. A
P, (72) is
Hence he spec al adius is equal o he eigen alue o
P�
⊗
[
(AP+B)
−1
A
]
wi h he
la ges magni ude. As he eigen alues o he K onecke p oduc o wo ma ices is
he se o all c ossp oduc s o he eigen alues o he espec i e ma ices, he spec-
al adius is he p oduc o he eigen alues o P and
(AP+B)−1A
wi h he la ges
magni udes. Following Lan and Meye -Gohde (2014,Co olla y 4.2.), (17) can be
ac o ed as
Hence, he la en oo s associa ed wi h
A𝜆+AP+B
a e hose oo s o
M(𝜆)
no con-
ained in se o eigen alues o P. By assump ion, he eigen alues o P a e inside he
closed uni ci cle and he oo s associa ed wi h
A𝜆+AP+B
a e ou side he open uni
ci cle. By inspec ion, he eigen alues o
(AP+B)−1A
a e he in e ses o he oo s
associa ed wi h
A𝜆+AP+B
and, he e o e, a e all inside he open uni ci cle. Hence
he p oduc o he la ges magni ude eigen alue o P and ha o
(AP+B)−1A
is less
han one in absolu e alue. Tha is, he spec al adius o he F éche de i a i e o F a
P, he unique bounded solu ion, is less han one, comple ing he p oo .
Acknowledgemen s I am g a e ul o Johanna Saecke , Pablo Winan , and pa icipan s o he 29 h In e -
na ional Con e ence on Compu ing in Economics and Finance (2023 CEF) o use ul commen s and sug-
ges ions and o Elena Schlipphack and Maximilian Thomin o in aluable esea ch assis ance. The code o
eplica e he analysis can be ound a h ps:// gi hub. com/ AlexM eye - Gohde/ Linea - DSGE- wi h- Be no ulli.
Any and all e o s a e en i ely my own.
Funding Open Access unding enabled and o ganized by P ojek DEAL. This esea ch was suppo ed
by he DFG h ough g an no. 465469938 “Nume ical diagnos ics and imp o emen s o he solu ion o
linea dynamic mac oeconomic models”.
Decla a ions
Compe ing In e es s The au ho s ha e no disclosed any compe ing in e es s.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License,
which pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as
you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Com-
mons licence, and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle
a e included in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he
ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is
no pe mi ed by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission
(A2)
ec
(DPF(ΔP))=
([
(AP+B)−1C
]�
⊗
[
(AP+B)−1A
])
ec (ΔP
)
(A3)
=(
P
�
⊗
[
(AP+B)
−1
A
])
ec (ΔP
)
(A4)
M(𝜆)
≡
A𝜆2+B𝜆+C=(A𝜆+AP+B)(𝜆−P)
641
Sol ing Linea DSGE Models wi hBe noulli I e a ions
di ec ly om he copy igh holde . To iew a copy o his licence, isi h p://c ea i ecommons.o g/
licenses/by/4.0/.
Re e ences
Adjemian, S., Bas ani, H., Juilla d, M., Mihoubi, F., Pe endia, G., Ra o, M., & Villemo , S. (2011).
Dyna e: Re e ence manual, Ve sion 4. Dyna e Wo king Pape s1. CEPREMAP.
Al-Sadoon, M. M. (2018). The linea sys ems app oach o linea a ional expec a ions models. Econome -
ic Theo y, 34(3), 628–658. h ps:// doi. o g/ 10. 1017/ S0266 46661 70001 60
Al-Sadoon, M. M. (2020). Regula ized solu ions o linea a ional expec a ions models. Discussion
pape .
Ande son, G. S. (2010). A eliable and compu a ionally e icien algo i hm o imposing he saddle poin p op-
e y in dynamic models. Jou nal o Economic Dynamics and Con ol, 34(3), 472–489. h ps:// doi. o g/ 10.
1016/ S1474- 6670(17) 40506-4
Ande son, G.S., Le in, A., & Swanson, E. (2006). Highe -o de pe u ba ion solu ions o dynamic disc e e-
ime a ional expec a ions models. Discussion Pape 2006-01. Fede al Rese e Bank o San F ancisco
Wo king Pape Se ies.
Ande son, G. S., & Moo e, G. (1985). A linea algeb aic p ocedu e o sol ing linea pe ec o esigh models.
Economics Le e s, 17(3), 247–252. h ps:// doi. o g/ 10. 1016/ 0165- 1765(85) 90211-3
A uoba, S. B., Fe nández-Villa e de, J., & Rubio-Ramí ez, J. F. (2006). Compa ing solu ion me hods o
dynamic equilib ium economies. Jou nal o Economic Dynamics and Con ol, 30(12), 2477–2508.
h ps:// doi. o g/ 10. 1016/j. jedc. 2005. 07. 008
Bai, Z.-Z., & Gao, Y.-H. (2007). Modi ied Be noulli i e a ion me hods o quad a ic ma ix equa ion. Jou nal
o Compu a ional Ma hema ics, 25(5), 498–511.
