Estimation and inference in games of incomplete information with unobserved heterogeneity and large state space

Fan, Yanqin; Jiang, Shuo; Shi, Xue ao
A icle
Es ima ion and in e ence in games o incomple e
in o ma ion wi h unobse ed he e ogenei y and la ge
s a e space
Quan i a i e Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Fan, Yanqin; Jiang, Shuo; Shi, Xue ao (2024) : Es ima ion and in e ence in games
o incomple e in o ma ion wi h unobse ed he e ogenei y and la ge s a e space, Quan i a i e
Economics, ISSN 1759-7331, The Econome ic Socie y, New Ha en, CT, Vol. 15, Iss. 4, pp. 893-938,
h ps://doi.o g/10.3982/QE2169
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Quan i a i e Economics 15 (2024), 893–938 1759-7331/20240893
Es ima ion and in e ence in games o incomple e in o ma ion
wi h unobse ed he e ogenei y and la ge s a e space
Yanqin Fan
Depa men o Economics, Uni e si y o Washing on
Shuo Jiang
MOE Key Lab o Econome ics, WISE, Depa men o S a is ics and Da a Science a School o Economics,
Xiamen Uni e si y
Xue ao Shi
School o Economics, Uni e si y o Sydney
Building on he sequen ial iden i ica ion esul o Agui egabi ia and Mi a (2019),
his pape de elops es ima ion and in e ence p ocedu es o s a ic games o in-
comple e in o ma ion wi h payo - ele an unobse ed he e ogenei y and mul i-
ple equilib ia. Wi h payo - ele an unobse ed he e ogenei y, sequen ial es ima-
ion and in e ence ace wo main challenges: he ma ching- ypes p oblem and a
la ge numbe o ma chings. We ackle he ma ching- ypes p oblem by cons uc -
ing a new minimum-dis ance c i e ion o he co ec ma ching and he payo
unc ion wi h bo h co ec and inco ec “momen s.” To handle la ge numbe s o
ma chings, we p opose a no el and compu a ionally as mul is ep momen selec-
ion p ocedu e. We show ha asymp o ically, i achie es a ime complexi y ha is
linea in he numbe o “momen s” when he occu ence o mul iple equilib ia
does no depend on he numbe o “momen s.” Based on his p ocedu e, we con-
s uc a consis en es ima o o he payo unc ion, an asymp o ically uni o mly
alid and easy- o-implemen es o linea hypo heses on he payo unc ion,
and a consis en me hod o g oup payo unc ions acco ding o he unobse ed
Yanqin Fan: [email p o ec ed]
Shuo Jiang: [email p o ec ed]
Xue ao Shi: [email p o ec ed]
This pape is a e ised e sion o he p e iously ci cula ed pape “Es ima ion and in e ence in games o in-
comple e in o ma ion wi h nonsepa able unobse ed he e ogenei y,” da ed Sep embe 17, 2020. We would
like o hank a senio membe o he edi o ial boa d and h ee anonymous e e ees o hei insigh ul
commen s, which subs an ially imp o ed his pape . We hank Au eo de Paula, Jin-Chuan Duan, A naud
Mau el, Xun Tang, and Ruli Xiao, and pa icipan s o he 2018 Sea le-Vancou e Econome ics Con e ence,
he 2019 No h Ame ican Summe Econome ics Socie y Mee ings, he 2021 Annual Con e ence o he In-
e na ional Associa ion o Applied Econome ics, and 2023 Econome ic Socie y Aus alasian Mee ing o
help ul discussions. This wo k was acili a ed by Hyak supe compu e sys em and CSDE simula ion clus e
a he Uni e si y o Washing on. Shuo Jiang acknowledges he suppo o Fujian Key Lab o S a is ics and
he inancial suppo om he Humani ies and Social Sciences Founda ion o he Minis y o Educa ion o
China [G an 23YJC790053], Basic Scien i ic Cen e P ojec [G an 71988101] o Na ional Science Founda-
ion o China, he 111 P ojec [G an B13028], and Na ional Na u al Science Founda ion o China [G an
72373127].
©2024 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h p://qeconomics.o g.h ps://doi.o g/10.3982/QE2169
894 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
he e ogenei y. Ex ensi e simula ions demons a e he ini e sample e icacy o ou
p ocedu es.
Keywo ds. Ma ching- ypes p oblem, minimum-dis ance cha ac e iza ion, mul-
is ep momen selec ion p ocedu e, ime complexi y.
JEL classi ica ion. C12, C13, C57.
1. In oduc ion
Mo i a ion and main con ibu ions
The sequen ial app oach o iden i ica ion and es ima ion o disc e e games o incom-
ple e in o ma ion is widely used in he li e a u e; see Agui egabi ia and Mi a (2007),
Baja i, Benka d, and Le in (2007), and Pesendo e and Schmid -Dengle (2008) o sem-
inal con ibu ions and Baja i, Hong, and Nekipelo (2013) o asu ey.In he i s s ep,
he equilib ium condi ional choice p obabili ies (CCPs he ea e ) a each obse ed s a e
a e iden i ied and es ima ed om he da a. In he second s ep, he payo unc ion is
iden i ied and es ima ed using a ia ions in he obse ed s a e a iable such as exclu-
sion es ic ions.1By a oiding he compu a ion o equilib ium o e e y gi en s a e and
pa ame e alue, he sequen ial app oach is compu a ionally less cos ly han he all-
solu ion o he join me hod such as he nes ed ixed-poin algo i hm. Howe e , he
sequen ial app oach elies c i ically on he assump ion ha he e is no common knowl-
edge payo - ele an unobse ed he e ogenei y (unobse ed he e ogenei y he ea e ) in
he payo unc ion.2As discussed ex ensi ely in Agui egabi ia and Mi a (2019), his
assump ion is e y es ic i e and o en iola ed in he da a, mo i a ing hem o s udy
iden i ica ion o a gene al class o games o incomple e in o ma ion wi h payo ele an
unobse ed he e ogenei y and mul iple equilib ia.
As poin ed ou in Agui egabi ia and Mi a (2019), he p esence o payo ele an
unobse ed he e ogenei y c ea es a majo challenge in he sequen ial iden i ica ion
e e ed o as he ma ching- ypes p oblem, ha is, he di icul y o co ec ly ma ch-
ing equilib ium CCPs o each unobse ed s a e ac oss di e en obse ed s a es. The
ma ching- ypes p oblem a ises because he i s -s ep iden i ica ion o equilib ium CCPs
a each obse ed s a e is equi alen o he iden i ica ion o a nonpa ame ic ini e mix-
u e model, which is known o be iden i ied only up o a label swapping o he mixing
componen s. Despi e his ma ching- ypes p oblem, Agui egabi ia and Mi a (2019)es-
ablish a necessa y and su icien condi ion o he sequen ial iden i ica ion o model
p imi i es in games wi h unobse ed he e ogenei y o ini e suppo and mul iple equi-
lib ia. Building on hei sequen ial iden i ica ion esul , his pape de elops a sequen ial
es ima ion app oach o he class o games o incomple e in o ma ion allowing o bo h
unobse ed he e ogenei y o ini e suppo and mul iple equilib ia.3
1Depending on he con ex s/speci ica ions, we will use payo unc ion, payo ec o , and payo pa am-
e e in e changeably h oughou his pape .
2In De Paula (2013), such unobse ed he e ogenei y is also called game-le el he e ogenei y o game-
le el shock.
