Polynomial Chaos Expansion: Efficient Evaluation and Estimation of Computational Models

Feh le, Daniel; Heibe ge , Ch is ophe ; Hube , Johannes
A icle — Published Ve sion
Polynomial Chaos Expansion: E icien E alua ion and
Es ima ion o Compu a ional Models
Compu a ional Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Feh le, Daniel; Heibe ge , Ch is ophe ; Hube , Johannes (2025) : Polynomial
Chaos Expansion: E icien E alua ion and Es ima ion o Compu a ional Models, Compu a ional
Economics, ISSN 1572-9974, Sp inge US, New Yo k, NY, Vol. 65, Iss. 2, pp. 1083-1146,
h ps://doi.o g/10.1007/s10614-024-10772-5
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/323340
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Compu a ional Economics (2025) 65:1083–1146
h ps://doi.o g/10.1007/s10614-024-10772-5
Polynomial Chaos Expansion: E icien E alua ion
andEs ima ion o Compu a ional Models
DanielFeh le1· Ch is ophe Heibe ge 2· JohannesHube 3
Accep ed: 25 Oc obe 2024 / Published online: 23 Janua y 2025
© The Au ho (s) 2025
Abs ac
We apply Polynomial chaos expansion (PCE) o su oga e ime-consuming epea ed
model e alua ions o di e en pa ame e alues. PCE ep esen s a andom a iable,
he quan i y o in e es (QoI), as a se ies expansion o o he andom a iables, he
inpu s. Repea ed e alua ions become inexpensi e by ea ing unce ain pa ame e s
o a model as inpu s, and an elemen o a model’s solu ion, e.g., he policy unc ion,
second momen s, o he pos e io ke nel as he QoI. We in oduce he heo y o PCE
and apply i o he s anda d eal business cycle model as an illus a i e example.
We analyze he con e gence beha io o PCE o di e en QoIs and i s e iciency
when used o es ima ion. The esul s a e p omising bo h o local and global solu-
ion me hods.
Keywo ds Polynomial chaos expansion· Pa ame e in e ence· Pa ame e
unce ain y· Solu ion me hods
JEL Classi ica ion C11· C13· C32· C63
* Johannes Hube
johannes1.hube @u .de
Daniel Feh le
[email p o ec ed]
Ch is ophe Heibe ge
ch is ophe [email p o ec ed]g.de
1 Depa men o Economics, Kiel Uni e si y, Wilhelm-Seelig-Pla z 1, 24118Kiel, Ge many
2 Depa men o Economics, Uni e si y o Augsbu g, Uni e si ä ss aße 16, 86159Augsbu g,
Ge many
3 Depa men o Economics, Uni e si y o Regensbu g, Uni e si ä ss aße 31, 93053Regensbu g,
Ge many
1084
D.Feh le e al.
1 In oduc ion
A an abs ac le el, compu a ional economic models a e mappings om inpu s
o ou pu s o he model. The o me a e he model’s pa ame e s, he la e a e
he quan i ies o in e es (QoIs) and depend on he esea ch ques ion. Typical
QoIs a e he economic agen s’ policy unc ions, he second momen s o he mod-
el’s a iables, o he likelihood implied by a gi en se o obse ed da a. In all
cases, he model’s pa ame e s a e ypically unknown, and plausible alues mus
be de i ed om obse ed da a o ea ed as andom a iables om he Bayesian
pe spec i e. Ei he way, he unce ain y o pa ame e s ansla es in o unce ain y
ega ding he model’s ou comes. Es ima ion me hods, such as minimum dis ance
es ima o s o likelihood-based me hods as well as a ca e ul s udy o he sensi i -
i y o he model’s ou comes o a se o di e en pa ame e alues equi e nume -
ous epea ed solu ions o he model. Depending on he complexi y o he model,
es ima ion and sensi i i y analyses can become a ime-consuming compu a ional
ask o e en excessi e. We show ha PCE o e s an elegan way o deal wi h his
p oblem and p o ide MATLAB® code o ease i s implemen a ion.
In a nu shell, PCE enables he ep esen a ion o a andom a iable— he QoI—
as a se ies expansion o o he andom a iables— he inpu s. Ou app oach is
o use PCE as a su oga e o he dis ibu ion o he model ou come gi en some
pa ame e unce ain y. The e o e, we depic di e en model ou comes as QoIs
(e.g., he policy unc ion o he pos e io s ke nel) in e ms o a se ies expansion
o he model’s unce ain pa ame e s. Gi en he espec i e o mulae, he equi ed
epea ed e alua ions a e ime-e icien compa ed o epea ed solu ions o he
en i e model. Wi hou limi ing he applicabili y o o he pu poses, we apply solu-
ion and es ima ion me hods o dynamic s ochas ic gene al equilib ium (DSGE)
models as we a e amilia wi h he equi ed echniques.
Mo e o he poin , a e in oducing he heo y o PCE and he cons uc ion
o a unca ed PCE, we apply he me hod o he benchma k eal business cycle
(RBC) model, since his model is sui ed as an illus a i e example due o i s well-
known and simplis ic na u e. We analyze he con e gence beha io o he PCE
o a ious model ou comes including he model’s linea solu ion, a p ojec ion
solu ion, he a iables’ second momen s, and he impulse esponse unc ion. Fu -
he , we conduc Mon e Ca lo expe imen s. We es ima e he pa ame e s om a
linea ized and non-linea ized model using a ious es ima ion echniques, namely
gene alized me hod o momen s (GMM), simula ed me hod o momen s (SMM),
maximum-likelihood es ima ion (MLE), and Bayesian es ima ion (BE).
We documen linea con e gence beha io . Conside ing h ee unknown pa am-
e e s, we ind ema kably well app oxima ions wi h only a ew model e alua-
ions. Suppose he model ou come, e.g., he linea ized policy unc ion, has o be
e alua ed o a sample o 100,000 pa ame e alues. In ha case, he PCE wi h
unca ion deg ee 7 p o ides an app oxima ion wi h
L2
e o o
10−3
while he
compu a ional ime is lowe by he ac o 30. We ex end he analysis o a highe -
dimensional p oblem whe e all six model pa ame e s a e assumed unknown.
As ou cons uc ion o he PCE applies a enso basis quad a u e ule, he
1085
Polynomial Chaos Expansion: E icien E alua ion and…
cons uc ion su e s om he cou se o dimensionali y. Thus, we compa e ull-
enso -g id quad a u e ules wi h wo emedies: spa se-g id quad a u e ules and
leas squa es. A compa ison o ime e sus accu acy shows ha hese long-es ab-
lished spa se me hods milden he cou se o dimensionali y. Beyond ou imple-
men a ions, we highligh ha he epea ed model e alua ions equi ed o PCE
a e pa allelizable because he pa ame e alues a e p ede e mined, dis inc om
he ecu si e na u e o mos applica ions. Addi ionally, we discuss he expanding
li e a u e on spa se PCE.
Ou analysis con inues wi h Mon e Ca lo expe imen s as in Ruge-Mu cia (2007),
whe e we gauge he quali y o he model’s PCE when applied o es ima ion. He e,
we use he linea ized solu ion o he eal business cycle (RBC) model o he da a-
gene a ing p ocess and he econome ic model since his p ocedu e allows us o cal-
cula e he analy ic second momen s and likelihood unc ions. Consequen ly, he di -
e ences in he es ima es owa ds he benchma k p ocedu e o epea ed solu ions a e
solely based on he PCE app oxima ion. The PCE based me hod is ema kably e i-
cien and accu a e. Es ima es de ia e only negligibly om he benchma k p ocedu e
and mos no able, he compu a ion ime can be educed by 99 pe cen o BE and by
50 pe cen o GMM, SMM, and MLE.
In he las s ep o ou analysis, we s ess PCE ou mo e by gauging he quali y
o he non-linea model’s PCE o likelihood-based es ima ions. Fo his pu pose,
we eplica e he indings o Fe nández-Villa e de and Rubio-Ramí ez (2005), who
ha e shown ha non-linea i y is al eady ele an o he es ima es o ou benchma k
RBC model. We show ha he use o PCE o he es ima ion o he non-linea model
enhances he accu acy o he es ima es conside ably in compa ison o epea ed,
linea ly-sol ed model es ima ion and educes ime up o 97 pe cen compa ed o
epea ed globally-sol ed model es ima ion. Also wo h no ing, wi h PCE as a su -
oga e o he likelihood, he likelihood om a pa icle il e becomes con inuously
di e en iable—allowing a g adien -based op imiza ion.
In i s gene al o m, he unde lying heo y o he me hod es s on he heo y in o-
duced by Wiene (1938) and he Came on and Ma in (1947) heo em o a amily
o s ochas ically independen and no mally dis ibu ed andom a iables and He -
mi e polynomials. The p ope y does no only hold o He mi e polynomials and
p obabili y measu es o no mally dis ibu ed andom a iables bu also ex ends o
o he commonly used dis ibu ions and he co esponding o hogonal polynomials
om he Askey scheme. This ex ension, ini ially p oposed by Xiu and Ka niadakis
(2002), is also known as gene alized polynomial chaos expansion. Ghanem and Spa-
nos (1991) p o ide he i s applica ions o he heo y o he p oblem o unce ain
model pa ame iza ion.
The e a e wo pionee applica ions in economics. P öhl (2017) uses PCE o dis-
c e ize he s a e-space o he benchma k he e ogeneous agen model and Ha enbe g
e al. (2019) use he polynomial coe icien s o global sensi i i y analysis o he
RBC model. Ge sbach e  al. (2021) ollow Ha enbe g e al. (2019) and use PCE
o iden i y decisi e pa ame e s. In inance PCE is e.g., applied by Albe e io e al.
(2019); Dias and Pe e s (2021); Ma coni (2016). The applica ion o PCE in Bayesian
in e ence was i s analyzed by Ma zouk e al. (2007) in enginee ing bu o he bes
o ou knowledge, he me hod has no ye been s udied o es ima e compu a ional
1086
D.Feh le e al.
economic models. Scheidegge and Bilionis (2019) inco po a e pa ame e unce -
ain y in hei solu ion o compu a ional economic models by ew i ing he pa am-
e e s as s a es. This way, he model is indi ec ly sol ed in he pa ame e domain.
The emainde o he pape is s uc u ed as ollows. Fi s , in sec ion2 we e iew
he basic heo y o he exis ence o polynomial chaos expansions, p esen common
p ac ical me hods o compu e he PCE coe icien s, and discuss he applica ion o
a poin wise app oxima ion o he mapping om he pa ame e s o he model ou -
come, i.e., he cons uc ion o a su oga e. In sec ion3, we apply ou app oach o
he benchma k RBC model and discuss ou esul s and po en ial d awbacks. Sec-
ion4 concludes. Mo e de ailed de i a ions, applica ions, e c. can be ound in he
appendix.
2 Gene alized Polynomial Chaos Expansions
We begin by e iewing he basic idea and heo y behind he concep o PCE. While
PCE p o ed use ul o a ious applica ions, we ocus on hei implemen a ion o
e icien ly e alua e compu a ionally expensi e model ou comes when one o mo e
o he model’s inpu s, i.e., model pa ame e s, a e unce ain. Fu he , we gi e an ana-
ly ically ac able example o ou line he concep o PCE in Appendix 1.
No a ion and P elimina ies We conside a compu a ional economic model whe e
𝜗i∈Θ
i,Θi⊂ℝ,i=1, …,k,
deno es an a bi a y selec ion o
k∈ℕ
pa ame e s o
he model. Mo eo e , we a e in e es ed in some model ou come(s) deno ed by a
ec o
y∈ℝm
,m∈ℕ
. The ela ion be ween he inpu pa ame e s
𝜗i
and he model
ou come(s) y is de e mined de e minis ically, i.e., epea ed compu a ion o y wi h
he same inpu s
𝜗i
o he model p oduces he same esul .1 This mapping be ween
he
𝜗i
and y is desc ibed by
whe e
.
Wi hou loss o gene ali y, we conside he case
m=1
in he ollowing and no e
ha o
m≥2
all de i a ions can be applied sepa a ely o each componen
yi
o y,
i=1, …,m
, in he same way.
Now u he conside he case whe e he alues
𝜗i
o he model pa ame e s a e sub-
jec o some unce ain y o he esea che . In o de o accoun o his unce ain y, we
swi ch om he de e minis ic ep esen a ion o he pa ame e s o he pe spec i e o
y=h(𝜗1,…,𝜗k)
1 E.g., i y deno es some second momen s o he model, hese a e de i ed ei he om a ailable analy ic
o mulae om he (app oxima ed) model solu ion o a e compu ed om simula ions wi h he same sam-
ple o shocks.