Binde , M., & Pesa an, M. H. (1997). Mul i a ia e linea a ional expec a ions models: Cha ac e iza ion o he
na u e o he solu ions and hei ully ecu si e compu a ion. Econome ic Theo y, 13(6), 877–88. h ps://
doi. o g/ 10. 1017/ S0266 46660 00063 07
Binde , M., & Pesa an, M. H. (1999). Mul i a ia e a ional expec a ions models and mac oeconome ic mod-
eling: A e iew and some new esul s. Mac oeconomics. In M. H. Pesa an & M. Wickens (Eds.), Hand-
book o applied econome ics (3 d ed., Vol. 1, pp. 111–155). London: Wiley. h ps:// doi. o g/ 10. 1111/b.
97806 31215 585. 1999. 00004.x
Blancha d, O. J., & Kahn, C. M. (1980). The solu ion o linea di e ence models unde a ional expec a ions.
Econome ica, 48(5), 1305–1311. h ps:// doi. o g/ 10. 2307/ 19121 86
Calda a, D., Fe nández-Villa e de, J., Rubio-Ramí ez, J., & Yao, W. (2012). Compu ing DSGE models wi h
ecu si e p e e ences and s ochas ic ola ili y. Re iew o Economic Dynamics, 15(2), 188–206. h ps://
doi. o g/ 10. 1016/j. ed. 2011. 10. 001
Chen, X. S., & L , P. (2018). On es ima ing he sepa a ion be ween (A, B) and (C, D) associa ed wi h he gen-
e alized Syl es e equa ion
AXD −BXC =E
. Jou nal o Compu a ional and Applied Ma hema ics, 330,
128–140. h ps:// doi. o g/ 10. 1016/j. cam. 2017. 07. 025
Dennis, J. E., J ., T aub, J. F., & Webe , R. P. (1976). The algeb aic heo y o ma ix polynomials. SIAM Jou -
nal on Nume ical Analysis, 13(6), 831–845. h ps:// doi. o g/ 10. 1137/ 071306
Dennis, J. E., J ., T aub, J. F., & Webe , R. P. (1978). Algo i hms o sol en s o ma ix polynomials. SIAM
Jou nal on Nume ical Analysis, 15(3), 523–533. h ps:// doi. o g/ 10. 1137/ 07150 34
Gan mache , F. R. (1959). The heo y o ma ices (Vol. I, II). Chelsea Publishing Company.
Golub, G. H., & an Loan, C. F. (2013). Ma ix compu a ions (4 h ed.). The Johns Hopkins Uni e si y P ess.
Hamma ling, S., Mun o, C. J., & Tisseu , F. (2013). An algo i hm o he comple e solu ion o quad a ic eigen-
alue p oblems. ACM T ansac ions on Ma hema ical So wa e, 39(3), 18:1-18:19. h ps:// doi. o g/ 10.
1145/ 24501 53. 24501 56
Higham, N. J. (2002). Accu acy and s abili y o nume ical algo i hms (2nd ed.). Socie y o Indus ial and
Applied Ma hema ics.
Higham, N. J. (2008). Func ions o ma ices: Theo y and compu a ion. Socie y o Indus ial and Applied
Ma hema ics.
Higham, N. J., & Kim, H.-M. (2000). Nume ical analysis o a quad a ic ma ix equa ion. IMA Jou nal o
Nume ical Analysis, 20, 499–519. h ps:// doi. o g/ 10. 1093/ imanum/ 20.4. 499
642
A.Meye -Gohde
Higham, N. J., & Kim, H.-M. (2001). Sol ing a quad a ic ma ix equa ion by New on’s me hod wi h exac line
sea ches. SIAM Jou nal on Ma ix Analysis and Applica ions, 23(2), 499–519. h ps:// doi. o g/ 10. 1137/
S0895 47989 93509 76
Ho n, R. A., & Johnson, C. R. C. R. (2012). Ma ix analysis (2nd ed.). Camb idge Uni e si y P ess.
Hube , J., Meye -Gohde, A., & Saecke , J. (2023). Sol ing linea DSGE models wi h s uc u e-p ese ing dou-
bling me hods. IMFS Wo king Pape Se ies 195. Goe he Uni e si y F ank u , Ins i u e o Mone a y and
Financial S abili y (IMFS).
Judd, K. L. (1992). P ojec ion me hods o sol ing agg ega e g ow h models. Jou nal o Economic Theo y,
58(2), 410–452. h ps:// doi. o g/ 10. 1016/ 0022- 0531(92) 90061-L
Judd, K. L. (1998). Nume ical me hods in economics. MIT P ess.
Klein, P. (2000). Using he gene alized Schu o m o sol e a mul i a ia e linea a ional expec a ions model.