3Al hough he join iden i ica ion does no su e om he ma ching- ypes p oblem, es ima o s based
on i o games wi h bo h unobse ed he e ogenei y and mul iple equilib ia ha e no been o mally de el-
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 895
To ackle he ma ching- ypes p oblem, we cons uc a no el cha ac e iza ion o he
co ec ma ching and he ue payo ec o ia a minimum-dis ance c i e ion wi h bo h
co ec and inco ec “momen s.” The se o co ec momen s co esponds o he co -
ec ma ching; and he ue payo ec o is uniquely de e mined h ough he co ec
ma ching. In he new minimum-dis ance c i e ion, he momen unc ions a e linea in
he unknown payo ec o wi h coe icien s depending on he equilib ium CCPs iden-
i ied in he i s s ep. Es ima ion o he equilib ium CCPs is s anda d and can be done
expedi iously using exis ing me hods such as hose in Bonhomme, Jochmans, and Robin
(2016)andXiao (2018). Using he plug-in es ima o s o he coe icien s, we ob ain a ec-
o o momen unc ions, based on which o he co ec se o momen s o he co ec
ma ching will be selec ed and he payo ec o will be es ima ed. Al hough his p oce-
du e alls wi hin he gene al amewo k o And ews (1999), he momen selec ion p o-
cedu es in And ews (1999) a e compu a ionally cos ly, and o en imes in easible e en in
games wi h mode a ely sized s a e spaces. The eason is ha he numbe o ma chings
g ows exponen ially wi h he size o he s a e space. Fo example, in he Simple Game in-
oduced in Sec ion 2wi h medium numbe s o playe s and obse ed and la en s a es,
he e can be housands o illions o ma chings.
To o e come his compu a ional challenge, we p opose a new and ingenious mul i-
s ep momen selec ion (MMS) p ocedu e o selec ing he co ec ma ching. I is based on
he insigh ha in he minimum-dis ance c i e ion, a co ec ma ching selec s he same
la en s a e ac oss all obse ed s a es. As a esul , a misma ch on any single obse ed
s a e esul s in a w ong ma ching ha needs no o be conside ed in he es ima ion. Ex-
ploi ing his ea u e, in he new MMS p ocedu e, we i s elimina e ma chings ha a e
inco ec wi h high p obabili y in mul iple s eps, hen es ima e he co ec ma ching
and he payo ec o using he emaining possible ma chings. By ca e ully designing
he s eps in ol ed, he new MMS p ocedu e selec s he co ec ma ching wi h p obabil-
i y app oaching one and is much as e o implemen han he momen selec ion p o-
cedu es in And ews (1999) o games wi h la ge s a e spaces. Theo e ically, we show ha
when he e is no mul iple equilib ia o when he numbe o obse ed s a es wi h mul i-
ple equilib ia does no inc ease wi h he numbe o momen s, he new MMS p ocedu e
achie es a linea ime complexi y in he numbe o momen s o la ge sample sizes.4
This is a signi ican imp o emen o e he exponen ial ime complexi y o he momen
selec ion p ocedu es in And ews (1999). P ac ically, he new MMS-based es ima o o
he payo ec o can be calcula ed wi hin asecondwhen he e a e housands o illions
o ma chings; while he es ima o in And ews (1999) equi es housands o seconds o
compu ee enwhen he ea eonlymillions o ma chings.
When he e a e mul iple equilib ia played in he da a, bo h mul iple equilib ia and
unobse ed he e ogenei y con ibu e o he mix u es o CCPs. To es ima e he ca dinal-
i y o he suppo o he unobse ed he e ogenei y and he payo ec o on each la en
s a e, we p opose a me hod o g ouping he payo ec o s by ex ending he k-means
oped and a e unexplo ed in p ac ice. Addi ionally, hey would ca y he hea y compu a ional bu den o he
exis ing nes ed ixed-poin algo i hm.
4The ime complexi y desc ibes he amoun o ime i akes o un an algo i hm. I is commonly es i-
ma ed by coun ing he numbe o elemen a y ope a ions pe o med by he algo i hm.
896 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
me hod o penalize mo e clus e s. The numbe o clus e s es ima e he numbe o la-
en s a es, and he payo s on each la en s a e a e es ima ed by he cen e s o each
clus e . We show ha ou es ima o s consis en ly es ima e he numbe o la en s a es
and he payo ec o on each la en s a e. This me hod o sepa a ing mul iple equilib ia
om unobse ed he e ogenei y is no el and can be used in o he con ex s such as he
dynamic game wi h bo h mul iple equilib ia and unobse ed he e ogenei y s udied in
Luo, Xiao, and Xiao (2022).
Las ly, we de elop a as - o-compu e and asymp o ically uni o mly alid in e ence
p ocedu e o linea hypo heses on he payo ec o . Despi e he di icul ies in gene al
pos -selec ion in e ence, ou es is asymp o ically uni o mly alid and is easy o imple-
men wi h known c i ical alues om he chi-squa ed dis ibu ion.
Al hough we ocus on he class o games wi h nonpa ame ic payo unc ions in
Agui egabi ia and Mi a (2019), we demons a e ha he no el minimum-dis ance cha -
ac e iza ion can be easily modi ied o games wi h pa ame ic payo unc ions com-
monly adop ed in empi ical wo k. As a esul , he MMS p ocedu e applies o hese
games as well. In he simula ion sec ion, we epo he ini e sample pe o mance o
he new MMS p ocedu e, he es ima o s o he payo s and he numbe o unobse ed
he e ogenei y ypes, and he es when applied o ou games wi h pa ame ic payo s.
Based on he simula ion esul s, we p o ide a ule-o - humb o choosing he uning pa-
ame e s o implemen a ion o he MMS p ocedu e, and demons a e i s e ec i eness
using di e en designs cons uc ed om he ou games. O e all, he simula ion esul s
con i m he ini e sample e icacy o bo h he es ima ion and in e ence p ocedu es.
Rela ed li e a u e
Ou pape connec s wi h se e al s ands o he li e a u e. Fi s , he pape is closely e-
la ed o wo ks on s a ic games o incomple e in o ma ion wi h/wi hou unobse ed he -
e ogenei y. In he analysis o s a egic iming incen i es among adio s a ions, Swee ing
(2009) es ima es a pa ame ic game o incomple e in o ma ion ha allows o mul iple
equilib ia bu no unobse ed he e ogenei y. He also s a es ha “es ima ion o a game
wi h many possible choices, mul iple equilib ia, and obse ed and possibly unobse ed
he e ogenei y is well beyond he cu en li e a u e.” Swee ing (2009) has spa ked impo -
an esea ch. De Paula and Tang (2012) p opose a o mal es o mul iple equilib ia
when he e is no unobse ed he e ogenei y. G ieco (2014) s udies iden i ica ion and es-
ima ion in a game wi h no mally dis ibu ed p i a e in o ma ion, allowing bo h o he
p esence o mul iple equilib ia and o no mally dis ibu ed unobse ed he e ogene-
i y. In G ieco (2014), he unobse ed he e ogenei y is o a nuisance na u e; and he pa-
ame e o in e es does no depend on unobse ed he e ogenei y. Xiao (2018)s udies
sequen ial iden i ica ion and es ima ion in a game wi h mul iple equilib ia and no un-
obse ed he e ogenei y. She de elops a new me hod based on eigendecomposi ion o
iden i ying and es ima ing CCPs ha come om mul iple equilib ia. Agui egabi ia and
Mi a (2019) p esen a gene al iden i ica ion amewo k allowing o bo h unobse ed
he e ogenei y and mul iple equilib ia, which nes s he se up in Xiao (2018). In Agui e-
gabi ia and Mi a (2019), he payo pa ame e o in e es is allowed o depend on un-
obse ed he e ogenei y, which di e s om he pa ame e o in e es in G ieco (2014).

Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 897
Luo, Xiao, and Xiao (2022) show ha he ma ching- ypes p oblem in dynamic games
o incomple e in o ma ion could be esol ed by making use o he special s uc u e o
Ma ko pe ec equilib ium and he longi udinal a ia ions o obse ed s a es. The e is
ano he s and o li e a u e on he iden i ica ion and es ima ion o comple e in o ma-
ion games such as Baja i, Hong, and Ryan (2010), in which all unobse ables a e com-
mon knowledge among playe s. A ecen pape by Magnol i and Ronco oni (2023)s ud-
ies iden i ica ion and es ima ion unde he solu ion concep o he Bayesian co ela ed
equilib ium, which is p oposed by Be gemann and Mo is (2013,2016). Magnol i and
Ronco oni (2023) allow o all in o ma ion s uc u es consis en wi h playe s knowing
hei own payo s and he dis ibu ion o opponen s’ payo s. Thei in o ma ion s uc-
u e nes s bo h he comple e and incomple e in o ma ion se ings.5
Ou pape also con ibu es o he li e a u e on momen selec ion and uni o m in e -
ence. In a seminal pape , And ews (1999) p oposes se e al consis en momen selec ion
p ocedu es o he gene alized me hod o momen s es ima ion wi h alid and in alid
momen s. And ews and Lu (2001) ex end hese p ocedu es and apply hem o dynamic
panel da a models. Two p oblems emain unsol ed ega ding he momen selec ion p o-
cedu es. Fi s , i is well known ha execu ing he p ocedu es in And ews (1999)and
And ews and Lu (2001) can be compu a ionally cos ly (see Liao (2013)).6Second, he
buil -in momen selec ion implies ha he in e ence p oblem is a pos -selec ion in e -
ence. Cons uc ing an asymp o ically uni o mly alid in e ence me hod is challenging
in his con ex (see Leeb and Pö sche (2005)andLeeb and Pö sche (2008)). In he pa-
pe , we sol e bo h p oblems by ully exploi ing he s uc u e o ou se up. Fi s , a co ec
ma ching selec s he same la en s a e o each obse ed s a e. As a esul , we know he
s uc u e o he se o co ec momen s. This ac enables us o design a mul is ep algo-
i hm ha is as o compu e. Second, in ou se ing, he minimum numbe o co ec
momen s is known. This allows us o cons uc an asymp o ically uni o mly alid and
easy- o-implemen in e ence me hod.
O ganiza ion o he es o his pape
The es o his pape is o ganized as ollows. Sec ion 2uses wo games o incomple e
in o ma ion wi h unobse ed he e ogenei y o in oduce ou no el minimum-dis ance
c i e ion o he payo ec o . The i s game is a membe o he class o games wi h non-
pa ame ic payo unc ions s udied in Agui egabi ia and Mi a (2019) and is e e ed o
as he Simple Game; and he second game is he same as he Simple Game excep ha i s
payo unc ion is pa ame e ized. Sec ion 3p oposes he no el MMS p ocedu e and he
es ima o o he payo ec o , p o es i s consis ency, and shows he asymp o ic linea
ime complexi y o he p ocedu e o he Simple Game. Sec ion 4de elops an asymp o -
ically uni o mly alid es o linea hypo heses on he payo ec o . Sec ion 5ex ends
he me hods de eloped o he Simple Game o a game wi h a gene al numbe o playe s,
5Haile and Tame (2003) and A adillas-López, Gandhi, and Quin (2016) s udy iden i ica ion and in e -
ence o auc ion models unde weak assump ions ha could allow o mul iple equilib ia.
6Unlike Liao (2013)o Cheng and Liao (2015), a known se o alid momen s ha gua an ee iden i ica ion
o he unknown pa ame e is no a ailable in ou se up.
898 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
ac ions, la en s a es, and mos impo an ly mul iple equilib ia e e ed o as he Gen-
e al Game. We ex end he MMS es ima ion p ocedu e and he asymp o ically uni o mly
alid es de eloped o he Simple Game o he Gene al Game. In Sec ion 6, we in o-
duce a ian s o he Simple Game and Gene al Game and in es iga e he ini e sample
pe o mance o ou es ima ion and in e ence me hods ia Mon e Ca lo simula ion. Sec-
ion 7concludes. Ma hema ical de ails o he esul s o he Gene al Game a e p o ided
in Appendix A. Addi ional ma e ials a e collec ed in he Supplemen al Appendix (Fan,
Jiang, and Shi (2024)). Supplemen al Appendix B con ains p oo s o he esul s in he
pape . Supplemen al Appendix C con ains u he de ails on Xiao (2018)’s CCP es ima-
o and he iden i ica ion in he Gene al Game. Supplemen al Appendix D con ains ad-
di ional de ails on he simula ion. The codes o implemen ing he mul is ep es ima ion
and in e ence p ocedu es a e a ailable a h ps://gi hub.com/FanJiangShi/MMSP.
We close his sec ion by in oducing some no a ion used h oughou his pape . Fo
any q×1 ec o E,le Edeno e i s Euclidean no m and E0deno e i s L0no m,
ha is, he numbe o nonze o elemen s in E.Fo being some q×qma ix, deno e
E2
≡EE. Fo any ini e se , |·|deno es i s ca dinali y. Fo any gi en q-dimensional
ec o co ze os and ones and some q×pma ix A,le Acdeno e he subma ix o A
gene a ed by dele ing he ows in Aco esponding o ze os in c.Le Iqbe a q×qiden i y
ma ix. “wp →1” deno es “wi h p obabili y app oaching one.”
2. A minimum-dis ance cha ac e iza ion o he payo ec o
In his sec ion, we use wo games o incomple e in o ma ion wi h unobse ed he e o-
genei y o in oduce ou no el minimum-dis ance cha ac e iza ion o he payo ec o .
The i s game is a membe o he class o games wi h nonpa ame ic payo unc ions
s udied in Agui egabi ia and Mi a (2019) and is e e ed o as he Simple Game. The
second game is he same as he Simple Game excep ha i s payo unc ion is pa ame-
e ized. We in oduce he Simple Game in Sec ion 2.1 and e iew i s sequen ial iden i i-
ca ion in Sec ion 2.2.InSec ion2.3, we cons uc a no el cha ac e iza ion o he payo
unc ion in he Simple Game ia a minimum-dis ance c i e ion wi h bo h co ec and
inco ec momen s. In Sec ion 2.4, we in oduce he payo unc ion o he second game
and show how he minimum-dis ance cha ac e iza ion accommoda es o he pa ame -
ic s uc u e o he payo unc ion o he second game.