1087
Polynomial Chaos Expansion: E icien E alua ion and…
desc ibing hem by app op ia ely dis ibu ed andom a iables. The e o e, le
(Ω,A,P)
deno e a su icien ly ich p obabili y space so ha any unce ain model inpu pa am-
e e can be desc ibed by some eal- alued andom a iable
𝜃i∶Ω
→
ℝ,i=1, …,k,
whe e he eal line is equipped wi h he Bo el sigma-algeb a
B(ℝ)
. Mo eo e , le
𝜉1,…,𝜉k
deno e a amily o s ochas ically independen andom a iables chosen by
he esea che as a basis o he desi ed polynomial expansions, he so-called ge ms. In
applica ions, as will be desc ibed la e , he ge ms a e mos commonly ei he se equal
o he unce ain model pa ame e s
𝜃i
o o some na u al and con enien ans o ma ion
o hem. We assume:
1. The ge ms
𝜉1,…,𝜉k
co e he same s ochas ic in o ma ion as he unce ain model
pa ame e s, i.e.,
whe e
𝜎(
⋅
)
deno es he sigma-algeb a gene a ed by he andom a iables.
2. All momen s o each
𝜉i
exis , i.e.,
𝔼[|𝜉i|n]<∞
o all
i=1, …,k
and
n∈ℕ0
.
Mo eo e , we w i e
𝜽∶= (
𝜃
1,…,
𝜃
k)∶Ω
→ℝ
k
and
𝝃∶= (
𝜉
1,…,
𝜉
k)∶Ω
→ℝ
k
o
he k-dimensional andom ec o o he unce ain model pa ame e s and o he an-
dom ec o o he ge ms, espec i ely, whe e
ℝk
is also equipped wi h i s Bo el sigma-
algeb a
B(
ℝ
k)
. Fo each
i=1, …,k,
le
P
𝜉
i
∶= P◦𝜉
−1
i
deno e he p obabili y measu e
o
𝜉i
on
(
ℝ
,B(
ℝ
))
and analogously le
P
𝝃∶= P◦𝝃
−1
=
⨂k
i=1
P𝜉
i
deno e he p oduc
p obabili y measu e o
𝝃
on
(
ℝ
k
,
B
(
ℝ
k))
. The Hilbe space (o equi alence classes) o
squa e-in eg able eal- alued unc ions on
(
ℝ
,B(
ℝ
),
P𝜉
i)
is deno ed by
whe e he inne p oduc is de ined by
We use he no a ion
‖
⋅
‖
L
2
i
o he induced no m on
L2
i
. We in oduce he analogous
no a ion, i.e.,
L2
∶= L
2
(ℝ
k
,B(ℝ
k
),dP
𝝃)
, o he space o squa e in eg able eal al-
ued unc ions on
(
ℝ
k
,B(ℝ
k
),P
𝝃)
and w i e
⟨
⋅
,
⋅
⟩L2
and
‖
⋅
‖L2
o he inne p oduc
and o he induced no m on
L2
. I he dis ibu ions o he andom a iables
𝜉i
pos-
sess p obabili y densi y unc ions
wi∶ℝ
→
ℝ+
, he inne p oduc s become
and
𝜎(𝜉1,…,𝜉k)=𝜎(𝜃1,…,𝜃k),
L2
i∶= L
2
(ℝ,B(ℝ),dP𝜉i)
∶=
{
∶ℝ→ℝ
||
is measu able and ∫ℝ
2dP𝜉i<∞
},
⟨
,g
⟩
L2
i
∶=
∫ℝ
gdP𝜉i=𝔼[ (𝜉i)g(𝜉i)] o ,g∈L2(ℝ,B(ℝ),P𝜉i)
.
⟨
,g
⟩
L2
i
=
∫ℝ
(s)g(s)wi(s)ds
,
1088
D.Feh le e al.
so ha
L2
i
=L
2
(ℝ,B(ℝ),w
i
(s)ds
)
and
L2=L2(
ℝ
k,
B
(
ℝ
k),w(s)ds)
whe e w is he
join p obabili y unc ion
w
(s) ∶=
∏k
i=1
w
i
(s
i)
. No e ha Assump ion 2 is equi alen
o he ac ha o each
i=1, …,k
all uni a ia e polynomials a e included in
L2
i
o ,
again equi alen ly, ha all k- a ia e polynomials a e included in
L2
.
Since, by Assump ion 1, each
𝜃i
is
𝜎(𝝃)
-measu able, he e exis measu able
𝜓i
∶ℝ
k
→
ℝ
which sa is y
We w i e
𝜓∶= (
𝜓
1,…,
𝜓
k)∶
ℝ
k
→ℝ
k
so ha
𝜽=𝜓
◦
𝝃
. Mo eo e , no e ha
𝜎(𝝃)=𝜎(𝜽)
also implies he exis ence o a measu able, in e se mapping
𝜓
−1
wi h
𝜓
◦
𝜓−1=𝜓−1
◦
𝜓=id
. A u he assump ion we make is ha
3. he second momen o each model inpu pa ame e exis s, i.e.,
𝔼
[𝜃
2
i
]<
∞
o
i=1, …,k
. Equi alen ly, each
𝜓i
is squa e in eg able on
(
ℝ
k
,B(ℝ
k
),P
𝝃)
, i.e.,
𝜓i
∈L
2
o all
i=1, …,k
.2
Mo eo e , as he model inpu pa ame e s
𝜃i
a e now ea ed as andom, he model
ou come o in e es is andom. We he e o e adap i s no a ion o
Y∶Ω→ℝ
. Ye ,
gi en any elemen a y e en
𝜔∈Ω
and co esponding ealiza ion
𝜃i(𝜔)
, he mapping
be ween he model pa ame e s and he model ou come is s ill de e mined de e min-
is ically by
Y(𝜔)=h(𝜃1(𝜔),…,𝜃k(𝜔))
, i.e.,
The inal assump ion is ha Y is a well-de ined andom a iable wi h ini e second
momen s, i.e.,
4. h is measu able and
h
◦
𝜓
is squa e in eg able on
(
ℝ
k
,B(ℝ
k
),P
𝝃)
, i.e.,
h
◦
𝜓∈L2
.
2.1 Single Unce ain Pa ame e andGe m (k=1)
We begin ou desc ip ion wi h he simples case wi h only one single unce ain inpu
pa ame e
𝜃
and one single ge m
𝜉
, i.e.,
k=1
. In gene al, any a bi a y choice o
he ge m ha sa is ies Assump ion 2 implies ha all polynomials a e included in
L2
, and he e o e allows he cons uc ion o an o hogonal sys em o polynomials
{
q
n
}
n∈
ℕ
0
⊂L
2
, i.e., a amily o polynomials whe e
qn
is o (exac ) deg ee n and
⟨
,g
⟩
L2=
∫ℝ
…
∫ℝ
(s1,…,sk)g(s1,…,sk)w1(s1)⋅…⋅wk(sk)ds1…dsk
,
𝜃i=𝜓i
◦
𝝃.
Y=h
◦
𝜽=h
◦
𝜓
◦
𝝃, o some h∶
ℝ
k
→ℝ
.
2 No e ha he hi d assump ion is al eady implied by he second i he ge ms a e se equal o (some
polynomial ans o ma ion o ) he model inpu pa ame e s.
1089
Polynomial Chaos Expansion: E icien E alua ion and…
whe e
𝛿m,n
deno es he Knonecke del a. This can gene ally be achie ed by applying,
e.g., he G am-Schmid p ocess o he sequence o monomials.
In p ac ice, he dis ibu ion o he unce ain inpu pa ame e is gi en and one is
ee o se he ge m. I is hen con enien o de ine he ge m in such a way ha i) an
easy ep esen a ion
𝜃=𝜓(𝜉)
o he pa ame e in e ms o he ge m a ises and ii)
he amily o o hogonal polynomials in
L2
co esponds o some well-known class
o polynomials. Table 1 summa izes he na u al choice o he ge m and he co -
esponding amily o o hogonal polynomials when he inpu pa ame e is no mal,
uni o m, Be a, o (in e se) Gamma dis ibu ed. Mo e de ails o hese classes a e
gi en in Appendix 2. Addi ionally, Xiu and Ka niadakis (2002) p o ide a simila
o e iew o disc e e dis ibu ions.
In all o he cases p esen ed in Table 1 he espec i e amilies o o hogo-
nal polynomials
{qn}n∈
ℕ
0
o m a comple e o hogonal sys em, i.e., lie densely in
L2
=L
2
(ℝ,B(ℝ),P𝜉)=L
2
(ℝ,B(ℝ),w(s)ds
)
whe e w is he co esponding p ob-
abili y densi y o
𝜉
.3 Mo e gene ally, i ollows om Riesz (1924) ha
{qn}n∈
ℕ
0
is a
comple e o hogonal sys em in
L2
i and only i he e exis s no o he measu e
𝜇
on
(ℝ,B(ℝ))
which gene a es he same momen s as
P𝝃
, i.e., i and only i he e is no
o he measu e
𝜇
such ha
I comple eness o
{qn}n∈
ℕ
0
in
L2
can be es ablished, hen Assump ions 3 and 4 gua -
an ee he exis ence o Fou ie se ies expansions o
𝜓
and
h
◦
𝜓
in he o hogonal
polynomials, i.e., he e a e coe icien s
{
𝜗
n
}
n∈
ℕ
0
and
{
y
n
}
n∈
ℕ
0
,

𝜗
n
,y
n
∈
ℝ
, so ha
No e ha iden i y and con e gence is unde s ood in
L2
which also implies poin wise
con e gence i.e., o a subsequence bu no poin wise con e gence.4 Mo eo e , since
P
𝜃=P𝜉◦𝜓
−1
, also
h
=
∑∞
n=0
y
n
(q
n
◦𝜓
−1)
in
L2(
ℝ
,
B
(
ℝ
),P𝜃)
.
Hence, he unce ain model inpu pa ame e
𝜃=𝜓
◦
𝜉
as well as ou model ou -
come
Y=h
◦
𝜓
◦
𝜉
can bo h be expanded exac ly by a polynomial se ies in he ge m,
i.e., by
⟨
q
n
,q
m⟩L
2=
‖
q
n‖2
L
2𝛿
m,n
o all m,n∈ℕ
0,
∫ℝ
snd𝜇=
∫
sndP𝝃=𝔼[𝜉n] o all n∈ℕ0
.
𝜓
=
∞
∑
n=0

𝜗nqnin L2=L2(ℝ,B(ℝ),P𝜉),
h
◦𝜓=
∞
∑
n=0
ynqnin L2=L2(ℝ,B(ℝ),P𝜉)
.
3 See Szegő (1939) o p oo s o comple eness.
4 Fo condi ions o poin wise con e gence see e.g., Jackson (1941).
1090
D.Feh le e al.
These se ies expansions a e called he polynomial chaos expansions (PCE) o
𝜃
and
Y wi h espec o he ge m
𝜉
. Mo eo e , o hogonali y o
{q
n
}
n∈ℕ
0
implies ha he
Fou ie coe icien s a e de e mined by
Now in p ac ice, equa ions (1a, b) jus i y app oxima ions o he unce ain model
inpu pa ame e
𝜃
as well as o he model ou come Y by hei unca ed PCE, i.e., by
The app oxima ions hen con e ge o he ue andom a iables,
SN(𝜃)
→
𝜃
and
SN(Y)
→
Y
in
L2
as
N
→
∞
. Ye , equa ions (2a, b) om which he coe icien s
a e de ined can in gene al no be e alua ed analy ically. This in ol es a second
(1a)
𝜃
=𝜓(𝜉)=
∞
∑
n=0

𝜗nqn(𝜉)in L2(Ω,A,P)
,
(1b)
Y
=h(𝜃)=h(𝜓(𝜉)) =
∞
∑
n=0
ynqn(𝜉)in L2(Ω,A,P)
.
(2a)

𝜗
n=
‖
qn
‖
−2
L2
⟨
𝜓,qn
⟩
L2=
‖
qn
‖
−2
L2
∫ℝ
𝜓qndP𝜉
,
(2b)

y
n=
‖
qn
‖
−2
L2
⟨
h◦𝜓,qn
⟩
L2=
‖
qn
‖
−2
L2
∫ℝ
(h◦𝜓)qndP𝝃
.
S
N(𝜃)=SN(𝜓◦𝜉) ∶=
N
∑
n=0