Jou nal o Economic Dynamics and Con ol, 24(10), 1405–1423. h ps:// doi. o g/ 10. 1016/ S0165-
1889(99) 00045-7
Kågs öm, B. (1994). A pe u ba ion analysis o he gene alized Syl es e equa ion
(AR −LB
,
DR −LE)=(C
,
F)
. SIAM Jou nal on Ma ix Analysis and Applica ions, 15(4), 1045–1060.
h ps:// doi. o g/ 10. 1137/ S0895 47989 32462 12
Kågs öm, B., & Po omaa, P. (1996). LAPACK-s yle algo i hms and so wa e o sol ing he gene alized Syl-
es e equa ion and es ima ing he sepa a ion be ween egula ma ix pai s. ACM T ansac ions on Ma h-
ema ical So wa e (TOMS), 22(1), 78–103. h ps:// doi. o g/ 10. 1145/ 225545. 225552
Lan, H., & Meye -Gohde, A. (2014). Sol abili y o pe u ba ion solu ions in DSGE models. Jou nal o Eco-
nomic Dynamics and Con ol, 45, 366–388. h ps:// doi. o g/ 10. 1016/j. jedc. 2014. 06. 005
Lubik, T. A., & Scho heide, F. (2003). Compu ing sunspo equilib ia in linea a ional expec a ions models.
Jou nal o Economic Dynamics and Con ol, 28(2), 273–285. h ps:// doi. o g/ 10. 1016/ S0165- 1889(02)
00153-7
Meye -Gohde, A. (2023). Nume ical s abili y analysis o linea DSGE model—Backwa d e o s, o wa d
e o s and condi ion numbe s. IMFS Wo king Pape Se ies 193. Goe he Uni e si y F ank u , Ins i u e
o Mone a y and Financial S abili y (IMFS).
Meye -Gohde, A., & Neuho , D. (2015). Sol ing and es ima ing linea ized DSGE models wi h VARMA
shock p ocesses and il e ed da a. Economics Le e s, 133, 89–91. h ps:// doi. o g/ 10. 1016/j. econl e . 2015.
05. 024
Meye -Gohde, A., & Saecke , J. (2024). Sol ing linea DSGE models wi h New on me hods. Economic Mod-
elling, 133, 106670. h ps:// doi. o g/ 10. 1016/j. econm od. 2024. 106670
Mole , C. B., & S ewa , G. W. (1973). An algo i hm o gene alized ma ix eigen alue p oblems. SIAM Jou -
nal on Nume ical Analysis, 10(2), 241–256. h ps:// doi. o g/ 10. 1137/ 07100 24
Polyak, B. (1964). Some me hods o speeding up he con e gence o i e a ion me hods. USSR Compu a ional
Ma hema ics and Ma hema ical Physics, 4(5), 1–17. h ps:// doi. o g/ 10. 1016/ 0041- 5553(64) 90137-5
Rendahl, P. (2017). Linea ime i e a ion. Discussion pape : IHS economics se ies.
Sims, C. A. (2001). Sol ing linea a ional expec a ions models. Compu a ional Economics, 20(1–2), 1–20.
h ps:// doi. o g/ 10. 1023/A: 10205 17101 123
Sme s, F., & Wou e s, R. (2007). Shocks and ic ions in US business cycles: A bayesian DSGE app oach. The
Ame ican Economic Re iew, 97(3), 586–606. h ps:// doi. o g/ 10. 1257/ ae . 97.3. 586
S ewa , G. W. (1971). E o bounds o app oxima e in a ian subspaces o closed linea ope a o s. SIAM
Jou nal on Nume ical Analysis, 8(4), 796–808. h ps:// doi. o g/ 10. 1137/ 07080 73
Taylo , J. B., & Uhlig, H. (1990). Sol ing nonlinea s ochas ic g ow h models: A compa ison o al e na i e
solu ion me hods. Jou nal o Business & Economic S a is ics, 8(1), 1–17. h ps:// doi. o g/ 10. 1080/ 07350
015. 1990. 10509 766
Tisseu , F., & Mee be gen, K. (2001). The quad a ic eigen alue p oblem. SIAM Re iew, 43(2), 235–286.
h ps:// doi. o g/ 10. 1137/ S0036 14450 03819 88
Uhlig, H. (1999). A oolki o analysing nonlinea dynamic s ochas ic models easily. In R. Ma imon & A.
Sco (Eds.), Compu a ional me hods o he s udy o dynamic economies (Vol. 3, pp. 30–61). Ox o d
Uni e si y P ess.
Villemo , S. (2011). Sol ing a ional expec a ions models a i s o de : Wha dyna e does. Dyna e Wo king
Pape s2. CEPREMAP.
Wieland, V., A anasye a, E., Kue e, M., & Yoo, J. (2016). New me hods o mac o- inancial model compa i-
son and policy analysis. In J. B. Taylo & H. Uhlig (Eds.), Handbook o mac oeconomics handbook o
mac oeconomics (Vol. 2, pp. 1241–1319). Else ie . h ps:// doi. o g/ 10. 1016/ bs. hesmac. 2016. 04. 004
643
Sol ing Linea DSGE Models wi hBe noulli I e a ions
Wieland, V., Cwik, T., Mülle , G. J., Schmid , S., & Wol e s, M. (2012). A new compa a i e app oach o mac-
oeconomic modeling and policy analysis. Jou nal o Economic Beha io & O ganiza ion, 83(3), 523–
541. h ps:// doi. o g/ 10. 1016/j. jebo. 2012. 01. 006
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.