Ko b, Nai a; B enneisen, Jan-Niklas; Lengnick, Ma hias; P oano, Ch is ian;
Wohl mann, Hans-We ne
A icle
Spillo e e ec s be ween he s ock ma ke and he
eal economy in a mixed- equency agen -based
mac o inancial model
Jou nal o Economics and S a is ics
P o ided in Coope a ion wi h:
De G uy e B ill
Sugges ed Ci a ion: Ko b, Nai a; B enneisen, Jan-Niklas; Lengnick, Ma hias; P oano, Ch is ian;
Wohl mann, Hans-We ne (2024) : Spillo e e ec s be ween he s ock ma ke and he eal economy
in a mixed- equency agen -based mac o inancial model, Jou nal o Economics and S a is ics, ISSN
2366-049X, De G uy e Oldenbou g, Be lin, Vol. 244, Iss. 4, pp. 331-350,
h ps://doi.o g/10.1515/jbns -2024-0017
This Ve sion is a ailable a :
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Nai a Ko b*, Jan-Niklas B enneisen, Ma hias Lengnick,
Ch is ian R. P oaño and Hans-We ne Wohl mann
Spillo e Effec s Be ween he S ock Ma ke
and he Real Economy in a Mixed-F equency
Agen -Based Mac ofinancial Model
h ps://doi.o g/10.1515/jbns -2024-0017
Recei ed Janua y 13, 2024; accep ed Sep embe 16, 2024
Abs ac : This pape illus a es a beha io al mixed equency mac o-finance model
whe e bo h eal and financial a iables a e gene a ed on a daily basis. Fu he , while
financial sec o da a is collec ed a he same equency as i is gene a ed (i.e. daily),
eal da a can only be collec ed on a qua e ly basis. Unde hese ci cums ances,
ou pu and infla ion, upon which da a is a ailable wi h a significan delay, become
unsui able as he sole in o ma ion guide o mone a y policy. We sugges ha policy
make s can deal wi h his in o ma ion p oblem by eac ing o he a iable on which
da a is collec ed on high equency basis: he s ock p ice.
Keywo ds: new Keynesian model; mixed- equency mac oeconomics; beha io al
mac oeconomics; op imal mone a y policy; mac o-finance in e ac ion; heu is ic
swi ching;
JEL Classifica ion: E44; E52; G01
1 In oduc ion
How much addi ional s abili y in he eal sec o jus ifies a leaning-agains - he-wind
mone a y policy whe e he policy a e decisi ely goes beyond he con en ional
We would like o hank Philipp Haube o he excellen esea ch assis ance.
This a icle is pa o he special issue “Ad ancing Agen -based Economics”published in he Jou nal o
Economics and S a is ics. Access o u he a icles o his special issue can be ob ained a www.deg uy e .
com/jbns .
*Co esponding au ho : Nai a Ko b, O o-F ied ich-Uni e si ä Bambe g, Bambe g, Ge many,
E-mail: [email p o ec ed]. h ps://o cid.o g/0000-0002-5441-0801
Jan-Niklas B enneisen, Ma hias Lengnick and Hans-We ne Wohl mann, Ch is ian-Alb ech s-
Uni e si ä zu Kiel, Kiel, Ge many
Ch is ian R. P oaño, O o-F ied ich-Uni e si ä Bambe g, Bambe g, Ge many
Jou nal o Economics and S a is ics 2024; 244(4): 331–350
Open Access. © 2024 he au ho (s), published by De G uy e . This wo k is licensed unde he
C ea i e Commons A ibu ion 4.0 In e na ional License.
wisdom and eac s o s ock p ice upda es? The baggage o appea ing o in e e e in
he s ock ma ke is no ligh o bea . Fu he mo e, he li e a u e on whe he o no ,
and o wha ex en , he policy a e should conside leaning, is no se led. In his
pape , we ocus on he in o ma ion p o ided by s ock ma ke upda es o he policy
making p ocess, p oposing ha a judicious u iliza ion o his in o ma ion could yield
s abili y benefi s ha would o he wise be o ei ed i his in o ma ion we e
o e looked.
The p ima y aim o his pape is o examine how a mone a y policy ule ha
esponds o high- equency s ock p ice upda es, which a e co ela ed wi h he eal
sec o , affec s he mac oeconomic and financial s abili y o an economy. This
app oach is con as ed wi h wai ing o eal da a o be collec ed and published a a
slowe pace, he eby losing in a-qua e in o ma ion.