𝜗nqn(𝜉),
S
N(Y)=SN(h◦𝜓◦𝜉) ∶=
N
∑
n=0
ynqn(𝜉)
.
Table 1 O e iew: common dis ibu ions and co esponding ge ms and o hogonal polynomials on
L2
a
We use he scale- a e no a ion
Dis ibu ion o
𝜃
Ge m O hogonal polynomials
Family Pa ame ic
𝜉
𝜓
qn
No mal
𝜃∼N(𝜇,𝜎2)
𝜉
∶=
𝜃−𝜇
√2𝜎
𝜓(s)=𝜇+√2𝜎s
(physicis s) He mi e Hn
Uni o m
𝜃∼U(0, 1)
𝜉∶= 2𝜃−1
𝜓
(s)=
s+1
2
Legend e Ln
Be a
𝜃∼Be a(𝛼,𝛽)
𝜉∶= 2𝜃−1
𝜓
(s)= s
+1
2
Jacobi
J
(𝛽−1,𝛼−1)
n
Gamma
𝜃∼Gamma(𝛼,𝛽)1
𝜉∶= 𝛽𝜃
𝜓
(s)=
s
𝛽
Gene al Lague e
La
(𝛼−1)
n
In e se Gamma
𝜃∼In -Gamma(𝛼,𝛽)1
𝜉
∶=
𝛽
𝜃
𝜓
(s)=
𝛽
s
Gene al Lague e
La
(𝛼−1)
n
1097
Polynomial Chaos Expansion: E icien E alua ion and…
he se ies expansion o he linea policy unc ion can be w i en as
Mo eo e , he

A
𝛼
coincides wi h he expansion coe icien s om he PCE o he
model ou come
A(𝜗)
. Hence, he PCE o a linea policy is again linea and is ep e-
sen ed by he polynomial expansion o he ma ix- alued unc ion
𝜗
↦
A(𝜗)
.
A second popula app oach o compu e he model’s policy unc ion a e p ojec ion
me hods.7 In his app oach g is cons uc ed as a linea combina ion o some sui able
basis unc ions
Φi
by
The coe icien s in he PCE o g wi h espec o
𝜗
hen sa is y
and he expansion o g can he e o e be w i en as
Now obse e ha he
ci𝛼
coincide wi h he coe icien s in he polynomial expansion
o he model ou come
ci(𝜗)
, i.e., wi h he coe icien s in he PCE o he coe icien s
o he p ojec ion solu ion. Consequen ly, he PCE o g is again a linea combina-
ion o he basis unc ions
Φi
and he coe icien s a e ep esen ed by he polynomial
expansion o
𝜗
↦
ci(𝜗)
.
3 Nume ical Analysis
This sec ion p esen s he nume ical implemen a ion o a PCE o he benchma k RBC
model. Fi s , we analyze he con e gence beha io o he se ies expansion o di e -
en model ou comes o in e es . Mo e speci ically, he model ou comes include he
solu ion, he second momen s, and he impulse esponse unc ions om he model’s

g
𝛼(x)=
�‖
q𝛼
‖
−2
L2∫ℝk
q𝛼(s)A(𝜓(s))dP𝝃(s)
�
x=∶ 
A𝛼x
,
g
(x,𝜗)=
�
𝛼∈ℕk
0
g𝛼(x)q𝛼(𝜓−1(𝜗)) =
⎛
⎜
⎜
⎝�
𝛼∈ℕk
0

A𝛼q𝛼(𝜓−1(𝜗))
⎞
⎟
⎟
⎠
x
.
g
(x;𝜗)=
d
∑
i=1
ci(𝜗)Φi(x)
.

g
𝛼(x)=
d
�
i=1�‖
q𝛼
‖
−2
L2∫ℝk
q𝛼(s)
�
ci(𝜓(s))
�
dP𝝃(s)
�
Φi(x) =∶
d
�
i=1
ci𝛼Φ(x)
,
g
(x,𝜗)=
�
𝛼∈ℕk
0
g𝛼(x)q𝛼(𝜓−1(𝜗)) =
d
�
i=1
⎛
⎜
⎜
⎝�
𝛼∈ℕk
0
ci𝛼q𝛼(𝜓−1(𝜗))
⎞
⎟
⎟
⎠
Φ(x)
,
7 See, o ins ance, Judd (1996), Chap e  11, Hee and Maußne (2024), Chap e  5, Judd (1992) o
McG a an (1999).

1098
D.Feh le e al.
linea app oxima ion. Addi ionally, we conside a global p ojec ion solu ion. Second,
we compa e di e en me hods o compu e he PCE coe icien s ega ding accu acy and
e iciency. Las ly, we pe o m Mon e-Ca lo expe imen s, whe e we e alua e he pe o -
mance o PCE o empi ical applica ions as ma ching momen s and likelihood-based
app oaches—bo h o linea and non-linea solu ions.
3.1 The Model
We conside a benchma k RBC model whe e he social planne sol es he ollowing
maximiza ion p oblem
whe e
Y ,C ,N ,
and
K
deno e ou pu , consump ion, wo king hou s, and he capi al
s ock, espec i ely. Mo eo e , he log o o al ac o p oduc i i y,
z
, e ol es acco d-
ing o he AR(1) p ocess
The p ede e mined s a e a iables
x
and he non-p ede e mined con ol a iables
y
a e
3.2 Con e gence Beha iou
Fi s , o s udy he basic con e gence beha io o he PCE o a ious model ou -
comes in he benchma k RBC model, we conside an example whe e we se he
unce ain pa ame e s o
𝜃
∶=
(𝜁𝜂𝜌
)
. Mo eo e , we assume he ollowing p obabil-
i y dis ibu ions o he (s ochas ically independen ) unknown pa ame e s
The p obabili y densi y unc ions wi h suppo
Θ∶=[0.15;0.45]×[1;8]×[0.85;0.99]
a e illus a ed in Fig.1.
The ans o ma ions
𝜓i
be ween unknown pa ame e s and ge ms a e ixed as in
Table1 and he emaining pa ame e s a e calib a ed as summa ized in Table2.
max
Y
,C ,N ,K +1
U0∶=𝔼0
[∞
∑
=0
𝛽 C1
−
𝜂
(1−N )𝛾(1−𝜂)
1−𝜂
],
s. . C =Y −K +1+(1−𝛿)K ,
Y =ez K𝜁
N1−𝜁
,
gi en K
0
,z
0
,
z +1=𝜌z +𝜖 +1,𝜖 ∼iidN(0, 𝜎2).
x ∶=
�
K
z
�
and y ∶=
⎛⎜⎜⎝
Y
C
N
⎞⎟⎟⎠
.
𝜁∼0.15 +0.3
⋅
Be a(5,7), 𝜂∼1+7
⋅
Be a(3,7), 𝜌∼0.85 +0.14
⋅
U(0, 1).
1099
Polynomial Chaos Expansion: E icien E alua ion and…
Linea Policy Func ion
The i s model ou come ha we conside is he model’s linea solu ion which is
o he o m
Gi en any pa ame e alues
𝜗∈Θ
he ma ix
A
(𝜗)=
(
a
ij
(𝜗)
)
, wi h i=1, …, 6 and j=1, 2 ∈ℝ
6
×
2,
can be easily compu ed
nume ically om a ailable me hods. As desc ibed in sec ion2.3, he expansion o
he linea policy unc ion is again linea and is ep esen ed by he polynomial expan-
sion o
A(𝜗)
. Hence, ou ask is o cons uc o each mapping
aij ∶𝜗
↦
aij(𝜗)
he
unca ed PCE8
Mo eo e , we i s wan o abs ac om e o s in he compu a ion o he expansion
coe icien s
aij𝛼
and o ocus on he con e gence beha io o
a(N)
ij
→aij in
L2
as
N→∞
. The e o e, we compu e he coe icien s om ull-g id Gauss-quad a u e
ules wi h a su icien ly la ge numbe o nodes which should gua an ee ha
(
x +1
y
)
=A(𝜗)x
.
(10)
a(N)
ij (𝜗) ∶= S o
N(aij◦𝜓)(𝜓−1(𝜗)) =
∑
𝛼∈ℕ3
0
,
|
𝛼
|
≤N
aij𝛼q𝛼(𝜓−1(𝜗))
.
Fig. 1 Dis ibu ions o unce ain pa ame e s I
Table 2 Calib a ion I
1
Ins ead o pinning down he alue o
𝛾
we se he s eady s a e alue o
N=0.3
and he model’s s eady
s a e de e mines
𝛾
Pa ame e Desc ip ion Value
𝛽
Discoun ac o 0.994
𝛿
Ra e o capi al dep ecia ion 0.014
NS eady s a e labo supply
1
0.300
𝜎
S anda d de ia ion 0.010
8 We only discuss he mappings
𝜗
↦
aij(𝜗)
o
i=1, 3, …,6
and
j=1, 2
since he expansion o he
exogenous AR(1)-p ocess (
i=2
) w. . .
𝜌
is i ial.
1100
D.Feh le e al.
in eg a ion e o s in (5b) (whe e now
h=aij
) emain insigni ican . Mo e conc e ely,
we apply
N+5
nodes in each o he h ee one-dimensional quad a u e ules. We
compu e he coe icien s om he quad a u e ules and de e mine he
L2
e o om
whe e we d aw
M=105
iid sample poin s
𝜗(i)
om he dis ibu ion o
𝜃
.
The esul s a e p esen ed in Fig.2a in
log10
-base o
N=1
o
N=19
and sugges
linea con e gence o he se ies expansions o each
aij
. The
L2
e o o all compo-
nen s o he ma ix al eady alls o he o de o magni ude o
−3
o
N=7
and is as
low as
−6
o
N=19
. Mo eo e , Fig.2b also shows he ime needed o all compu-
a ions. In case o he PCE, he o al ime epo ed includes i) he compu a ion o
expansion coe icien s
aij𝛼
om he ull-g id quad a u e ules which equi e
(N+5)3
model e alua ions and ii) he subsequen ( i ial) e alua ion o he unca ed PCE
a(N)
ij
(𝜗(i)
)
a he 100,000 sample poin s. Fo compa ison, we also show he compu a-
ional ime ha is equi ed o de e mine he model solu ion
aij
(𝜗
(i))
epea edly a all
100,000 sample poin s. Mos impo an ly, since e en o
N=19
he numbe o
model e alua ions o he cons uc ion o he PCE is signi ican ly smalle a 13824
han he numbe o e alua ion poin s, he ime equi ed by he PCE emains less
han one- hi d o he ime needed o epea edly sol ing he model.
Second Momen s
The second model ou comes we conside a e he model’s second momen s. Mo e
speci ically, we conside he a iables’ s anda d de ia ions and he co ela ions
ob ained om he model’s linea policy. Ins ead o elying on simula ions, we
employ a ailable o mulae o momen s o i s -o de au o eg essi e p ocesses o he
linea solu ion. We p oceed he same way as in he p eceding pa ag aph and com-
pu e o each momen , say x, a se ies expansion x
(N)
∶=
∑
𝛼∈ℕ3
0
,
�
𝛼
�≤
Nx𝛼q𝛼(𝜓
−1
(𝜗
))
.
Impo an ly, no e ha we di ec ly cons uc he PCE o he second momen s, i.e., o
he mapping
𝜗
↦
x(𝜗)
. An al e na i e app oach o employ PCE o he second
momen s would be o i s cons uc he PCE o he linea policy and subsequen ly
use his PCE o he linea policy o compu e he second momen s.
Figu e2c again shows linea con e gence o he PCEs o each second momen .
The
L2
e o in he app oxima ion o he model’s momen s has allen o he o de
o magni ude o
−3
by
N=7
and u he declines o
−6
by
N=19
. Mo eo e ,
he compu a ion ime o he PCE e sus he ime o epea ed compu a ions o he
model’s momen s is illus a ed in Fig.2d. Fo he same easons as be o e, he ime
needed by he PCE emains h oughou signi ican ly lowe han he ime equi ed o
epea ed calcula ions.
(11)
‖
a(N)
ij −aij‖L2=
�
∫ℝ3
�
a(N)
ij (𝜗)−aij(𝜗)
�
2
dP𝜽
�1∕2
≈
�
1
M
M
�
i=1
�
a(N)
ij (𝜗(i))−aij(𝜗(i))
�
2
�
1∕
2
1101
Polynomial Chaos Expansion: E icien E alua ion and…
Impulse Response Func ion
The nex model ou comes we discuss a e he a iables’ impulse esponse unc-
ions in esponse o a one- ime shock o TFP by one condi ional s anda d de ia ion.
Fo he sake o exposi ion, we only conside he a iables’ ou comes o he nex
ou pe iods a e he shock hi s he economy and add he ema k ha he se ies
Fig. 2
L2
con e gence o PCE and compu a ion ime on an In el® Co e™i7-7700 CPU @ 3.60GHz
1102
D.Feh le e al.
expansions become mo e i ial o la e pe iods whe e he a iables con e ge back
o hei s a iona y alues. Hence, we cons uc PCEs o all a iables’ ou comes,
say
X +s
, o pe iods
s=0, …,4
. No e again ha he PCE is cons uc ed di ec ly o
each mapping
𝜗
↦
X +s(𝜗)
.
We show he
L2
e o s o e he unknown pa ame e s’ suppo in Fig.2e. Con e -
gence is again linea as
N→∞
and he
L2
e o s o all a iables’ ou comes all o
he o de o magni ude o
−5
by
N=19
. Fu he mo e, he compu a ion ime o he
PCE emains a below he ime equi ed o epea ed compu a ions o he model’s
IRFs.
P ojec ion Solu ion
The las model ou come o which we wan o illus a e he con e gence beha io
is he model’s p ojec ion solu ion compu ed om Chebyshe polynomials as basis
unc ions. Mo e speci ically, we de ine
k
∶= ln(K
∕K
⋆
(𝜗
))
whe e
K⋆(𝜗)
is he capi-
al s ock’s s a iona y solu ion and app oxima e he policy unc ion o wo king hou s
by
whe e we u he in oduce he ans o ma ion
n ∶= ln(N ∕(1−N ))
. The
Ti
a e
Chebyshe polynomials o deg ee i and
[
k;