2.1 The simple game
In he Simple Game, he e a e h ee playe s, wo ac ions, one exclusi e obse ed s a e
a iable, and one dicho omous unobse ed s a e a iable.7Each playe , deno ed as
i=1, 2, 3, chooses an ac ion di∈{0, 1}. Be o e choosing his ac ion, playe id aws his
p i a e in o ma ion i(di) o wo ac ions di=0anddi=1 om a bi a ia e dis ibu-
ion. Fo i=1, 2, 3, deno e zi∈Zias an obse able exclusi e s a e a iable, which does
7I ollows om Allman, Ma ias, and Rhodes (2009) and Agui egabi ia and Mi a (2019) ha when he
numbe o mixing componen s is 2, he minimum numbe o playe s equi ed o iden i ying CCPs up o a
label swapping is 3.
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 899
no en e he payo s o o he playe s, whe e Ziis a ini e se wi h ca dinali y |Zi|.Le
k∈K≡{A,B}be a common knowledge s a e a iable ha is known by all playe s bu
unobse ed by he econome ician.8Playe i’s payo om choosing ac ion diis gi en
by πi(di,d−i,zi,k,i(di)), whe e he ec o d−ideno es he join ac ions o all he o he
playe s excep i.
Following Agui egabi ia and Mi a (2019), we assume ha playe i’s payo is addi-
i ely sepa able in his p i a e in o ma ioni(di)and can be w i en as
πidi,d−i,zi,k,i(di)=πi(di,d−i,zi,k)−i(di),
whe e πi(di,d−i,zi,k)cap u es how playe i’s payo o choosing dichanges wi h e-
spec o his opponen s’ ac ions and s a e a iables.9Since he op imal ac ion is in a ian
unde mono onically inc easing ans o ma ions o payo s, we no malize he payo s
using πi(0, d−i,zi,k,˜i(0)) and de ine he no malized payo o di=1as
πi(1, d−i,zi,k)≡πi(1, d−i,zi,k)−πi(0, d−i,zi,k)
sd ,
whe e sd deno es he s anda d de ia ion o i(1)−i(0). We e e o πi(di,d−i,zi,k)as
he payo unc ion o playe ihe ea e . By no maliza ion, πi(0, d−i,zi,k)=0.
De ine he no malized p i a e in o ma ion o playe ias i≡1
sd (i(1)−i(0)).Le
z≡(z1,z2,z3)∈Z≡Z1×Z2×Z3. We adop he assump ion in Agui egabi ia and Mi a
(2019)onis a ed below.10
Assump ion 2.1. (i){i}3
i=1
i.i.d.
∼F(·),whe e F(·)is an absolu ely con inuous dis ibu ion
unc ion wi h a p obabili y densi y unc ion deno ed as (·)and is known o he econo-
me ician.(ii)The suppo o (·)is R.(iii)1,2,and 3a e independen o he s a e
a iables (z,k).
A(pu e)s a egyin hisgameisde inedas ollows.
De ini ion 2.1 (S a egy). Fo gi en zand k, a (pu e) s a egy o playe iis a mapping
σi(i,z,k):R×Z×K→{1, 0}.
Fo no a ional compac ness, we use σ≡(σ1(1,z,k),σ2(2,z,k),σ3(3,z,k)) o de-
no e a s a egy p o ile gi en (z,k).Le 1(·)deno e he indica o unc ion. Any gi en σis
8The assump ion o a ixed and ini e suppo o unobse ed he e ogenei y is no only used in Agui e-
gabi ia and Mi a (2019) (p. 1663), bu also in single agen dynamic disc e e choice models such as Kasaha a
and Shimo su (2009).
9The addi i e sepa abili y o he p i a e in o ma ion is commonly assumed in he li e a u e on he
econome ics o games o incomple e in o ma ion. Examples include Swee ing (2009), De Paula and Tang
(2012), Baja i, Hong, and Nekipelo (2013), and G ieco (2014).
10O he pape s ha main ain such an assump ion on p i a e in o ma ion include Zhu and Singh (2009),
Li, Liu, and Deininge (2013), and Xiao (2018). Pape s ha allow o unknown dis ibu ion o p i a e in o -
ma ion include A adillas-López (2010) and Lewbel and Tang (2015). Pape s ha allow o co ela ed p i a e
in o ma ion among playe s include Wan and Xu (2014), Xu (2014), and Liu, Vuong, and Xu (2017).
900 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
comple ely cha ac e ized by he ollowing CCPs:
pi≡1σi(i,z,k)=1 (i)di, o i=1, 2, 3.
Deno e jand qas he wo playe s o he han playe i.Theexpec ed payo unc ion o
playe iwi h di=1 o gi en(z,k)and σis compu ed as11
πi(1, z,k,σ)=
dj,dq∈{0,1}
pdj
j(1−pj)1−djpdq
q(1−pq)1−dqπi1, (dj,dq),zi,k. (2.1)
Bayesian Nash Equilib ium (BNE) is hen de ined as ollows.
De ini ion 2.2 (Equilib ium). Fo any gi en (z,k), a BNE o he game is a s a egy p o-
ile σ∗such ha o any playe iand o any i,
σ∗
i(i,z,k)=a g max
di∈{0,1}πidi,z,k,σ∗−i.
In he Simple Game, we assume ha he da a a e a ionalized by a single equilib ium.
This assump ion will be disca ded in Sec ion 5.
Assump ion 2.2. A single equilib ium is played in he da a o each (z,k)∈Z×K.
Deno e he equilib ium CCP o choosing ac ion 1 o playe ias pi(z,k)≡P (di=
1|z,k).Theni holds ha
pi(z,k)=1σ∗
i(i,z,k)=1 (i)di.
Fo any (z,k), he BNE o he game is equi alen ly cha ac e ized by he equilib ium
CCP ec o
p(z,k)≡p1(z,k),p2(z,k),p3(z,k), (2.2)
whe e
pi(z,k)=Fπi1, z,k,σ∗ o i=1, 2, 3.
Unde Assump ion 2.2, he equilib ium expec ed payo unc ion is unique and only de-
pends on zand k. Fo b e i y, we deno e i as πi(1, z,k)≡πi(1, z,k,σ∗).
Rema k 2.1. Fo any gi en (z,k)and s a egy p o ile σwi h co esponding CCP ec-
o (p1,p2,p3), he p obabili y ha choice 1 is op imal o playe iin he Simple Game
is gi en by F(πi(1, z,k,σ)) o i=1, 2, 3. This de ines he bes esponse mapping o
playe s i=1, 2, 3 on (z,k)as ollows:
izkπi(1, z,k,σ)=Fπi(1, z,k,σ). (2.3)
11Because o he no maliza ion, πi(0, z,k,σ)=0.
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 907
whe e
πz =π11, z ,k 
π11, z ,k
and z =z1p2z ,k +p3z ,k 
z1p2z ,k
+p3z ,k
.
The minimum-dis ance cha ac e iza ion o he payo ec o we cons uc ed o he
Simple Game in Sec ion 2.3 is alid wi h
π=⎡
⎢
⎣
πz1
.
.
.
πzl⎤
⎥
⎦,=⎡
⎢
⎣
z1
.
.
.
zl⎤
⎥
⎦,andπ=θ1
δ1,
whe e he ec o πis o dimension 2l×1, is he coe icien ma ix o dimension 2l×2,
and πis he ec o wi h dimension 2 ×1. Gi en some selec ed la en s a e o z1,we
could ma ch his la en s a e ac oss all obse ed s a es unde he same condi ions o
hose o S ep 2 iden i ica ion o he Simple Game in Sec ion 2.3.