Mo e p ecisely, we illus a e a beha io al mixed equency mac o-finance
model whe e bo h eal and financial a iables a e gene a ed on daily basis. Fu he ,
while financial sec o da a is collec ed a he same equency as i is gene a ed
(i.e. daily), eal da a can only be collec ed on a qua e ly basis. This si ua ion, in
which a a iable’s e olu ion h ough ime canno be obse ed a all da es i is
gene a ed, is known as empo al agg ega ion.
Ma ce (1991) explains ha empo al agg ega ion equen ly occu s in eco-
nomics because collec ing high- equency da a on ce ain a iables is o en p o-
hibi i ely expensi e. Howe e , he e is no eason o assume ha economic ime
se ies a e collec ed a a equency sufficien o ully cap u e he mo emen s o he
economy. Fo example, while we ypically only ha e qua e ly obse a ions on he
g oss na ional p oduc (GNP), i is easonable o belie e ha he beha iou o he GNP
wi hin a qua e ca ies ele an in o ma ion abou he s uc u e o he economy,
e en i his beha iou is unobse able.
The p oblem o ( empo al) agg ega ion is no only ele an o economic ques-
ions and applica ions, bu ex ends o a ious o he fields, including poli ical science
(Shellman 2004), heal h s udies (Lindo 2015; Phelps e al. 2018), and (disc e e) choice
modelling (Basu and Sulli an 2017; Wong, B owns one, and Bunch 2019).
The diffe en ia ion be ween he equency a which agen s make hei economic
decisions and he equency a which he in o ma ion ele an o hem becomes
a ailable is by no means i ial, as he la e may condi ion he o me . Fo ins ance,
as we will discuss below, i he policy in e es a e se by he mone a y au ho i ies is
specified as a unc ion o he eal g oss domes ic p oduc (GDP) and p ice infla ion,
and he ealiza ions o hese wo a iables become obse able on a qua e ly basis,
he policy a e will adop ha qua e ly equency oo, e en hough in p inciple i
could be ese on any gi en day. Mo eo e , he g ea e he disc epancy be ween
he da a gene a ing p ocess (DGP) and he da a collec ion p ocess (DCP), he mo e
significan nowcas and o ecas biases will become.
332 N. Ko b e al.
In models whe e agen s a e boundedly a ional and backwa d-looking, like ou s,
he disc epancy be ween he DGP and he DCP is pa icula ly significan . This is due
o he dual na u e o in o ma ion and cogni i e cons ain s: on he one hand, hese
agen s mus o m expec a ions based on his o ically a ailable da a a he han a
model-consis en app oach; on he o he hand, his da a is a ailable o hem wi h a
delay due o empo al agg ega ion. Ou pape is closely ela ed o li e a u e whe e
decisions (expec a ions) a e made unde cogni i e cons ain s and ely on simple
disc e e choice models in en i onmen s wi h issues such as da a a ailabili y,
collec ion delays, o agg ega ion. No able examples include he disc e e choice
app oach o en i onmen al and ene gy decision-making (G illi and Fe ini 2022) and
g aph lea ning (Tomlinson and Benson 2024).
The li e a u e mos ele an o ou pape includes Kon onikas and Ioannidis
(2005), Kon onikas and Mon agnoli (2006), Bask (2012), Wes e hoff(2012), Naimzada
and Pi eddu (2013), Lengnick and Wohl mann (2013, 2016), F anke and Sach (2014),
and mos ecen ly, P oaño and Ko b (2024). Excep o he la e , none o hese
au ho s, howe e , models he disc epancy o he DGP and he DCP explici ly. To
add ess his, we modi y he heo e ical model by Lengnick and Wohl mann (2016) o
explici ly diffe en ia e be ween he DGP and DCP, and s udy he implica ions o
policy making. In con as o P oaño and Ko b (2024), who concen a e on analysing
eal shocks, ou ocus lies on financial shocks. We in es iga e how hese shocks
p opaga e o he eal economy wi hin a amewo k whe e eal da a a e empo ally
agg ega ed, ende ing eal a iables alone unsui able o he pu pose o high-
equency policy design.
The emainde o his pape is o ganized as ollows. Sec ion 2 desc ibes ou
mixed- equency beha iou al mac oeconomic model. Sec ion 3 p o ides he esul s
o he simula ion and he associa ed analysis. Sec ion 4 concludes.