k]×[z;z]=[ln(0.8);−ln(0.8)] × [−3
𝜎
√1−
𝜌2;3
𝜎
√1−
𝜌2
]
is he domain o he app oxi-
ma ion g. The emaining a iables a e compu ed analy ically om
k ,n
and
z
and
he coe icien s
ci,j(𝜗)
a e de e mined in such a way ha he model’s Eule equa ion
holds exac ly a 13 app op ia ely selec ed colloca ion poin s.9
We discussed in sec ion2.3 ha he expansion o he p ojec ion solu ion is again
a linea combina ion o he same basis unc ions, i.e., o
T
i
1
T
i
2
wi h
i1+i2≤4
, and
he coe icien s a e gi en by he se ies expansions o he mappings
𝜗
↦
ci,j(𝜗)
.
Hence, we cons uc unca ed PCEs,
c(N)
i
,
j∶=
∑
𝛼∈ℕ3
0
,
�
𝛼
�≤
Ncij𝛼q𝛼(𝜓−1(𝜗
))
om ull-
g id quad a u e ules wi h
N+5
nodes in each dimension. The
L2
e o ,
‖
c
(N)
i,j
−cij
‖
L
2
, in log10-basis is again dec easing linea ly as
N→∞
as displayed in
Fig.2g and he ime o cons uc ion and e alua ion o he PCEs in Fig.2h emains
h oughou signi ican ly smalle han he ime o epea ed compu a ions o he
global solu ion.
n
=g(k ,z ;𝜗)=
∑
i+j≤4
ci,j(𝜗)Ti
(
2k
−
k

k−k−1
)
Tj
(
2z
−
z
z−z−1
),
9 The colloca ion poin s a e combina ions o he ze os o he Chebyshe polynomials in he app oxima-
ion.

1103
Polynomial Chaos Expansion: E icien E alua ion and…
3.3 Compu a ion o PCE Coe icien s
In he p e ious subsec ion, ou ocus was on he con e gence beha io o he PCE
when he deg ee o unca ion N was inc eased. We he e o e abs ac ed om pos-
sible e o s in he compu a ion o he PCE coe icien s and employed a ull-g id
quad a u e ule wi h su icien ly many nodes. While ull-g id quad a u e ules ha e
he a o able p ope y ha he numbe o nodes can be easily chosen in such a way
ha hey p o ide exac in eg a ion ules o polynomials up o he desi ed deg ee,
he numbe o nodes g ows exponen ially in he dimension o he pa ame e ec-
o . Hence, hey may p o ide he mos con enien way o compu a ion o he PCE
coe icien s when he numbe o unknown pa ame e s is no oo la ge, bu hey
become quickly ine ec i e in highe dimensional p oblems. I he PCE coe icien s
a e de e mined om al e na i e me hods, he app oxima ion e o o he easible
PCE does no only include he e o om unca ion o he se ies expansion bu also
om a po en ially less accu a e app oxima ion o he PCE coe icien s ha becomes
necessa y.
In his sec ion, we now swi ch pe spec i e and analyze he con e gence beha io
o he PCE when i s coe icien s a e compu ed om di e en me hods. Nex o he
benchma k ull-g id quad a u e ule, he PCE coe icien s a e addi ionally app oxi-
ma ed by a spa se-g id Smolyak quad a u e ule and by leas squa es.
We apply ou analysis o he PCE o he model’s linea solu ion bu now con-
side a highe dimensional p oblem. The ec o o unknown pa ame e s expands o
𝜃
∶=
(𝜁𝜂𝜌𝛽
𝛿𝛾
)
.10 The assumed dis ibu ions o
𝜁,𝜂
and
𝜌
emain as be o e in
Fig.1, and he dis ibu ions o he addi ional unknown pa ame e s a e chosen as
The p obabili y densi ies o
𝛽,𝛿
and
𝛾
a e isualized in Fig.3.
We compu e he unca ed PCE (10) o each mapping
aij ∶𝜗
↦
aij(𝜗)
in he
linea policy A(𝜗)=
(
a
ij
(𝜗)
)
, wi h i=1, …, 6 and j=1, 2 ∈ℝ
6×2
. The PCE coe i-
cien s a e now de e mined ei he by i) a ull-g id Gauss quad a u e ule wi h
N+1
𝛽∼0.9 +0.09
⋅
Be a(7,4), 𝛿∼0.01 +0.01
⋅
Be a(3,3), 𝛾∼1.5 +1
⋅
Be a(5,4).
Fig. 3 Dis ibu ions o unce ain pa ame e s II
10 These a e all o he model’s pa ame e s excep he s anda d de ia ion
𝜎
which does no a ec he
model’s linea policy.
1104
D.Feh le e al.
nodes o each pa ame e (FGQ), ii) a spa se-g id Smolyak-Gauss quad a u e ule
wi h linea g ow h whe e he le el is se in such a way ha he one-dimensional
quad a u e ules include he nodes up o deg ee
N+1
(SGQ), iii) leas squa es
Fig. 4
L2
Con e gence o PCE wi h app oxima ed coe icien s and compu a ion ime on an In el®
Co e™i7-7700 CPU @ 3.60GHz I
1105
Polynomial Chaos Expansion: E icien E alua ion and…
whe e he numbe o sample poin is se ei he wice (LSMC1) o i ) h ee imes as
la ge as he numbe o unknown PCE coe icien s (LSMC2). A e cons uc ion o
he unca ed PCE by each o he ou me hods, we compu e he PCE’s
L2
e o as
in (11) om a d aw o
M=105
iid sample poin s om he pa ame e ’s dis ibu ion.
Figu e 4 shows he con e gence o he unca ed (app oxima ed) PCEs wi h
app oxima ed coe icien s o inc easing N. As expec ed, he PCE cons uc ed om
a ull-g id quad a u e ule, which should p o ide he mos accu a e de e mina ion
o he coe icien s, also shows he as es con e gence. I is ollowed by he PCE
cons uc ed om he spa se-g id Smolyak quad a u e ule while he PCEs whe e
he coe icien s a e compu ed by leas squa es pe o m wo s . Since inaccu acies in
he coe icien s o highe deg ee polynomials may ha e la ge impac on he
L2
e o
o he PCE,11 he PCEs compu ed om leas squa es e en show inc easing e o s
o la ge N. Ye , he necessa y compu a ions o he ull-g id quad a u e me hod
also equi e by a he mos ime. Figu e4k shows ha by
N=5
he cons uc ion
and e alua ion o he PCE al eady consumes mo e ime han 100,000 epea ed com-
pu a ions o he model solu ion. In compa ison, he spa se-g id quad a u e ule is
al eady signi ican ly less compu a ionally cos ly while he leas -squa es me hods a e
leas expensi e o compu e and emain less ime-consuming han epea ed compu a-
ions o he model solu ion up o
N=10
.
Finally, Fig.5 p o ides a mo e con enien illus a ion o he di e en me hods’
e iciency and plo s he PCEs’
L2
e o e sus he equi ed compu a ion ime, bo h
in
log10
-basis. Acco ding o his me ic he ull-g id quad a u e me hod al eady
pe o ms wo s and equi es he mos compu a ion ime o each he same quali y
o app oxima ion as he o he me hods. The mos e icien me hod is he spa se-
g id Smolyak quad a u e ule. In he p esen case wi h six unknown pa ame e s, i
eaches an app oxima ion wi h
L2
e o o he o de o magni ude o
−4
be o e he
equi ed ime o he PCE’s cons uc ion exceeds he ime o 100,000 epea ed com-
pu a ions o he model solu ion.
3.4 Mon e Ca lo expe imen s o empi ical me hods
3.4.1 Es ima ion Based onLinea ized Models
Design
Ou Mon e Ca lo s udy o linea ized models ollows Ruge-Mu cia (2007) and
analyzes he pe o mance o PCE when applied o di e en es ima ion me hods. We
se he ec o o unce ain pa ame e s o
𝜃∶= (𝛽,𝜌,𝜎)
and choose he ollowing p ob-
abili y dis ibu ions wi h suppo
Θ∶=[0.97;0.999]×[0.75;0.995]×[0.004;0.012]
o he unknown pa ame e s:
𝛽∼0.97 +0.029
⋅
Be a(2,2), 𝜌∼0.75 +0.245
⋅
Be a(2,2), 𝜎∼0.004 +0.009
⋅
U(0, 1).
11 No e ha he no m o he o hogonal polynomials,
‖q𝛼‖L2
, is inc easing in
|𝛼|
.
1106
D.Feh le e al.
Figu e6 illus a es he unce ain pa ame e s’ p obabili y densi ies and he emaining
pa ame e s a e calib a ed as summa ized in Table3.
The simula ed da a and he subsequen es ima ion o he pa ame e s a e bo h om
a linea ized model. While he ad an age o PCE inc eases wi h mo e sophis ica ed
Fig. 5
L2
Con e gence o PCE wi h app oxima ed coe icien s and compu a ion ime on an In el®
Co e™i7-7700 CPU @ 3.60GHz II
1113
Polynomial Chaos Expansion: E icien E alua ion and…
Table6 displays he esul s om MLE. Fi s , de ia ions be ween he es ima es
om he me hod based on he policy unc ion’s PCE, he likelihood unc ion’s PCE,
and he benchma k e sion emain ema kably small. The a e age e o conce ning
he policy unc ion’s PCE es ima ion is smalle han one pe mille compa ed o he
benchma k and ela i e o he ange o he pa ame e . Fu he mo e, as he 95 pe -
cen ile is smalle han he a e age, he e o is mos ly smalle han on a e age. The
same holds o he es ima ion wi h he likelihood unc ion’s PCE. The a e age e o
is less han a hal pe cen and he median is less han one pe mille. Using he PCE
o he policy unc ion does no educe he compu a ion ime signi ican ly, because
he e alua ion o he likelihood unc ion is he ime-consuming pa . Fo his eason,
using he PCE o he likelihood unc ion is much mo e e icien . The o al p ocedu e
is abou 50 pe cen as e han he benchma k on a e age and he pu e maximiza ion
p ocedu e akes less han hal a second on a e age.
Finally, Table7 and Table8 summa ize he esul s om he PCE-based me h-
ods—app oxima ion o he policy unc ion o he ke nel o he pos e io —in BE.
Fi s , he e o s be ween he wo app oxima ions a e i ually he same. The a e age
e o s o he means and he medians a e less han o equal o one- ou h o a pe cen .
While de ia ions sligh ly inc ease o es ima es o he pos e io ’s lowe and uppe
quan iles, hey emain almos always less han 1.25 pe cen . Recognizing ha e o s
may be pa ly caused by he RWMH algo i hm i sel , he de ia ions be ween he
me hods a e negligible. Using he PCE o he policy unc ion does no educe he
compu a ion ime signi ican ly, because he e alua ion o he likelihood unc ion is
likewise he ime-consuming pa . Fo his eason, he PCE o he likelihood unc-
ion is much mo e e icien and nea ly 99 pe cen as e han he benchma k.15
3.4.2 Es ima ion Based on heGlobal Solu ion
We p oceed wi h ou analysis by conduc ing he p e ious likelihood-based es i-
ma ion o global, i.e., non-linea model solu ions. On he one hand, he model’s
linea solu ion allowed an analy ical de i a ion o he objec i e unc ion o he
es ima ions and, consequen ly, an exac assessmen o he goodness o hei PCE
app oxima ion. On he o he hand, he solu ion and he de i a ion o he objec i e
unc ions a e as by hemsel es. Consequen ly, ime is no c i ical. Non-linea
solu ions and likelihood unc ion e alua ion wi h pa icle il e s ely on nume i-
cal, pa ly Mon e Ca lo, me hods, which makes he assessmen ague. Howe e ,
hese me hods a e ime-consuming, making PCE an in e es ing me hod o o e -
come hese bu dens.
We ollow Fe nández-Villa e de and Rubio-Ramí ez (2005). The au ho s show
ha he non-linea i ies a e c ucial o pa ame e in e ence, e en o ou benchma k
RBC model. We de ia e om ou p e ious s udy and ollow Fe nández-Villa e de
and Rubio-Ramí ez (2005) by conside ing only one ue alue o he pa ame e s
𝜃o
and he p io dis ibu ion choice, which is now uni o m in all dimensions. The la e
allows us o ocus on he e ec s o he non-linea solu ion. The o me is o e alua e
15 I mus be men ioned ha a highe numbe o pa ame e s leads o a dec ease in e iciency.