3. Mul is ep momen selec ion es ima ion o he payo ec o in he
simple game
As we demons a e in Sec ion 2.4, he pa ame e s o in e es in he a ian s o he Simple
Game sha e he same minimum-dis ance cha ac e iza ion as he Simple Game wi h e-
de ined π,,andπin he momen unc ion. Wi hou loss o gene ali y, we ocus on de-
eloping es ima ion and in e ence p ocedu es o he Simple Game in Sec ions 3and 4.
F om Lemma 2.1, i ollows ha he ue selec ion ec o c0and he ue pa ame e
ec o π0sa is y
(c0,π0)=a g min
c∈C,π∈Gc(π)2
W(c), (3.1)
whe e Gc(π)=πc−cπand W(c)a e a posi i e de ini e weigh ing ma ix ha can de-
pend on c. The ma ices πand depend on equilib ium CCPs and can be es ima ed
by he plug-in app oach using equa ions (2.6)and(2.9) once he equilib ium CCPs on
all he obse ed and la en s a es a e es ima ed. Exis ing me hods such as hose in Bon-
homme, Jochmans, and Robin (2016)andXiao (2018) can be used o es ima e he equi-
lib ium CCPs. We ocus on he CCP es ima o de eloped by Xiao (2018) in his pape and
p esen he s eps in Appendix C.1. Deno e he esul ing es ima o s as πnand n.Weno e
ha his s ep is s anda d in he li e a u e and can be implemen ed as e en o la ge l.
This is because he eigendecomposi ion p ocedu e is done o each obse ed s a e sep-
a a ely; and each p ocedu e is as o compu e.13 We ocus on he second s ep om now
on.
Le he sample momen unc ions be
Gn(π)≡πn−nπ, o π∈.
13Using 1000 simula ions, he a e age ime needed o compu e he eigendecomposi ion and ob ain
playe 1’s CCP on one obse ed s a e is less han 10−4second.

908 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
We use Gn,c(π) o deno e he sample momen unc ions selec ed by c. The sample e -
sion o (3.1)is
(
c,π)≡a g min
c∈C,π∈Gn,c(π)2
Wn(c), (3.2)
whe e Wn(c)is he sample weigh ing ma ix. This is equi alen o he momen selec ion
p ocedu es in And ews (1999) when applied o he Simple Game. Sol ing (3.2) equi es
pe o ming disc e e op imiza ion o e C. Since |C|=2l−1, implemen ing (
c,π)is com-
pu a ionally challenging o la ge l. Fo example, when he e a e se en exclusi e s a es
o each playe , l=|Z2|×|Z3|=49. The size o Cbecomes 248 ≈2.8 ×1014.Thismo-
i a es ou compu a ionally less cos ly mul is ep momen selec ion (MMS) p ocedu e
p oposed in Sec ion 3.1.InSec ion3.2, we show consis ency o he MMS p ocedu e and
i s ime complexi y esul . In con as o he es ima o (
c,π), which has an exponen-
ial ime complexi y in l, he ime complexi y o he MMS is asymp o ically linea in l.
Sec ion 3.3 p o ides some guidance on he p ac ical implemen a ion o he MMS p oce-
du e. P oo s o he heo ems in his sec ion a e p o ided in he Supplemen al Appendix.
3.1 Mul is ep momen selec ion p ocedu e
As he numbe o exclusi e s a es |Z2|o |Z3|inc eases, he alue o l ises sha ply. The
MMS p ocedu e explo es an impo an ea u e o he game o educe he compu a ion
ime: he ue payo ec o π0is only de ined o he sys em selec ed by c0, and none o
he o he sys ems selec ed by c∈Cand c= c0has a solu ion. Speci ically, in he MMS
p ocedu e, we elimina e he ma chings/selec ion ec o s c∈C ha a e ce ainly inco -
ec in mul iple s eps ins ead o one s ep as in he compu a ion o (
c,π). Wi h ca e ul
design o he s eps in ol ed, we a e able o cons uc an e ec i e pa ame e space o
c0o a much smalle size han C(see he las s ep in he p ocedu e). The MMS p o-
cedu e selec s he co ec ma ching and es ima es he payo ec o using he e ec i e
pa ame e space o c0. To succinc ly in oduce ou idea, we p esen a wo-s ep momen
selec ion (TMS) p ocedu e i s and hen ex end i o he gene al MMS p ocedu e.
Fo any ec o sc o dimension 2lconsis ing o ze os and ones, we use Gn,sc(π) o
deno e he momen unc ions selec ed by sc om Gn(π). Di e en om he selec ion
ec o s in C,wele sc selec ewe han lmomen s and call i a subselec ion ec o . We
pa i ion sc in o lsub ec o s, whe e each sub ec o con ains wo elemen s. Deno e sc
o =1, ,las he h sub ec o o sc such ha sc ≡[sc1,,scl]. De ine
Jn(sc)≡min
π∈Gn,sc(π)2.
3.1.1 The TMS p ocedu e Le l1∈{lπ,lπ+1, ,l}. Heu is ically, i we know ha he
i s l1momen s selec ed by some selec ion ec o con ains inco ec momen s, hen all
he 2l−l1selec ion ec o s in C ha selec he same i s l1momen s can be igno ed in
he es ima ion o (c0,π0), because a ma ching is co ec only when all he lmomen s
a e selec ed co ec ly. S ep 1 below iden i ies ma chings o he i s l1momen s ha a e
inco ec wi h high p obabili y (wp →1). In S ep 2, we es ima e he co ec ma ching
and he ue payo ec o by minimizing an objec i e unc ion o e he p oduc space
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 909
o he e ec i e pa ame e space, which excludes he inco ec ma chings iden i ied in
S ep 1 and he pa ame e space . We p esen he de ailed s eps below.
S ep 0: Se l1∈{lπ,lπ+1, ,l},α1∈(0, 1],andλ∈(−1, 0).
S ep 1: De ine he collec ion o subselec ion ec o s in S ep 1 as
SC1≡[sc1,,scl]∈R2l:sc1=[1, 0];sc ∈[1, 0],[0, 1],
o ∈{2, ,l1};andsc =[0, 0] o ∈{l1+1, ,l}.
By de ini ion, sc1∈SC1selec s none o he las 2(l−l1)momen s. So Jn(sc1) o all
sc1∈SC1, and deno e Jα1
nas he alue o he 100α1% smalles . Compa e Jα1
nwi h nλ.I
Jα1
n>n
λ, hen collec all sc1such ha Jn(sc1)≤Jα1
n;o he wise,collec allsc1such ha
Jn(sc1)≤nλ. Deno e he collec ion as SC1
n:
SC1
n≡sc1∈SC1:Jnsc1≤maxJα1
n,nλ.
The se SC1
nis he ou pu o S ep 1.
S ep 2: De ine he e ec i e pa ame e space o c0as
Cn≡[c1,,cl]∈R2l:[c1,,cl1]=sc1
1,,sc1
l1 o some
sc1∈SC1
n;andc ∈[1, 0],[0, 1] o ∈{l1+1, ,l}.
The TMS es ima o is de ined by he ollowing minimiza ion p oblem:14
(
c,π)≡a g min
c∈Cn,π∈Gn,c(π)2
Wn(c).
Fo each c∈Cn, he i s 2l1componen s o ca e he same as he i s 2l1componen s
o some sc1∈SC1
n;and helas 2
(l−l1)componen s can selec any combina ion o he
las 2(l−l1)momen s allowed by C. Since e e y c∈Cnselec s lmomen s ou o he l
pai s, Cn⊆C. In he special case whe e SC1
n=SC1,weha eCn=C.