2 The Theo e ical F amewo k
In he ollowing, we se up a beha iou al mac oeconomic model whe e all economic
ac i i ies, bo h in he financial and he eal sec o s, ake place on a daily basis.
Fu he , we assume ha eal mac oeconomic a iables a e obse able on a qua -
e ly basis gi en he da a collec ion cos s associa ed wi h a iables such as he GDP
and i s componen s in he eal wo ld. As we will discuss la e , his assump ion has
impo an implica ions since i affec s he agen s’in o ma ion se s, and hus hei
o ecas s and economic decisions.
To be as clea as possible in ou exposi ion, we use dis inc indices ha esemble
he diffe en equencies a which economic decisions a e made as well as he
Mixed-F equency Agen -Based Model 333
equencies a which da a is collec ed. The index e e s o he daily equency in ou
amewo k. We deno e by q he qua e ly ime index. Fu he , le Tqdeno e he
numbe o “ ading days”, i.e. days wi h economic ac i i y, in a qua e . Following
Lengnick and Wohl mann (2013, 2016), we assume ha he e is no economic ac i i y
(s ock ma ke ading, goods p oduc ion, e c.) on he weekends, so ha Tq=
64 ≈3⋅30 ⋅5
7days pe qua e . This implies ha a qua e , q,isdefined o con ain
he days 64(q−1) +1, …,64q. This is illus a ed in Figu e 1. The uppe pa o he
figu e depic s he e olu ion o he da a collec ion p ocess, which occu s a a qua e ly
equency. The lowe pa shows he e olu ion o he da a gene a ing p ocess, which
occu s a a daily equency.
2.1 The Real Sec o
To acili a e a be e unde s anding o ou high- equency modelling app oach and
he ole o diffe en in o ma ion se s a diffe en equencies, we desc ibe fi s he
qua e ly law o mo ions o he Lengnick and Wohl mann (2016) model. Then we
de i e he co esponding o mula ions a he daily equency ollowing F anke and
Sach (2014).
The IS equa ion de i ed in Lengnick and Wohl mann (2016) eads
xq=E
qxq+1
[]
−1
σiq−E
q[πq+1]
[]
+c1E
qΔsq+1−πq+1
[] (1)
whe e x
q
ep esen s he ou pu gap, i
q
he policy nominal in e es a e, s
q
he
a e age s ock p ice, σ he isk a e sion coefficien , and E
q[xq+1]and E
q[πq+1]
he agen s’agg ega e ou pu gap and infla ion expec a ions a qua e qcon-
ce ning he nex qua e q+1, espec i ely. The ilde deno es ha expec a ions
a e o med in a boundedly a ional way in con as o he a ional expec a ions
ope a o E
[⋅].
The new Keynesian Phillips cu e (NKPC) eads
πq=βE
q[πq+1]+γxq−κsq(2)
Figu e 1: Time scale as indexed by ading days ( ) and qua e s (q).
334 N. Ko b e al.
whe e β=1/(1+
), whe e
is he s eady s a e eal in e es a e in qua e ly e ms,
and γ=(1−θ)(1−βθ)(σ+η)
θ, whe e ηis he in e se F isch elas ici y o labou supply and θ
ep esen s he deg ee o p ice s ickiness (measu ed a he qua e ly equency).
1
As in F anke and Sach (2014), we deno e now wi h h he ime in e al o a
equency o in e es ela i e o he qua e ly ime in e al (wi h 0 < h≤1 o ime
in e als o a highe - han-qua e ly equency). In case o he daily equency we
ha e h≡1/Tq, such ha he daily IS-equa ion is gi en by
x =E
[x +1]−h
σi −E
[π +1]
[]
+c1E
Δs +1−hπ +1
[] (3)
whe e x
ep esen s he ou pu gap a day ,i
he nominal in e es a e in qua e ly
e ms, which may be ese a any day wi hin a qua e , s
he s ock p ice (obse able
a day ), and E
[Δs +1] he a e age expec a ion o he s ock p ice change be ween
and +1.
No e ha , ollowing F anke and Sach (2014), and o main ain consis ency in ou
defini ions, he flow a iables a e uni o mly exp essed as ‘qua e ized’magni udes.