1114
D.Feh le e al.
ou es ima o s by compa ing he es ima ed a e age wi h he ue alues, as an exac
objec i e unc ion o he assessmen is missing. We se he ue pa ame e alues
𝜃o={𝛽o,𝜌o,𝜔o}={0.985, 0.9725, 0.0085}
and he p io s
No e ha he domain o he p io s o
𝜌
and
𝜔
emains while o
𝛽
, he domain
sh inks. The la e is because
𝛽
is well-iden i ied. Ou side his domain, he likeli-
hood is oo small (
<exp(−1000)
) o an accu a e pa icle il e e alua ion. In he
discussion below, we de o e ou sel es o cases whe e he model ou come is no
well-de ined o canno be compu ed in a nume ically s able way a all nodes o he
quad a u e ules.
Las ly, some in o ma ion on he non-linea solu ion and he pa icle il e : we
apply he p ojec ion solu ion desc ibed abo e wi h
[
k;

k]×[z;z]=[ln(0.9);−ln(0.9)] × [−2
𝜎
√1−
𝜌2;2
𝜎
√1−
𝜌2
]
and use a gene alized boo -
s ap pa icle il e wi h 2,000 pa icles (see He bs & Scho heide, 2016 Algo-
i hm14). We conduc he exe cises o
M=96
di e en da ase s, each simula ed
using he globally sol ed model. I no o he wise s a ed, we s ill obse e
T=200
pe iods o
Y
.
Maxmimum Likelihood
In he maximum likelihood analysis, we can only compa e he maximum o he
likelihood om he Kalman il e using a linea solu ion and o he PCE app oxi-
ma ed likelihood as he likelihood di ec ly om he pa icle il e is no di e en i-
able— uling ou g adien -based op imize . The li e a u e e e s o he use o di e -
en iable likelihood su oga es o non-g adien -based op imize s. While he la e is a
esea ch opic i sel , we con ibu e o he o me idea by assessing he possibili y o
su oga e he likelihood wi h PCE.16
Figu e7 p esen s he esul s dependen on he unca ion le el (
N∈{8, 9, ..., 14}
).
The uppe h ee panels ((a)–(c)) display he bias o he es ima o s ela i e o he ue
pa ame e alues, and he middle h ee panels ((d)–( )) he ela i e s anda d de i-
a ions o he es ima o s. The las wo panels ((g) and ( )) indica e he amoun o
a success ul PCE app oxima ion, i.e., inne maxima (g), and he ime di e ences
( ). I u ns ou ha bo h app oxima ions (linea solu ion and PCE su oga ed likeli-
hood) es ima e on a e age
𝛽
well. Bo h a e on a e age wi hin he ange o
±0.02%
.
The es ima es o
𝜌
a e mo e biased. Ye , o unca ions
N≥10
, he PCE es ima o
becomes no iceably less biased. The bigges di e ence be ween he es ima ion s a -
egies is conce ning
𝜎
. While o
N≥10
he PCE es ima es luc ua e close a ound
he ue alue, he es ima e om he linea solu ions de ia es on a e age by 3.25%
om he pa ame e ’s ue alue. The analysis shows, ha he es ima o s o he PCE
su oga e a e less o equal biased. Ye , he es ima o ’s luc ua ion is highe . Howe e ,
𝛽∼0.98 +0.01
⋅
U(0, 1),𝜌∼0.75 +0.245
⋅
U(0, 1),𝜎∼0.004 +0.009
⋅
U(0, 1).
16 No e ha in ou example he PCE likelihood su oga e MLE is on a e age mo e accu a e han he
a e age pos e io modes om he epea ed global solu ion sample .
1115
Polynomial Chaos Expansion: E icien E alua ion and…
he es ima o ’s s anda d de ia ion con e ges wi h N o he s anda d de ia ion o he
linea solu ion es ima es and is al eady simila o
𝛽
and
𝜎
o
N≥13
.
The amoun o success ul PCE, i.e., likelihood maxima a he bounds, inc eases
om
85%
o
N=8
abo e 95% o
N≥8
and equals 100% o
N=14
. One maxi-
miza ion wi h he PCE app oxima ion akes on a e age be ween 9 min (
N=8
) and
40 min (
N=14
) and akes much longe han wi h he use o he linea solu ion (14
Fig. 7 ML om a ious likelihood app oxima ions (
M=96
). PCE: PCE app oxima ed likelihood om a
pa icle il e , SPCE: Only success ul PCE app oxima ions (
MS
), i.e., exclusion o maxima a he pa am-
e e bounds. LinRep: Repea ed likelihood e alua ion using he Kalman Fil e om he linea model solu-
ion. N equals he unca ion le el, he quad a u e le el equals N+1. Compu a ion ime on one co e o an
AMD® EPYC™7313 (Milan) CPU @ 3.00GHz
1116
D.Feh le e al.
Fig. 8 Obse able: Ou pu
Y
.
𝜖 j
: mean e o . E o s a e exp essed as de ia ions om he benchma k
me hod o epea edly sol ing he (global) policy unc ion in pe cen o he ange o he pa ame e ’s dis-
ibu ion.
N=13
equals he unca ion le el, he quad a u e le el equals
N+1
. Pe o med on one co e o
an AMD® EPYC™7313 (Milan) CPU @ 3.00GHz
Table 9 Compu a ional ime compa ison
N=13
equals he unca ion le el, and he quad a u e le el equals
N+1
. Pe o med on one co e o an
AMD® EPYC™7313 (Milan) CPU @ 3.00GHz
linea epea ed pos e io PCE policy c . PCE non-linea epea ed
hh:mm:ss 00:05:45 00:32:27 17:45:11 18:17:37
1117
Polynomial Chaos Expansion: E icien E alua ion and…
sec). Howe e , he du a ion o he likelihood e alua ion o he non-linea model is
s ill quick and can be educed easily and d as ically ia pa alleliza ion.17
Finally, he p oblem a ises in whe he he es ima o ’s s anda d de ia ion and he
emaining bias a ise gene ally om he maximum likelihood me hod and he pa i-
cle il e o om limi a ions o he PCE app oxima ion. We can iden i y he easons
by imp o ing he p ope ies o he ue MLE and he pa icle il e . To dec ease he
bias and s anda d de ia ion o he ue MLE, we inc ease he numbe o obse a-
ions (T=500) c.p., and o dec ease he noise in he pa icle il e , we inc ease he
numbe o pa icles o 10,000 c.p. Appendix 7 p esen s he esul s (Fig.12 and 13).
The addi ional in o ma ion (T=500) leads o a simila dec easing s anda d de ia-
ion o bo h es ima o s he PCE su oga e likelihood and he likelihood om he
model’s linea app oxima ion. Howe e , while he bias o he MLE om he PCE
su oga e likelihood sh inks u he , he bias o he MLE om he linea solu ion
only dec eases o
𝜌
. The bias o
𝛽
and
𝜎
emain o e en inc ease. Fu he , wi h
mo e in o ma ion, he PCE app oxima ion becomes mo e s able. Rega ding he
highe amoun o pa icles, he e is unsu p isingly no imp o emen in he bias o he
es ima es om he PCE app oxima ed likelihood. Howe e , he es ima o ’s s and-
a d de ia ion dec eases o all conside ed unca ion le els. Wi h hese wo esul s,
we conclude ha PCE app oxima ion e o s a e nei he he d i e s o he emaining
inaccu acies no limi s a highe accu acy.
Bayesian Es ima ion
No e ha in a Bayesian con ex besides he mode, we canno obse e he ue
s a is ics o he pos e io dis ibu ion. Since he pos e io s a is ics ob ained om
he global p ojec ion solu ion and he gene ic boo s ap pa icle il e should be a
leas unbiased (see Fe nández-Villa e de & Rubio-Ramí ez, 2005) we use hem as
a benchma k case and compa e i wi h h ee o he me hods o e alua e he pos e-
io : i) he linea app oxima e solu ion combined wi h a likelihood ob ained om
he Kalman-Fil e , ii) he PCE su oga e o he pos e io ke nel, iii) and he PCE
app oxima ion o he global p ojec ion solu ion oge he wi h he likelihood om
he gene ic pa icle il e . Fo bo h QoIs, we se he unca ion and he quad a u e
le el o
N=13
and
M=N+1
, espec i ely.
As in he linea se up, we use he RWMH algo i hm o gene a e 100,000 d aws
om he pos e io dis ibu ion. Howe e , since ini ializing he algo i hm a he pos-
e io mode is di icul when he likelihood is app oxima ed by a pa icle il e (see
he discussion abo e), we depa om he linea se up and speci y he algo i hm’s
p oposal densi y using es ima es o he pos e io mean and a iance. We ob ain
hese es ima es om 10,000 addi ional d aws om a RWMH algo i hm based on a
p oposal densi y pinned down by he p io ’s mean and a iance.
Figu e 8 displays he mean absolu e de ia ions ( ela i e o he ange o he
pa ame e ’s dis ibu ion) o he h ee compe ing me hods o he benchma k case o
17 We use he e only one co e. Hence he compu a ional ime o he PCE app oxima ion can be oughly
di ided by he amoun o a ailable co es, e.g., wi h
>160
co es, he
N=14
app oxima ion should
become as e han he linea app oxima ion, igno ing wo ke s’ alloca ion ime.
1118
D.Feh le e al.
a ious pos e io s a is ics and Table9 gi es an o e iew o e he a e age compu a-
ional ime o one es ima ion.18
Fo all h ee es ima ion pa ame e s and all displayed s a is ics o he pos e io ,
he PCE ex ension o he global p ojec ion solu ion yields he es ima ion esul s ha
come closes o he benchma k case. While he a e age absolu e de ia ion is well
below hal a pe cen o all pa ame e s, he a e age ime equi ed o one es ima-
ion (17h:45m:11s) is only a ound hal an hou less compa ed o he benchma k
(18h:17m:37s). Howe e , his di e ence depends on he ac ion o compu a ional
ime o he solu ion on he o al ime. No e ha he solu ion becomes quickly mo e
ime-consuming han he il e wi h an inc easing numbe o s a es.
Fo
𝛽
, he de ia ions be ween he PCE expansion o he pos e io ke nel and he
benchma k case a e simila o hose o he linea app oxima ion o he model solu-
ion. Howe e , o
𝜌
and in pa icula
𝜎
, he esul s a e signi ican ly close o he
benchma k me hod, wi h abou one and almos wo pe cen age poin s lowe de ia-
ion. This, oge he wi h he ac ha he compu a ional ime equi ed (00h:32m:27s)
is signi ican ly lowe , makes he PCE su oga e o he Pos e io ke nel a p omising
al e na i e o he benchma k me hod.
In con as , he es ima es based on he linea app oxima ion o he model solu ion
a e compu a ionally much mo e a o able (00h:05m:45s) bu also de ia e he mos
om he benchma k case wi h an a e age absolu e de ia ion om jus unde wo
o almos ou pe cen . In line wi h he esul s by Fe nández-Villa e de and Rubio-
Ramí ez (2005), we documen ha o he pa ame e
𝜎
he de ia ions a y sys em-
a ically o di e en pe cen iles o he pos e io , as he mean absolu e de ia ions
be ween he linea and he global benchma k es ima ion me hod dec ease by mo e
han one and a hal pe cen age poin s om he 5- h o he 95- h pe cen ile.
Discussion
Ou s udy o PCE o es ima ing a s anda d RBC model shows ha he PCE-based me h-
ods deli e su icien accu a e su oga es o ep oduce he esul s o es ima es om he
benchma k p ocedu e— epea edly sol ing he model. Gains in e iciency a e la ge han
50 pe cen o ma ching momen s i he PCE o he policy unc ion is used and o MLE
i he PCE o he likelihood unc ion is used. Addi ionally, he PCE o he likelihood
om a pa icle il e is di e en iable and, hus, enables a g adien -based op imiza ion.
Gains in e iciency a e la ge han 95 pe cen o BE wi h he chosen numbe s o pa am-
e e s, unca ion deg ee, and quad a u e le el i he PCE o he pos e io ’s ke nel is used.
In ou speci ica ion o he p io dis ibu ions we shape and shi he dis ibu-
ions o achie e compac ness o he suppo . This p ocedu e is uncon en ional in
he Bayesian es ima ion o DSGE Models bu helps o PCE. Fi s and o emos , he
compac ness o he suppo helps o c ea e a se ing whe e he mapping om pa am-
e e s o he model ou come is squa e-in eg able. Second, i is indispensable o he
cons uc ion o he PCE coe icien s ha he model ou come is well-de ined and can
18 We p o ide he comple e es ima ion esul s (incl. a ious e o pe cen iles) in Tables12, 13 and 14 in
Appendix 7.