Le sc1
0∈SC1deno e he subselec ion ec o whose i s 2l1elemen s a e he same
as c0. In S ep 1, we de e mine i a subselec ion ec o sc is pa o an inco ec ma ching
by compa ing Jn(sc)wi h nλ, because Jn(sc1
0)<n
λoccu s wi h high p obabili y when n
is la ge. A he same ime, we keep a leas 100α1%elemen sinSC1 o p e en sc1
0 om
being elimina ed because o ini e sample e o .
The size o SC1is 2l1−1, which is much smalle han he size o Cwhen l1is smalle
han l. As a esul , S ep 1 can be implemen ed e y as . The se SC1
nhas α
12l1−1el-
emen s whe e α
1≡|SC1
n|/|SC1|; and he size o he e ec i e pa ame e space Cnis
(α
12l1−1)×2(l−l1)=α
12l−1.Whenα
1is small and lis mode a ely la ge, (
c,π)is compu-
a ionally much as e han (
c,π).
Rema k 3.1. Se ing α1=1 eco e s (
c,π)in (3.2), and π=πwhene e 
c=
c.
14The de ini ion o (
c,π)implici ly assumes ha he solu ion o he minimiza ion p oblem is unique.
This can be shown o hold wi h p obabili y app oaching one.
910 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
3.1.2 The MMS p ocedu e When lis e y la ge, implemen ing S ep 2 abo e may s ill
be ime consuming, because α
12l−1can be la ge. To u he educe he compu a ional
ime, we ex end he abo e TMS o MMS wi h any ini e numbe o s eps as needed. Fo
example, in an MMS wi h h ee s eps, he i s s ep is he same as he i s s ep in TMS.
Ins ead o selec ing om all he possible combina ions o (2l1+1)- h o 2l- h momen s
in he second s ep, we selec om he (2l1+1)- h o 2l2- h momen s, whe e l2≡l1+
o some p especi ied ∈{1, ,l−l1}. In he hi d s ep, we selec om he (2l2+1)- h
o 2l- h momen s. Below, we p esen he de ailed p ocedu e o implemen ing he MMS
p ocedu e wi h (S+1)s eps.
Le xdeno e he smalles in ege g ea e han o equal o x.
S ep 0: Se l1∈{lπ,lπ+1, ,l},α1∈(0, 1],λ∈(−1, 0),and∈{1, ,l−l1}.Le
S=l−l1
and α=2−.
S ep 1: Apply he same p ocedu e as S ep 1 in he TMS p ocedu e. The ou pu is he
se SC1
n.
S eps2,3,...S:Fo s=2, ,S, de ine ls≡ls−1+. The inpu o S ep sis he collec-
ion o subselec ion ec o s de ined as
SCs≡⎧
⎪
⎨
⎪
⎩
[sc1,,scl]∈R2l:[sc1,,scls−1]=scs−1
1,,scs−1
ls−1 o some
scs−1∈SCs−1
n;sc ∈[1, 0],[0, 1] o ∈{ls−1+1, ,ls};
and sc =[0, 0] o ∈{ls+1, ,l}
⎫
⎪
⎬
⎪
⎭.
By de ini ion, each scs∈SCs
nconsis s o h ee pa s: he i s 2ls−1componen s o scs
a e he same as he i s 2ls−1componen s o some scs−1∈SCs−1
n; he(2ls−1+1)- h o
2ls- h componen s o scsselec any combina ions allowed by C;andscsselec s none o
he las 2(l−ls)momen s. So Jn(scs) o all scs∈SCs, and deno e Jα
nas he alue o
he 100α% smalles . Cons uc he ou pu o S ep s,SCs
n,as
SCs
n≡scs∈SCs:Jnscs≤maxJα
n,nλ.
S ep (S+1):De ine he e ec i e pa ame e space o c0as
Cn≡[c1,,cl]∈R2l:[c1,,clS]=scS
1,,scS
lS o some
scS∈SCS
n;andc ∈[1, 0],[0, 1] o ∈{lS+1, ,l}.
The MMS es ima o is de ined by he ollowing minimiza ion p oblem:
(
c,π)≡a g min
c∈Cn,π∈Gn,c(π)2
Wn(c). (3.3)
Fo each c∈Cn, he i s 2lScomponen s o ca e he same as he i s 2lScomponen s
o some scS∈SCS
n;and helas 2
(l−lS)componen s can selec any combina ion o he
las 2(l−lS)momen s allowed by C.
Le α
s≡|SCs
n|/|SCs|.InS eps o s=1, ,S, he inpu se SCshas he ca dinal-
i y 2ls−1!s−1
i=1α
i,and heou pu se SCs
nhas he ca dinali y 2ls−1!s
i=1α
i.InS ep(S+1),
|Cn|=2l−1!S
i=1α
i.Fo la gel, we usually ha e la ge Sand small α
s o s=1, ,S.The
numbe o op imiza ions om S ep 1 o S ep (S+1)is much ewe han he numbe o
op imiza ions equi ed o compu ing (
c,π).
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 911
Rema k 3.2. The choice o he uning pa ame e s l1,α1,λ,anda e independen o l
and n.SeeSec ion3.3 o mo e discussion.
Rema k 3.3. By eplacing he ze os wi h ones and he ones wi h ze os in 
c,weob ain
an es ima o o he ue selec ion ec o in C2. Based on i , we can es ima e he payo
ec o . Al e na i ely, we can apply he abo e MMS p ocedu e o C2 o ob ain he es i-
ma o s.
3.2 Asymp o ic p ope ies o he MMS p ocedu e
The consis ency o (
c,π)is p o ed unde he ollowing assump ions.
Assump ion 3.1. The space is compac .
Assump ion 3.2. Fo ∀c∈Cn,Wn(c)p
→W(c) o some posi i e de ini e ma ix W(c).
Assump ion 3.2 imposes a s anda d assump ion on he weigh ing ma ix. No e ha
he weigh ing ma ix Wn(c)is only used in he las s ep o he MMS p ocedu e. The ol-
lowing heo em s a es consis ency o he es ima o s 
cand π.
Theo em 3.1. Unde Assump ions 2.1–2.6 and 3.1–3.2,i holds ha 
c=c0wp →1and
πp
→π0 o any l1∈{lπ,lπ+1, ,l},α1∈(0, 1],λ∈(−1, 0),and ∈{1, ,l−l1}.
Theo em 3.1 shows ha he MMS p ocedu e is consis en o any l1,α1,λ,and ha
sa is y he equi emen s in he heo em. The alues o he uning pa ame e s only a ec
he ini e sample pe o mance o he es ima o .
The heo em below shows ha asymp o ically he ime and space complexi ies o
he MMS a e linea in l.15
Theo em 3.2. Le Assump ions 2.1–2.6 and 3.1 hold.Then wi h p obabili y app oaching
one as n→∞, o all payo s excep o a se o Lebesgue measu e ze o,bo h he ime and
spacecomplexi ieso heMMSp ocedu ea elinea inl.
Conside he space o payo s o all h ee playe s such ha he assump ions in he
heo em a e sa is ied. Theo em 3.2 shows ha , excep o a subse o Lebesgue measu e
ze o in his space, bo h he compu a ion ime and memo y s o age equi ed o pe -
o ming he MMS p ocedu e a e linea in lwi h p obabili y app oaching one as n→∞.