Acco dingly, a nominal in e es a e i
o alue ape qua e means ha , in an
h-economy, he in e es is ha o e he pe iod ( , +h). The same applies o he
infla ion a e: p ice infla ion om pe iod o pe iod +his equal o he p oduc o he
qu a e ized infla ion in pe iod +h(π
+h
) and h. Fu he mo e, he high equency
ep esen a ion o he model equi es he adjus men o p e e ence pa ame e s wi h
a ime dimension. Acco dingly, he discoun ac o βbecomes βd=1
1+h , he Cal o
pa ame e θd=1−h⋅(1 −θ),
2
and γd=(σ+η)(1−θd)(1−βdθd)
θd. These conside a ions imply
o he dynamics o he qua e ized daily infla ion a e.
hπ =βdhE
[π +1]+γdx −κs .(4)
2.2 The Da a Collec ion P ocess
As p e iously men ioned, in ou model, while financial a iables, such as s ock p ices,
a e obse able daily, da a on eal sec o a iables suchas he agg ega e ou pu gap and
agg ega e p ice infla ion can only easibly be ga he ed on a qua e ly basis.
1The alue o γis ob ained by combining he fi s o de condi ion o he fi m p oblem unde Cal o
p ice mechanism, and he households’fi s o de condi ion. Fo mo e cla ifica ion, consul F anke
and Sach (2014).
2Fo ins ance, i θ= 0.55, i means ha 45 % o fi ms ese hei p ices each qua e . Calcula ing o
daily ese s, θd=1−1
64 ⋅(1−0.55)≈0.993. Thus, abou 0.7 % o fi ms ese hei p ices daily, as 1 −0.
993 ≈0.007.
Mixed-F equency Agen -Based Model 335
In o de o model his non- i ial issue in a s ylized manne , we assume ha he
daily alues o eal a iables a e unobse able un il he cu en qua e has ended.
Once he qua e is comple e, all daily alues o he eal a iables om ha pe iod
become a ailable o he s a is ical office. The office hen collec s hese alues o
compu e a qua e ly agg ega e, which is subsequen ly made a ailable o he public.
DCP : xq≔1
Tq∑
Tqq
=Tq(q−1)+1
x (5)
DCP : πq≔1
Tq∑
Tqq
=Tq(q−1)+1
π .(6)
2.3 Expec a ions o he Real Va iables
Diffe en ia ing he da a a ailabili y o he model a iables significan ly impac s he
in o ma ion se s a ailable o agen s and he expec a ions hey o m abou u u e
a iables based on hese in o ma ion se s. This diffe en ia ion is pa icula ly c ucial
in a amewo k whe e agen s a e assumed o be boundedly a ional and whe e
he e ogenei y may play a significan ole. In he ollowing, we use ules o humb
employed by Lengnick and Wohl mann (2016), adjus ing hem howe e o he DCP
discussed abo e, i.e.
Ta ge ing expec a ions : E
a
qyq+1=y
y∈{x,π}(7)
S a ic expec a ions : E
s a
qyq+1=yq−1(8)
Ex apola ing expec a ions : E
ex
qyq+1=yq−1+αyyq−1−yq−2
()
(9)
wi h α
y
> 0. Agen s using he fi s expec a ional ule-o - humb (i.e. a ge e s) simply
assume ha he a iables o in e es (xand π) will be a hei explici ly announced
a ge s o he cen al bank (x
and π, espec i ely). Following, De G auwe and Mac-
chia elli (2015) and Lengnick and Wohl mann (2016), we no malise hese o ze o.
Agen s using he second heu is ic (i.e. s a ic) simply assume ha he a iables o
in e es will no change in he nex qua e . Finally, agen s using he hi d ule
(i.e. ex apola o s) add o he mos ecen qua e ly alue a momen um e m. No e
ha , while he expec a ions o he a ge e s emain cons an o e ime, s a ic and
ex apola i e expec a ions change only once pe qua e when new da a is
published.
336 N. Ko b e al.
The ac ions o agen s ωy,j
q o he h ee heu is ics j∈{ a , s a, ex } a e de e -
mined by a disc e e choice app oach wi h he in ensi y o choice pa ame e ϕ.
ωy,j
q=exp ϕAy,j
q
()
exp ϕAy, a
q
()
+exp ϕAy,s a
q
()
+exp ϕAy,ex
q
() y∈x,π
{}
.(10)
The a ac i i y Ay,j
qo a heu is ic jis defined as a geome ic sum o pas squa ed
expec a ion e o s (c. . De G auwe (2010, 2011)).