1119
Polynomial Chaos Expansion: E icien E alua ion and…
be compu ed in a nume ically s able way a all nodes o he quad a u e ules.19 He e,
impo ance sampling o leas squa es, adap i e spa se g ids, o g id domain educ-
ions p oduces a emedy.20
In non-Bayesian app oaches, he applica ion o PCE demands he o he wise
unnecessa y speci ica ion o p io dis ibu ions. As
L2
con e gence o he se ies
expansion is achie ed w. . . his p io dis ibu ion o he pa ame e s, es ima ion
esul s become less accu a e i he ue pa ame e alue is a odds wi h he choice o
p io s, especially i he ue pa ame e is ou side he p io domain.21
Simila ly, Lu e al. (2015) show ha using PCE o BE may be inaccu a e in wo
cases. Fi s , he QoI is ep esen ed poo ly by a low-o de polynomial. Second, he
pos e io mass is in o he egions han he p io mass. To sol e hese p oblems, hey
sugges an adap i e inc easing polynomial o de by e i ying he accu acy a he
nex e alua ion poin . As ou manual adap ion is usually no easible as i equi es
he benchma k esul s, his is also a p ac ical me hod o de e mining he unca ion
le el in gene al. In addi ion, a small magni ude o he N h Fou ie coe icien indi-
ca es a su icien ly high unca ion le el.
As PCE is a spec al decomposi ion app oxima ed wi h a unca ed polynomial
expansion, gene ally, Runge’s and Gibbs’ phenomena could a ise. Bo h esul in spu i-
ous oscilla ion. Ye , using Gaussian quad a u es and nodes p e en s he o me phe-
nomenon, and he la e phenomenon only appea s in he p esence o discon inui y
jumps. P oblems wi h he app oxima ion o a la unc ion a e unknown. Thus, he
equen lack o iden i ica ion o DSGE models does no challenge PCE i sel .
Conce ning ime, he success o PCE is de e mined by he a io o he numbe o
model e alua ions necessa y o compu e he coe icien s and he numbe o model
e alua ions o he exe cise a hand. Hence, PCE wo ks bes in cases wi h a small
numbe o unknown pa ame e s whe e he exe cise demands many model e alua-
ions. On he one hand, PCE loses e iciency in highe dimensional p oblems. On
he o he hand, mos exe cises a e ecu si e (Mon e Ca lo sample , g adien -based
op imize , e c.), whe e he model e alua ions a e independen o each o he o con-
s uc ing PCE. This independence makes he cos ly e alua ions pa allelizable—
educing he cu se o dimensionali y d as ically wi h clus e o cloud compu ing. In
addi ion, Soize and Descelie s (2010) de elop ools o educe he e alua ion ime o
he cons uc ed PCE.
Finally, ou analysis is limi ed o an e godic, s able s ochas ic p ocess. Howe e ,
Ozen and Bal (2016) show ha , wi h some adap ions, PCE becomes sui able o
ime-dependen solu ions and Jacquelin e al. (2015) o models wi h de e minis ic
eigen equencies.
19 Fo example, la ge alues o he capi al sha e quickly esul in nume ical p oblems o he compu-
a ion o he linea app oxima ion o he policy unc ion, and oo la ge dis ances o he ue pa ame e
esul in minus in ini y log-likelihood alues.
20 Fo he la e , no e ha he p io s mus no change as o he wise in o ma ion om he da a would en e
he p io .
21 To pu i simply, he p io dis ibu ion in such cases is only a guess ha de e mines he accu acy o he
solu ion in di e en anges o pa ame e alues.
1120
D.Feh le e al.
4 Conclusion
The p esen a icle discusses he sui abili y o PCE o compu a ional models in eco-
nomics. Fo his pu pose, we i s p o ide he heo e ical amewo k o PCE, e iew he
basic heo y, and gi e an o e iew o common dis ibu ions and co esponding o hogo-
nal polynomials. We show how o use he expansion as a poin wise app oxima ion o
he QoI, e.g., o su oga e he linea ized policy unc ion o a policy unc ion based on
p ojec ion me hods.
Second, we analyze PCE when applied o a s anda d RBC model and p o ide
p ac ical insigh s. We s udy con e gence beha io o a ious QoIs and compa e he
mos common me hods o compu e he PCE coe icien s o a lowe dimensional
and a highe dimensional p oblem. Mon e Ca lo expe imen s o di e en empi i-
cal me hods show ha he PCE-based me hods can accu a ely ep oduce he esul s
o he benchma k me hod o epea edly sol ing he model. Gains in e iciency a e
la ge, especially o Bayesian in e ence.
Ou discussion add esses po en ial d awbacks o he me hod. Fi s , he e iciency
o PCE su e s om he cu se o dimensions in p oblems wi h nume ous unknown
pa ame e s. Fu he , poo ly chosen p io s may a ec he accu acy o he es ima es.
PCE is a powe ul ool o a b oad se o applica ions and he ecen li e a u e
add esses he highligh ed d awback. We hope his a icle can encou age applica ions
o PCE in economics. Especially, o pa ame e in e ence in complex models whe e
nume ous epea ed solu ions a e in easible o when ime is c i ical as in eal- ime
analysis o high- equency da a.
5 Supplemen a y in o ma ion
MATLAB® code and eplica ion ile a e a ailable a www. johan neshu be . de/ PCE.
Appendix1 ASimple Example
He e, we wan o ou line he concep a a simple bu analy ically ac able cons uc-
ion o a PCE. Since ou nume ical analysis ocuses on disc e ely- imed models, ou
example conside s he ollowing sys em o linea i s -o de di e ence equa ions in
wo eal- alued a iables
x1,
and
x2,
,
o all
∈ℕ
, and gi en
x1,0
and
x2,0
. Mo eo e ,
𝜗∈(0, 1)
is an unknown pa ame e .
While he a iables’ explici ecu sion can be de i ed s aigh o wa dly he e by
he mapping
𝜗
↦
H(𝜗)
om he unknown pa ame e o he (linea ized) policy can
ypically no be de i ed analy ically, bu can only be compu ed nume ically i he
sys em o di e ence equa ions is non-linea and s ochas ic. In consequence, i
H(𝜗)
𝜗x
1, +1
+x
2, +1
=x
1,
,
x1, +1
+x
2, +1
=x
2, ,
(
x1, +1
x2, +1
)
=H(𝜗)
(
x1,
x2,
)
, whe e H(𝜗) ∶=
(
h11(𝜗)h12(𝜗)
h21(𝜗)h22(𝜗)
)
=
(−1
1−𝜗
1
1−𝜗
1
1−𝜗
−𝜗
1−𝜗),
1121
Polynomial Chaos Expansion: E icien E alua ion and…
needs o be compu ed o di e en pa ame e alues, he unde lying nume ical me h-
ods mus e en ually be applied epea edly. PCE, on he o he hand, aims o ep esen
he mapping
𝜗
↦
H(𝜗)
as a unca ion om he Fou ie se ies
whe e
qn
is he n- h polynomial om a amily o o hogonal polynomials,
𝜓−1(𝜗)
is
a ans o ma ion o he pa ame e space in o he space o he polynomial o hogonal
coun e pa ’s a gumen , and

h(n)
ij
is he co esponding Fou ie coe icien o he poly-
nomial. The unca ed se ies expansion is cons uc ed om a limi ed numbe o
nume ical e alua ions o he mapping as ollows.
Fi s , he unce ain y abou he pa ame e is aken in o accoun by desc ibing i by
a andom a iable
𝜃
wi h sui able p obabili y dis ibu ion
P𝜃
. Fo he p esen example,
suppose ha
𝜃
is uni o mly dis ibu ed o e he in e al
(0, b),0<b≤1
. Second, he
se ies expansion is cons uc ed in a well-known amily o o hogonal polynomials, which
sa is ies o hogonali y w. . . some weigh ing unc ion w. The eby, he app op ia e amily
o o hogonal polynomials is mos con enien ly chosen in such a way ha he weigh -
ing unc ion w coincides wi h he p obabili y densi y unc ion o he unknown pa ame e .
Howe e , in o de o achie e con o mi y be ween he weigh ing unc ion and he densi y
unc ion, a (linea ) ans o ma ion o he pa ame e ypically becomes necessa y. In he
p esen case, Legend e polynomials
{Ln}n≥0
a e o hogonal w. . . he weigh ing unc ion
w(s)=1(−1,1)(s)
, i.e., hey sa is y
Hence, ans o ma ion o he unknown pa ame e
𝜃
o he so-called ge m
𝜉
by
yields he desi ed esul , and Legend e polynomials a e o hogonal w. . . he p oba-
bili y dis ibu ion
P𝜉
o
𝜉
. Gi en ha
b<1
, he mapping
s
↦
hij(𝜓(s))
o each en y
hij
o he ma ix H is squa e in eg able w. . .
P𝜉
and can be ep esen ed by a Fou ie
se ies o he o m22
Mo eo e , o hogonali y implies ha he Fou ie coe icien s

h(n)
ij
sa is y
h
ij(𝜗)=
∞
∑
n=0

h(n)
ij qn(𝜓−1(𝜗))
,
�
ℝ
Ln(s)Lm(s)w(s)ds=
�0, i n≠m,
‖
Ln
‖
2∶= 2
2n+1
, i n=m
.
𝜉
∶= 𝜓−1(𝜃) ∶= 2𝜃
b
−1⇔𝜃=𝜓(𝜉)=
(𝜉+1)b
2,
(12)
h
ij(𝜓(s)) =
∞
∑
n=0

h(n)
ij Ln(s)
.
22 The de ails in which sense con e gence o he se ies can be es ablished a e discussed in he nex sec-
ion.
1122
D.Feh le e al.
Finally, nume ical in eg a ion me hods a e gene ally equi ed o compu e he coe i-
cien s

h(n)
ij
. Fo example, using Gauss-Legend e-quad a u e wi h M nodes
si
and
weigh s
𝜔i
yields23
Table10 shows o
b=0.9
and
M=5
he quad a u e weigh s
𝜔i
, he nodes
si
, he
co esponding e ans o med pa ame e alues
𝜗i∶= 𝜓(si)
, and o he ma ix en y
h11
he e alua ion
h
11(𝜗i)=
−1
1−𝜗
i
.
Toge he wi h
L0(si)=1
,
L1(si)=si
,
‖
L
0‖2
=
2
, and
‖
L1
‖
2=
2
3
, one can he e o e
compu e, e.g.,24

h
(n)
ij =
‖
Ln
‖
−2
∫1
−1
hij(𝜓(s))Ln(s)ds
.