In o he wo ds, excep o ce ain “excep ional” payo s, he linea ime and space com-
plexi ies hold wi h high p obabili y when nis la ge.
Al hough Theo em 3.2 is an asymp o ic esul , he simula ion esul s in Sec ion 6
show ha he MMS is ex emely as o compu e o all DGPs and sample sizes
conside ed. To gua an ee he consis ency, he MMS canno elimina e c∈Ci
15The space complexi y measu es he amoun o memo y space equi ed o pe o ming an algo i hm.
I is ano he impo an ac o when e alua ing he e iciency o an algo i hm.
912 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
minπ∈Gn,c(π)2<n
λ. Howe e , o a small sample size, he e migh be some c= c0
such ha he inequali y holds.16 When nbecomes la ge , ewe selec ion ec o s would
sa is y he inequali y, because minπ∈Gn,c(π)2<n
λholds o λ<0onlyi c=c0when
n→∞.Asa esul ,SCsin each s ep con ains ewe elemen s o la ge n.Thecompu-
a ion ime o he MMS dec eases and becomes linea in lin he limi .
3.3 P ac ical implemen a ion
We summa ize he compu a ion o he MMS es ima o (
c,π)in Algo i hm 1and p o-
ide guidance on he choice o uning pa ame e s l1,α1,λ,andin ini e samples. We
discuss he oles o he uning pa ame e s based on he ascending o de o hei ela i e
impo ance o he compu a ional ime. Fo s=1, ,S,le scs
0deno e he subselec ion
Algo i hm 1: The MMS p ocedu e.
16E en i in a e cases whe e minπ∈Gn,c(π)2is small o all c=c0, by se ing an agg essi e λ, he MMS
p ocedu e can s ill imp o e upon (3.2) in unning ime.

Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 913
ec o whose i s 2lselemen s a e he same as c0and he emaining elemen s a e ze os.
Namely, scs
0selec s he co ec i s lsmomen s.
We i s discuss he ole o λ. In each s ep, nλse es as a h eshold o iden i y scs∈
SCs ha is su ely (wp →1) di e en om scs
0. The h eshold is conse a i e o la ge λ
and agg essi e o smalle λ. By he p ope y o scs
0and Gn,scs
0(π),minπ∈Gn,scs
0(π)2=
Op(n−1).Thus,i minπ∈Gn,scs(π)2does no con e ge o ze o a a e nλ o λ>−1
as n→∞, henP (scs= scs
0)→1. Excluding such scs om he ou pu o S ep s,SCs
n,
educes he numbe o elemen s in he inpu o S ep (s+1),SCs+1, and he inpu s o
all he nex s eps. We ecommend λ=−0.01 based on he simula ion s udy.
The pa ame e α1ac s as a sa e y ne o keeping sc1
0in SC1
nin ini e samples. I also
indi ec ly p e en s scs
0 om being excluded om SCs
nin ini e samples o s=2, ,S.
Because minπ∈Gn,scs
0(π)2may no be small due o he ini e sample e o , enough
subselec ion ec o s need o be included in SCs
nso ha scs
0is no elimina ed in each
s ep. We delibe a ely se α=2−, so ha he numbe o elemen s in he ou pu se does
no dec ease as he algo i hm p oceeds. The e a e a leas α12l1−1elemen s in SCs
na e
each s ep. When α1is la ge , SCs
n ends o ha e mo e elemen s, which inc eases he
chance ha scs
0∈SCs
n. Ex ensi e simula ion sugges s se ing α1=0.5%.
The uning pa ame e l1a ec s he compu a ional ime o (
c,π), because he se
SC1
ndi ec ly a ec s SC2in S ep 2 and SCsin all he ollowing s eps. The e a e α
12l1−1
elemen s in SC1
n,whe eα
1≡|SC1
n|/|SC1|, and he same o de o elemen s in SCs
n
o s=2, ,S. The alue o α
1∈[α1,1
]is de e mined by he pe cen age o sc1’s in SC1
such ha minπ∈Gn,sc1(π)2is small. I Jα1
n>n
λ, henα
1=α1, while i Jα1
n≤nλ, hen
100α
1%o sc1’s in SC1sa is y ha Jn(sc1)≤nλ. In consequence, when only a small
po ion o elemen s in SC1
nmake minπ∈Gn,sc1(π)2small, α
1is small. In ui i ely,
minπ∈Gn,sc1(π)2 ends o be small i no o a ew inco ec momen s a e selec ed by
sc1. Because he pe cen age o such sc1’s in SC1dec eases wi h l1,α
1is smalle o
la ge l1, and ice e sa. Fo example, he pe cen age o subselec ion ec o s in SC1
ha selec only one inco ec momen is (l1−1)/2l1−1, which dec eases d ama ically as
l1inc eases. A he same ime, enough momen s ela i e o lπshall be included in S ep 1.
Since he numbe o elemen s in SC1
nis a p oduc o 2l1−1and α
1, we need o balance
he wo e ec s o l1 o achie e as e unning ime. Based on ex ensi e simula ions, we
sugges se ing l1=5lπ.17
The alue o a ec s he o al numbe o s eps and he numbe o op imiza ions
in each s ep. La ge leads o ewe s eps, because S=l−l1
. On he o he hand, low-
e ing dec eases he compu a ion ime o each s ep. Gi en he numbe o elemen s in
he ou pu se o S ep (s−1), he inpu se SCso S ep shas 2 imes mo e elemen s.
Dec easing would hen educe he ca dinali y o SCsand he numbe o ope a ions
in each s ep. We ecommend se ing =2 based on ex ensi e simula ions. Once is
chosen, he alue o αis de e mined acco dingly as 2−so ha mo e momen s we add
in each s ep, mo e agg essi e we a e in he elimina ion o inco ec ma chings.
In summa y, we ecommend se ing λ=−0.01, α1=0.5%, l1=5lπ,and=2in he
MMS p ocedu e. We call such a choice he ule-o - humb. See mo e discussion on he
oles o he uning pa ame e s and he ule-o - humb in Sec ion 6.1.
17Since lπis small, o cases whe e l<5lπ,(
c,π)can be employed di ec ly.
914 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
4. In e ence on he payo ec o in he Simple Game
This sec ion de elops a es o he ollowing linea hypo hesis:
H0:Rπ0= agains H1:Rπ0= , (4.1)
whe e Ris o dimension lR×lπwi h ank(R)=lRand is o dimension lR×1. A simple
es s a is ic would be
min
c∈C,Rπ= √nGn,c(π)2
Wn(c),
which is expec ed o be la ge i he null is inco ec . Howe e , his es s a is ic can be
compu a ionally challenging when he pa ame e space Cis la ge. Simila o he mul i-
s ep es ima o p oposed in Sec ion 3.1, we p opose he mul is ep es s a is ic:
Tn≡min
c∈Cn,Rπ= √nGn,c(π)2
Wn(c), (4.2)
whe e Cnis he e ec i e pa ame e space o c0in he las s ep o he MMS.18 See Sec-
ion 3.1.
4.1 Asymp o ic alidi y and consis ency
No e ha we a e dealing wi h a pos -selec ion in e ence p oblem because o he buil -
in momen selec ion in ou es s a is ic. Albei he many challenges aced wi h gen-
e al pos -selec ion in e ence documen ed in Leeb and Pö sche (2005)andLeeb and
Pö sche (2008), we a e able o cons uc an asymp o ically uni o mly alid and compu-
a ionally simple es based on Tn.