Ay,j
q=− yq−1−E
j
q−2[yq−1]
⎛
⎝⎞
⎠2
−ζAy,j
q−1y∈x,π,
{} (11)
whe e he memo y coefficien 0 ≤ζ< 1 de e mines he speed o agen s’ o ge ing
abou he pas .
Ma ke expec a ions a e hen gi en by a weigh ed a e age o he h ee
heu is ics.
E
qyq+1=ωy, a
qE
y, a
qyq+1+ωy,s a
qE
y,s a
qyq+1+ωy,ex
qE
y,ex
qyq+1.(12)
No e ha he se ings explained abo e cap u e he no ion ha , while he eal
a iables hemsel es a e gene a ed daily, he qua e ly na u e o he collec ion and
publica ion p ocess leads o expec a ions, weigh s and a ac i i y alues ha ollow
a qua e ly equency (i.e. change only once pe qua e ). Hence, he subsc ip qin
equa ions (7)–(12).
Figu e 2 illus a es he p ocesses o da a gene a ion and collec ion o he eal
a iables. The solid black line ep esen s he ue gene a ing p ocess o he a iable,
which is only obse ed a specific ime poin s (e.g. he las day o each qua e when
da a is collec ed). Va ious agen s employ diffe en ules o humb o in e p e his
obse ed da a and o m expec a ions abou he u u e.
2.4 The Financial Sec o
As in Wes e hoff(2008) and Lengnick and Wohl mann (2016), ou heo e ical
amewo k depic s he financial sec o wi h wo ypes o ade s: cha is s and
undamen alis s. These ade s o mula e hei expec a ions ega ding u u e s ock
p ices based on he ollowing ules.
E
C
[s +1]=s −1+kc[s −1−s −2](13)
Mixed-F equency Agen -Based Model 337
E
F
[s +1]=s −1+k s
−1−s −1
[] (14)
whe e kcand k a e bo h posi i e and s
ep esen s he agen s’pe cep ion abou he
“ undamen al”s ock p ice. Since he ue undamen al s ock p ice canno be known
wi h ce ain y, agen s ha e o o m belie s abou i .
3
Following ea lie wo k,
4
we
model he agen s’pe cep ion o be posi i ely co ela ed wi h eal economic
condi ions
s
=gx
q−1wi h g≥0.(15)
The excess s ock demand unc ions a e gi en by
Di
=ℓE
i
[s +1]−s
()
i∈{C,F}.(16)
We also allow o a hi d excess demand unc ion DNT
=0 which p esc ibes a
“no- ading”posi ion in ha pe iod.
T ade s choose be ween hese h ee diffe en ules acco ding o hei espec i e
a ac i i y which is de e mined as a unc ion o pas p ofi s.
Figu e 2: Illus a ion o boundedly a ional expec a ions in a model wi h daily DGP bu qua e ly DCP.
3See also De G auwe and Kal wasse (2012) and Be nanke and Ge le (2000).
4Consul Lengnick and Wohl mann (2013), Sec ion 2.3 o mo e de ails.
338 N. Ko b e al.
policy, we can see ha he leaning policy minimizes he sha p a ia ions caused by
in ense swi ching beha iou . This leads o smoo he business cycles o booms and
bus s o he s ock p ice and infla ion. Howe e , his comes a he cos o a sligh ly
mo e uns able ou pu gap and an ex emely a iable in e es a e.
Figu e 7 epea s he exe cise wi h a highe alue o he s ock p ice shock
s anda d de ia ion. The esul s a e e en mo e p onounced. A leaning mone a y
policy minimizes he swi ching beha iou in he s ock ma ke and infla ion expec-
a ions, leading o smoo he cycles in bo h infla ion and he s ock p ice. The ou pu
Figu e 6: A simula ion o 400 qua e s (25,600 days). Blue solid line: δ
s
= 0. Red do ed line: δ
s
= 18.
σ
s
= 0.01.
Mixed-F equency Agen -Based Model 345
cycles a e also sligh ly mo e smoo hed. Howe e , he in e es a e abso bs all o his
emo ed ins abili y.
Compa able findings can be de i ed by examining he impulse esponses o he
model a iables ollowing a one s anda d de ia ion s ock p ice shock.
9
Figu e 8
shows he esul s o a posi i e shock. Figu e 9 shows he esul s o a nega i e shock. In
he fi s case (second case), he infla ion a e eac s sligh ly nega i e (posi i e) as pe
Figu e 7: A simula ion o 400 qua e s (25,600 days). Blue solid line: δ
s
= 0. Red do ed line: δ
s
= 20.