h
(n)
ij ≈
‖
Ln
‖
−2
M
�
i=1
hij(𝜓(si))Ln(si)𝜔i
.
Table 10 Example
i
𝜔i
si
𝜗i
h11(𝜗i)
1 0.2369
−
0.9062 0.0422
−
1.0441
2 0.4786
−
0.5385 0.2077
−
1.2621
3 0.5689 0 0.4500
−
1.8182
4 0.4786 0.5385 0.6923
−
3.2500
5 0.2369 0.9062 0.8578
−
7.0314
23 I we addi ionally w i e he ans o ma ion
𝜓
be ween pa ame e and ge m in e ms o he Legend e
polynomials, i.e.,
we equi alen ly a i e a
No e ha his exp ession is iden ical o he mo e gene al o m in (3).
𝜓
(s)=
b
2
���
=∶ 
𝜗0
L0(s)+
b
2
���
=∶ 
𝜗0
L1(s)
,

h
(n)
ij ≈
‖
Ln
‖
−2
M
�
i=1
hij�
𝜗0L0(si)+ 
𝜗1L1(si)�Ln(si)𝜔i
.
24 Fo compa ison, exac in eg a ion yields

h
(0)
11 =1
2∫
1
−1
−1
1−(s+1)b
2
ds=ln(1−b)
b=−2.56,

h
(1)
11 =3
2∫1
−1
−s
1−(s+1)b
2
ds=6−3b
b2ln(1−b)+ 6
b=−
2.71.
1129
Polynomial Chaos Expansion: E icien E alua ion and…
and ha he gene alized Lague e polynomials
{
La
(𝛼−1)
n
}
n∈
ℕ
0
also o m a comple e
o hogonal sys em in
L2
(ℝ,B(ℝ),dP
𝜉)
wi h
Mo eo e , gi en he nodes
sj
and weigh s
𝜔 j
om he common Gauss-Lague e-
quad a u e ule o weigh ing unc ion
w(., 𝛼−1)
, he Gauss-quad a u e ule in
e ms o weigh ing unc ion
w(., 𝛼,𝛽)
has he same nodes while he weigh s a e
scaled by
𝜔
j=
𝜔
j
Γ(𝛼)
.
Appendix3 In usi e model expansion
S ochas ic Gale kin
Fo bo h me hods discussed in sec ion2.1.2, he compu a ion o he expansion
coe icien s is de ached om he unde lying p ocedu e om which he model ou -
come is compu ed. This is di e en om he hi d me hod. Ins ead o a mo e gene al
discussion, we he e o e only illus a e his me hod o he case whe e he PCE o a
model’s policy unc ion is cons uc ed. To simpli y he no a ion, suppose ha he
equa ions de ining he model’s solu ion can be educed o a sole Eule equa ion in a
single a iable. Le
S⊂ℝs
deno e he model’s s a e space and le
g∶S
→
ℝ
deno e
he a iable’s policy unc ion. The Eule equa ion is ypically ansla ed in o a unc-
ional (in eg al) equa ion o g, say
I he unc ional equa ion can no be sol ed analy ically, a common app oach is o
cons uc an app oxima ion
g
om linea combina ions o some basis unc ions,28
say
Φj,j=1, …,d
, i.e.,
In o de o de e mine he coe icien s
yj
in he app oxima ion, which now se es
as ou model ou come o in e es and should no be con used wi h he Fou ie
∫
ℝ
La(𝛼−1)
n(s)La(𝛼−1)
m(s)dP𝜉(s)=
=∫ℝ
La𝛼−1)
n(s)J(𝛼−1)
m(s)w(s;𝛼,𝛽)ds
=1
Γ(𝛼)∫∞
0
La(𝛼−1)
n(s)La(𝛼−1)m(s)w(s;𝛼−1)d
s
=Γ(n+𝛼)
Γ(𝛼)n!
𝛿nm.
R(g,x)=0 o all x∈S.

g
(x)=
d
∑
j=1
yjΦj(x)
.
28 Mos commonly hese a e selec ed ei he as ( enso p oduc s o ) Chebyshe polynomials o as piece-
wise linea o cubic polynomials.

1130
D.Feh le e al.
coe icien s o he PCE, one can, o example, selec d app op ia e colloca ion poin s
x1,…,xd∈S
and sol e he non-linea sys em o equa ions gi en by
o
y1,…,yd
.
Now conside he case whe e one pa ame e is unce ain and hence desc ibed by
he andom a iable
𝜃
. I he model’s ( educed) Eule equa ion in ol es
𝜃
, hen so
does he unc ional equa ion o g, i.e., we now w i e
Mo eo e , i one employs he abo e-men ioned solu ion me hod, he coe icien s
yj
will ypically also depend on
𝜃
, i.e., we ha e, in sligh abuse o no a ion,
Yj=hj(𝜃)
.
In pa icula , he mappings
hj
be ween he
Yj
and
𝜃
a ise implici ly om he non-
linea sys em o equa ions
In o de o a oid he necessi y o epea ed and po en ially compu a ionally expen-
si e solu ions o his sys em o equa ions o di e en alues o
𝜃
, one may wan o
ind o each
Yj
a PCE in e ms o some chosen ge m
𝜉
29
The PCE o he model’s (app oxima ed) policy unc ion wi h espec o he ge m
𝜉
is hen gi en by
Mo eo e , he Fou ie coe icien s
yjn
in he PCE can be de i ed by a Gale kin
me hod i we subs i u e he
Yj
in hei implici de ini ion in (13) wi h hei PCE and
impu e he co esponding condi ions
R(d
∑
j=1
yjΦj,xi
)
=0 o all i=1, …,d
R(g,x;𝜃)=0 o all x∈S.
(13)
R(d
∑
j=1
YjΦj,xi;𝜃
)
=0 o all i=1, …,d
.
𝜃
=𝜓(𝜉)=
∞
∑
n=0

𝜗nqn(𝜉),
Y
j=hj(𝜃)=hj(𝜓(𝜉)) =
∞
∑
n=0
yjnqn(𝜉)
.

g
(x;𝜉)=
d
∑
j=1
YjΦj(x)=
d
∑
j=1
∞
∑
n=0
yjnqn(𝜉)Φj(x)
.
29 No e ha in his case we ha e d model ou comes o in e es , namely he coe icien s
Yj=hj(𝜃)
in
g
.
1131
Polynomial Chaos Expansion: E icien E alua ion and…
Hence, we can sol e o he
d(N+1)
unknown coe icien s
yjn
in he unca ed PCE
Yj
≈
∑N
n=0
y
jn
q
n
(𝜉
)
om he sys em o equa ions
o
i=1, …,d
and
m=0, …,N
. The in eg al is compu ed nume ically, ei he
om Mon e-Ca lo d aws o om an app op ia e Gauss quad a u e. Mo eo e ,
𝜓(𝜉)
can be subs i u ed by i s unca ed se ies expansion as p e iously desc ibed in
subsec ion2.1.1.
Appendix4 Smolyak‑Gauss‑Quad a u e
Suppose ha o e e y
i=1, …,k
he dis ibu ion
P
𝜉
i
o
𝜉i
possesses a p obabili y
densi y unc ion
wi
, so ha
w
∶=
∏k
i=1
w
i
is he p obabili y densi y o
P𝝃
. Then (6b)
becomes
Fu he , suppose ha one-dimensional Gauss-quad a u e ules co esponding o
weigh ing unc ions
wi
and o hogonal polynomials
{q
in
}
n∈ℕ
0
a e a ailable. Fo
i=1, …,k
le
Qi(Mi)
deno e his one-dimensional Gauss-quad a u e ule wi h
Mi
nodes
{
s
(j)
i
,
M
i
}j=1,…,M
i
and weigh s
{
𝜔
(j)
i
,
M
i
}j=1,…,M
i
, i.e.,
Then choose o each
i=1, …,k
an inc easing sequence o na u al numbe s
{
M
ij}j∈
ℕ⊂ℕ
,
M
ij+1
>M
ij
and de ine he di e ence ope a o by
R
(d
∑
j=1
∞
∑
n=0
yjnqn(𝜉)Φj,xi;𝜓(𝜉)
)
=0 in L2,∀i=1, …,d
⇔⟨
R
(
d
∑
j=1
∞
∑
n=0
yjnqn(𝜉)Φj,xi;𝜓(𝜉)
)
,qm(𝜉)
⟩
L2
=0∀i=1, …,d,∀m∈ℕ0
.
0
≈
⟨
R
(d
∑
j=1
N
∑
n=0
yjnqn(𝜉)Φj,xi;𝜓(𝜉)
)
,qm(𝜃)
⟩
L2
=∫ℝ
R
(
d
∑
j=1
N
∑
n=0
yjnqn(𝜉)Φj,xi;𝜓(𝜉)
)
qm(𝜉)dP𝜉(𝜉
)
(14)
y
𝛼=
‖
q𝛼
‖−2
L2×
∫ℝ
…
∫ℝ
h(𝜓(s1,…,sk))q1𝛼1(s1)…qk𝛼k(sk)w1(s1)…wk(sk)ds1…dsk
.
Q
i(Mi)g∶=
M
i
∑
j=1
𝜔(j)
i,Mi
g(s(j)
i,Mi
) o g∈L2
i
.
Δi1∶= Qi(Mi1)and Δij ∶= Qi(Mij)−Qi(Mij−1),j≥2.
1132
D.Feh le e al.
The Smolyak-Gauss-quad a u e ule o o de
l∈ℕ
and wi h g ow h ules gi en by
{Mij}j∈ℕ
is de ined by
o equi alen ly aking ca e o duplica e e ms in he di e ence ope a o s
Applying he Smolyak-Gauss-quad a u e ule o (14) in pa icula yields he
app oxima ion
This p ocedu e equi es o e alua e he model ou come o in e es
h(
𝜓
(
s(j1)
1,M1,𝜈
1
…s(jk)
k,Mk,𝜈
k))
a all spa se-g id poin s.
Appendix5 Monomial ules
S oud (1971) in oduces spa se nume ical in eg a ion wi h monomial ules and p e-
sen s a ious ules o in eg a e in di e en spaces. In his sec ion, we p esen some
nume ical esul s o he calcula ion o he PCE coe icien s.
Rosenb ock unc ion
To show he gene al unc ioning o he monomial quad a u e ules, we i s eplica e
he exe cise o Bhusal and Subba ao (2020), i.e., app oxima e he Rosenb ock unc ion
Q
l∶=
∑
𝜈∈ℕk
|𝜈|≤k+l
k
⨂
i=1
Δi𝜈i
.
Q
l=
∑
𝜈∈ℕk
max{k,l+1}≤|𝜈|≤k+l
(−1)k+l−1
(
k−1
k+l−
|
𝜈
|)k
⨂
i=1
Qi(Mi𝜈i)
.
y𝛼≈
�
k
�
i=1‖qi𝛼i‖2
L2
i
�−1
�
𝜈∈ℕk
max{k,l+1}≤�𝜈�≤k+l
(−1)k+l−1
�
k−1
k+l−�𝜈�
�
M1,𝜈1
�
j1=1
…
Mk,𝜈k
�
jk=1
𝜔(j1)
1,M1,𝜈1
…𝜔(jk)
k,Mk,𝜈k
×h�𝜓�s(j1)
1,M1,𝜈1
…s(jk)
k,Mk,𝜈k��
×q1𝛼1
�
s(j1)
1,M1,𝜈
1�
…qk𝛼k
�
s(jk)
k,Mk,𝜈
k�
.
1133
Polynomial Chaos Expansion: E icien E alua ion and…
wi h PCE. We conside he cases whe e
xi∼U(−2, 2)
and
d∈{5, 6}
. As Bhusal
and Subba ao (2020), we conside he CUT-8 and CUT-6 ules om Adu hi e al.
(2018) and ull enso g id, spa se g id, and leas squa es om he main body o
he pape . We conside a unca ion a le els 4, 5, and 6. Las ly, no e ha o hose
dimensions (
d∈{5, 6}
), we could no ind any monomial ules p esen ed by S oud
(1971) highe deg ee 5 ha ha e solely non-nega i e weigh s and a e in he a i-
ables space, e.g., he i s weigh o he i h-deg ee ule p esen ed in Judd (1998) (
S oud (1971)
Cn5−5
) becomes
−
60.44 o
d=5
. Fu he , he deg ee 5 ules wi h
solely posi i e weigh s app oxima e he Rosenb ock unc ion poo ly. Las ly, he
CUT-8 ule nodes lea e he bounda ies o space o
xi
o
d>6
and has al eady one
nega i e weigh o
d=6
.
Table11 p esen s he esul s. The CUT-8 ule pe o ms well o
d=5
. How-
e e , he CUT-8 ule is ou pe o med by Leas Squa es. The pe o mance o he
CUT-8 becomes wo se wi h
d=6
, whe e one weigh becomes nega i e (
≈−.5
),
ye he app oxima ion seems s ill good.
(x)=
d−1
∑
i=1
100
(
xi+1−x2
i
)
2+(1−xi)
2
Table 11 Rosenb ock PCE app oxima ion
Tenso g id l l. = T unc. l l.=4 +1, Smolyak, min
NG id
, gi en
log10
L
2
E o <-5, Leas Squa es, wice
PCE coe icien s
5-d Rosenb ock PCEapp oxima ion
T unc. l l 456
log10 L2
E o
NG id
log10 L2
E o
NG id
log10 L2
E o
NG id
Tenso G id
−
10.98 3,125
−
10.87 7,776
−
11.15 16,807
Spa se G id
−
9.74 781
−
9.70 781
−
9.37 2,203
Leas Squa es
−
10.91 252
−
10.81 504
−
10.58 924
CUT-6 1.90 155 2.49 155 3.20 155
CUT-8
−
10.44 425
−
10.45 425 1.45 425
6-d Rosenb ock PCE app oxima ion
T unc. l l 456
log10 L2
E o
NG id
log10 L2
E o
NG id
log10 L2
E o
NG id
Tenso G id
−
10.87 15,625
−
10.72 46,656
−
10.84 117,649
Spa se G id
−
9.31 1,433
−
9.28 1,433
−
8.71 4,541
Leas Squa es
−
10.94 420
−
10.77 924
−
10.60 1,848
CUT-6 2.07 301 2.61 301 3.35 301
CUT-8
−
6.93 973
−
6.89 973 1.79 937
1134
D.Feh le e al.
RBC Model
Now we eplica e he in eg a ion analysis o he main pape (Figu e4 and 5 he e)
o he CUT ules. Gi en he esul s o he p e ious sec ion on monomial ules, we
Fig. 10
L2
Con e gence o PCE wi h app oxima ed coe icien s and compu a ion ime on an In el®
Co e™i7-7700 CPU @ 3.60GHz

1135
Polynomial Chaos Expansion: E icien E alua ion and…
educe he space o 5 dimensions (
𝛽
is now ixed) and assume
𝜃−𝜑
0
𝜑 1
=𝜓(s)∼B(1, 1)=U(0, 1
)
o all
𝜃
.
Fig. 11
L2
Con e gence o PCE wi h app oxima ed coe icien s and compu a ion ime on an In el®
Co e™i7-7700 CPU @ 3.60GHz
1136
D.Feh le e al.
Figu es10 and 11 illus a e he analyses. I u ns ou ha he monomial ule CUT8
ou pe o ms all o he spa se me hods o unca ion a
N=5
and all me hods in ime a
his unca ion le el. Howe e , a highe unca ion leads o mo e imp ecise app oxima-
ions. The p oblem is he lack o high-deg ee, high-dimensional monomial ules o di -
e en dis ibu ions. Howe e , sui able cases o exis ing monomial ules seem o wo k
well, which mo i a es u he esea ch o ind high-deg ee, high-dimensional monomial
ules o mixed dis ibu ions.
Appendix6 Fu he applica ions o Gene alized Polynomial Chaos
Expansions
We p esen he e addi ional applica ions o PCE. Fi s , we show how o use PCE as
su oga es o he g adien s. Fu he , s a is ical p ope ies o he model ou come, as
induced by he p ede ined dis ibu ion o he unce ain inpu pa ame e s, can be de i ed
di ec ly om he PCE. Addi ionally, in somewha o he con ex s, PCE can be used o
disc e ize he space o c oss-sec ional dis ibu ions.
Su oga e o G adien s
The unca ed PCE in (9) may also be used o app oxima e he de i a i es o he
mapping h be ween pa ame e alues and model ou comes. Mo e speci ically, he PCE
p o ides he app oxima ion
This app oxima ion can be use ul i such de i a i es mus be e alua ed a a po en-
ially la ge numbe o poin s. One example may be he me hod p oposed by Isk e
(2010) o conduc ing local iden i ica ion analysis which equi es di e en ia ion o
he linea ized policy unc ion conce ning he pa ame e s.
E alua ion o S a is ical P ope ies
Con e gence in
L2(Ω,
A
,P)
o he se ies expansion in (5b) implies ha he dis ibu-
ion o he model ou come Y can be equi alen ly cha ac e ized by i s polynomial
expansion. In pa icula , he mean and a iance o Y ollow di ec ly om he ac
ha con e gence in
L2
also implies con e gence o he mean and a iance so ha
o hogonali y o he polynomials (and
q0=1
o
0
∶= (0, …,0)∈ℕ
k
0
) yields
and
𝜕h
𝜕𝜗
i
(𝜗)≈
∑
𝛼∈ℕk
0
,
|
𝛼
|
≤N
y𝛼
k
∑
j=1
𝜕q𝛼
𝜕sk
(𝜓−1(𝜗))
𝜕𝜓
−1
j
𝜕𝜗i
(𝜗)
.
𝔼
[Y]=
�
𝛼∈ℕk
0
y𝛼𝔼[q𝛼(𝝃)] =
�
𝛼∈ℕk
0
y𝛼𝔼[q𝛼(𝝃)q0(𝝃)] =
�
𝛼∈ℕk
0
y𝛼
⟨
q𝛼,q0
⟩
L2=y0
,
1137
Polynomial Chaos Expansion: E icien E alua ion and…
Mo eo e , o he s a is ical p ope ies can be compu ed by Mon e Ca lo me hods.
La ge samples o Y can be e icien ly cons uc ed by d awing om he ge m’s dis-
ibu ion and inse ing he sample in o he expansion o Y. Compa ed o adi ional
me hods, epea ed and cos ly model e alua ions can hus be a oided.
Sobol’s indices o global a iance-based sensi i i y analysis The decomposi ion
o he model’s ou come a iance om abo e also lays he ounda ion o he sensi-
i i y analyses o Ha enbe g e al. (2019). Mo e speci ically, conside a unca ed
PCE
S o
N(
Y
)
o
Smax
N(Y)
o he model ou come Y as in (7b) o (8b). By eo de ing, one
can hen equi alen ly w i e he unca ed PCE as
i.e., o any collec ion
{𝜉i}i∈I
whe e
I⊂{1, …,k}
we now explici ly g oup he
polynomials
q𝛼(𝝃)
wi h non-ze o deg ee in each
𝜉i,i∈I
bu ze o-deg ee in all
𝜉,i∉I
. O hogonali y o he polynomials hen implies o any nonemp y collec ion
I⊂{1, …,k},I≠�
ha
and
The Sobol indices hen desc ibe he sha es o he a iance ha a e explained by a
collec ion
{𝜉i}i∈I
o ge ms o
I⊂{1, …,k},I≠�
The i s o de Sobol indices
S{i}
o single ge ms
𝜉i
a e in e p e ed as he ac ion o
he o al a iance which would disappea when
𝜉i
would be pe ec ly known. On he
o he hand, he o al con ibu ion indices a e de ined by
Va
[Y]=𝔼
⎡
⎢
⎢
⎢
⎣
⎛
⎜
⎜
⎝�
𝛼∈ℕk
0
y𝛼q𝛼(𝝃)−y0⎞
⎟
⎟
⎠
2
⎤
⎥
⎥
⎥
⎦
=𝔼
⎡
⎢
⎢
⎢
⎣
⎛
⎜
⎜
⎝�
𝛼∈ℕk
0⧵{0}
y𝛼q𝛼(𝝃)⎞
⎟
⎟
⎠
2
⎤
⎥
⎥
⎥
⎦
=
=
�
𝛼,𝛽∈ℕk
0
⧵{0}
y𝛼y𝛽
⟨
q𝛼,q𝛽
⟩
L2=
�
𝛼∈ℕk
0
⧵{0}
y2
𝛼
‖
q𝛼
‖
2
L2.
S
o
N(Y)=
∑
I⊂{1,…,k}
∑
𝛼∈ℕk
0,
|
𝛼
|
≤N
𝛼i≠0∀i∈I
𝛼
i
=0∀i∉I
y𝛼q𝛼(𝜉)
,
V
I∶= Va
��
𝛼∈ℕk
0,
�
𝛼
�
≤N
𝛼i≠0∀i∈I
𝛼
i=0∀i∉I
y𝛼q𝛼(𝝃)
�
=
�
𝛼∈ℕk
0,
�
𝛼
�
≤N
𝛼i≠0∀i∈I
𝛼
i=0∀i∉I
y2
𝛼
‖
q𝛼
‖
2
L
2
V
∶= Va [S o
N(Y)] =
∑
I⊂{1, …,k},
I≠�
VI
.
S
I∶=
V
I
V
.
1138
D.Feh le e al.
and desc ibe he ge m’s o al con ibu ion o he ou come’s a iance.
Rele ance o (Bayesian) es ima ion As Ha enbe g e al. (2019) no e, a su icien
size o he o al Sobol’ index o he pa ame e
𝜗i
is a necessa y condi ion o he
iden i iabili y o
𝜗i
using Y. In e ms o Bayesian es ima ion, PCE also acili a es he
compa ison o he model ou come’s p io and pos e io dis ibu ion. Once we ha e
ob ained he pa ame e s’ pos e io dis ibu ion, PCE enables he ep esen a ion o
he co esponding pos e io dis ibu ion o he model’s ou come. We can hen com-
pa e he PCE implied a iances and he con ibu ion o an a bi a y se o pa am-
e e s, which deli e s an indica o o he educed unce ain y o he model ou come
Y subjec o his se o pa ame e s.
Disc e izing space o c oss-sec ional dis ibu ions
He e, we b ie ly p esen he possibili y o using PCE o disc e ize he s a e space in mod-
els whe e a c oss-sec ional dis ibu ion o e he e ogeneous agen s becomes a s a e a i-
able o indi idual decision ules as sugges ed by P öhl (2017). He examples a e mod-
els ha combine idiosync a ic income isk wi h agg ega e p oduc i i y isk as Aiyaga i
(1994). In such models, households need o know he decision ules o o he households
o o m a ional expec a ions abou u u e agg ega es and p ices o hei own decisions.
Ye , since he decisions o o he households depend on hei espec i e indi idual s a es,
households need o ac o in he whole c oss-sec ional dis ibu ion o e indi idual s a es
o hei own decision. In consequence, he c oss-sec ional dis ibu ion o indi idual
s a es becomes an a gumen o he indi iduals’ policy unc ion in such models.
The li e a u e o e s di e en app oaches in o de o disc e ize he s a e space.
K usell and Smi h (1998) sugges a bounded a ionali y app oach and base he
indi iduals’ policy unc ion only on pa ial in o ma ion om he c oss-sec ional
dis ibu ion, e.g., a ini e numbe o momen s, and a pa ame ic law o mo ion o
hese measu es. The me hod o Rei e (2009) disc e izes he s a e space by piece-
wise uni o m dis ibu ions o e a ini e numbe o his og am bins. Di e en ly, P öhl
(2017) eplaces he c oss-sec ional dis ibu ion as an a gumen o he decision ule
by he coe icien s o i s ( unca ed) PCE gi en a choice o ge ms. Mo e p ecisely,
i
𝜉
deno es he ge m wi h cumula i e dis ibu ion unc ion
F𝜉
and
𝜇
is he c oss-
sec ional dis ibu ion o e indi idual s a es in pe iod , hen he andom a iable
is dis ibu ed acco ding o
𝜇
.30 One can hen compu e he coe icien s

𝜗n,
o i s PCE
S
T
i∶=
∑I⊂{1, …,k},
i∈IVI
V
𝜃
∶= 𝜇
−1
◦F
𝜉
◦
𝜉
30 No e ha
𝜃
does no deno e a model pa ame e in his con ex as in he es o he p esen pape .
Ins ead,
𝜃
is a andom a iable ha is dis ibu ed acco ding o he c oss-sec ional dis ibu ion
𝜇
and ha
is a unc ion o he ge m. Hence,
𝜃
can be in e p e ed as he andom a iable cons uc ed om he basis
𝜉
ha desc ibes a andom d aw om he mass o he e ogenous agen s in pe iod .
1145
Polynomial Chaos Expansion: E icien E alua ion and…
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