σ
s
= 0.04.
9Appendix A desc ibes how he impulse esponse unc ions a e compu ed.
346 N. Ko b e al.
equa ion (4). When he policy a e is leaning, he s ock p ice is ins an aneously
s abilized, and he swi ching beha iou is minimized. The balance be ween unda-
men al and cha is expec a ions esul s in expec ed s ock p ices ha a e only
ma ginally nega i e (posi i e). Consequen ly, ou pu also esponds sligh ly nega-
i ely (posi i ely) as desc ibed in equa ion (3).
I is no ewo hy o men ion ha he s epped o m pa e n o he infla ion a e
and ou pu (blue solid lines) a ises om he ac ha he mos significan changes o
hese a iables occu a he end o each qua e when da a is collec ed and expec-
a ions a e upda ed. When he policy a e leans ( ed do ed lines), his s epped o m
pa e n becomes smoo he due o he daily equency o in e es a e impac s on he
mo emen o hese a iables.
Figu e 8: Impulse esponses o a posi i e one s anda d de ia ion s ock p ice shock unde δ
s
= 0 (blue
solid line) and δ
s
= 18 ( ed do ed line).
Mixed-F equency Agen -Based Model 347
4 Concluding Rema ks
“F om he poin o iew o mac oeconomic s abili y, and pa icula ly o equilib ium
de e minacy, he con en ional wisdom appea s o be ha an explici esponse o
s ock p ices is a bad idea”(Ai audo, Nis icó, and Zanna 2015, p. 1274). Ai audo, Nis icó,
and Zanna (2015) showed ha a policy esponse o he s ock p ice could be indeed
s abilizing, when he eal sec o and he financial sec o a e s uc u ally ela ed. We
show ha such a policy esponse can be e en mo e jus ified, when eal da a is
empo ally agg ega ed. Unde hese ci cums ances, ou pu and infla ion, upon which
da a is a ailable wi h a significan delay, become unsui able as he sole in o ma ion
guide o mone a y policy. The policy make s can ex ac he missing in o ma ion by
conside ing leaning agains he s ock p ice.
We demons a e ha a posi i e esponse o s ock p ices can enhance s abili y in
bo h he eal and financial sec o s when a s uc u al ela ionship exis s be ween
Figu e 9: Impulse esponses o a nega i e one s anda d de ia ion s ock p ice shock unde δ
s
= 0 (blue
solid line) and δ
s
= 18 ( ed do ed line).
348 N. Ko b e al.
hem. Howe e , his benefi comes wi h he po en ial d awback o an uns able and
highly a iable policy a e. Addi ionally, unde he assump ion o bounded a io-
nali y, agen s swi ching be ween diffe en heu is ics leads o nonlinea eac ions o
shocks and mo e p onounced dynamic spikes, especially as he swi ching in ensifies.
Mone a y policy can mi iga e his effec by eac ing con empo aneously o s ock
p ice upda es, leading o mo e s able s ock p ices and a minimized swi ching
beha iou , which in u n esul s in smoo he eal and financial cycles. These benefi s
mus be weighed agains he cos s o a ia ion in he policy a e.
Ou model is a s ylized a emp o cap u e he in o ma ion gains o leaning
agains he wind in an en i onmen o empo al agg ega ion and boundedly a ional
beha iou . Fu he esea ch can p o ide an econome ic basis o his analysis,
analyse eal shocks, and s udy models unde diffe en eal-financial spillo e
channels han hose examined in ou model.
Appendix A: Impulse Response Analysis
To calcula e impulse esponse unc ions, we ollow he s eps o he expe imen
discussed in Lengnick and Wohl mann (2013). These s eps a e desc ibed as ollows:
1. Gene a e model dynamics o one pa icula andom seed.
2. Gene a e he dynamics again wi h he same andom seed, bu wi h ϵs
129 inc eased
(dec eased) by 0.01. In o he wo ds, a ime = 129, which is he fi s day o he
hi d qua e , he alue o he s ock p ice shock is highe (lowe ) han he same
shock a he same ime in he p e ious s ep wi h an amoun 0.01.
3. Calcula e he diffe ence be ween he ajec o ies o s eps 1 and 2 which gi es he
isola ed impac o he addi ional shock.
4. Repea s eps 1–3 o 500 imes.
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