Assume ha he model is ully cha ac e ized by ξ∈,whe eis some compac pa-
ame e space and is possibly in ini e-dimensional. Fo he Simple Game, ξincludes he
dis ibu ion o p i a e in o ma ion, condi ional p obabili y o la en s a e on obse ed
s a e, and payo ec o s o each indi idual. Deno e Ras he pa ame e space con-
sis en wi h he null hypo hesis and P ξ(·)as he p obabili y calcula ed unde ξ.The
objec i e is o ind a c i ical alue CV ha con ols he asymp o ic size de ined as
AsySize ≡lim sup
n→∞
sup
ξ∈R
P ξ(Tn>CV). (4.3)
We conside he d i ing model pa ame e sequence ξnand he se o d i ing model
pa ame e sequences unde H0wi h limi ξas
R(ξ)={ξn∈R:n≥1}:ξn→ξ∈R. (4.4)
The impo an ole o he analysis unde d i ing (sub)sequences has been emphasized
in And ews and Cheng (2012), Cheng (2015), and And ews, Cheng, and Guggenbe ge
18A es analogous o he J- es can be applied o es ing he join alidi y o he model assump ions
including he pa ame ic o m o he payo unc ion such as ha in Game 1. De ails a e omi ed due o
space conside a ions.
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 915
(2020). I s in oduc ion is no in ended as a li e al desc ip ion o eal-wo ld da a, bu
me ely a de ice ha helps us s udy he asymp o ic p ope y o he es s a is ic ha mim-
ics i s ini e-sample beha io . Fo he sys em o momen unc ions G(π),aselec ionis
co ec i Gc(π)=0 o some π, and is inco ec i Gc(π)=0 o any π. By Lemma 2.1,
he only ue selec ion is c0. Howe e , unde d i ing sequence o model pa ame e s, i
is possible ha Gξn
c(π)=0bu Gξn
c(π)→0 o some c=c0and π∈as n→∞,whe e
Gξn
c(π)deno es he momen unc ions selec ed by cunde he d i ing model pa ame e
sequence ξn. We call such selec ion a nea ly ue selec ion. When nea ly ue selec ions
exis , he p obabili y ha c0is he solu ion o he minimiza ion p oblem (4.2) does no
app oach one, so ha inco ec momen unc ions may be selec ed e en when n→∞.
Such phenomenon occu s in he pos -selec ion in e ence p oblem and o en compli-
ca es he in e ence p ocedu e. Howe e , since he numbe o co ec momen s in he
Simple Game is known o be l, he null asymp o ic dis ibu ion o Tnunde d i ing se-
quence is s ochas ically domina ed by he chi-squa ed dis ibu ion wi h (l−lπ+lR)-
deg ees o eedom. This allows us o cons uc an asymp o ically uni o mly alid es
using c i ical alue om he chi-squa ed dis ibu ion wi h (l−lπ+lR)-deg ees o ee-
dom e en in he p esence o nea ly ue selec ions. Mo eo e , by Assump ion 2.6, o any
c=c0, he sys em Gc(π)=0does no ha e a solu ion. The es s a is ic di e ges o in in-
i y unde he al e na i e hypo hesis because limn→∞minRπ= Gn,c(π)2
Wn(c)>0 o all
c∈Ci Rπ0= .
Assump ion 4.1. (i)The de i a i e o (·)is bounded.(ii)Fo any ξ∈R,z akes
each alue in Zwi h p obabili y bounded below by ε>0. (iii)Fo any ξ∈Rand
he pa ame e sequence {ξn}∈R(ξ),gi en each c∈Cn,Wn(c)=W(c)+op(1)wi h
W(c)being posi i e de ini e.(i )W(c0)=−1
0 o 0being he asymp o ic a iance o
√n(Gn,c0(π0)−Gc0(π0)).
Assump ion 4.1(i) is sa is ied by commonly used dis ibu ions and is needed o he
uni o m linea ep esen a ion o he momen unc ion gi en a uni o m linea ep esen-
a ion o he CCP es ima o . Assump ion 4.1(ii) assumes ha he suppo o zis he same
o all ξ∈Rand is one o he su icien condi ions needed o he uni o m linea ep-
esen a ion o Xiao (2018)’s CCP es ima o . Assump ion 4.1(iii) and (i ) equi e ha he
p obabili y limi o Wn(c)be posi i e de ini e and ha he op imal weigh ing ma ix be
used o c0. The asymp o ic a iance ma ix 0being nonsingula is sa is ied au oma i-
cally by he se ing o he Simple Game and is e i ied in he p oo o Theo em 4.1.
Deno e χ2
[d ],1−αas he (1−α)- h quan ile o he chi-squa ed dis ibu ion wi h d -
deg ees o eedom. The ollowing heo em s a es he asymp o ic alidi y and consis-
ency o he es based upon he es s a is ic Tnand c i ical alue χ2
[l−lπ+lR],1−α.The
p oo is p o ided in he Supplemen al Appendix.
Theo em 4.1. Le Assump ions 2.1–2.6 hold.Fo any l1∈{lπ,lπ+1, ,l},α1∈(0, 1],
λ∈(−1, 0),and ∈{1, ,l−l1},(i)i in addi ion Assump ions 3.1 and 4.1 hold, hen
limsup
n→∞
sup
ξ∈R
P ξTn>χ
2
[l−lπ+lR],1−α=α;
916 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
(ii)i in addi ion Assump ions 3.1–3.2 hold, hen o any ξ/∈R,
lim
n→∞P ξTn>χ
2
[l−lπ+lR],1−α=1.
Rema k 4.1. We can also es hypo heses in ol ing c oss-playe es ic ions on some la-
en s a e o c oss-la en s a e linea es ic ions o some playe s. Bo h can be achie ed
by s acking he momen unc ions and adjus ing he pa ame e space o he ue selec-
ion ec o . Fo example, le he null hypo hesis be ha he payo s o playe s 1 and 2 a e
equal on some la en s a e. Fo i=1, 2, we can cons uc he sample momen unc ions
Gni(πi)=πni −niπi o playe i. De ine
Gn(π1,π2)≡Gn1(π1)
Gn2(π2)=πn1
πn2−n10
0n2π1
π2and
C≡[c1,,c2l]∈R4l:c =c +l∈[1, 0],[0, 1] o ∈{1, ,l}.
In he cons uc ion o C, we keep he o de ing o mixing componen s he same ac oss
playe s because he CCPs o di e en playe s a e iden i iable up o he same label swap-
ping. The es can be ca ied ou by he es s a is ic
min
c∈Cn,π1=π2√nGn,c(π1,π2)2
Wn(c)
and he c i ical alue χ2
[2l−lπ],1−α,whe eCnis he e ec i e pa ame e space cons uc ed
om a simila MMS p ocedu e.
4.2 Boo s ap es ima ion o he weigh ing ma ix
To implemen he p oposed es o he Simple Game, he weighing ma ix needs o sa -
is y Assump ion 4.1(iii) and (i ), which in ol es es ima ing he asymp o ic co a iance
ma ix o √n(Gnc0(π0)−Gc0(π0)). Since πnand na e ob ained om plugging in es i-
ma o s o he equilib ium CCPs ia he eigendecomposi ion p ocedu e, es ima ing 0
om i s analy ical exp ession can be di icul . We p opose a nonpa ame ic boo s ap
es ima o o 0in his sec ion.
Any π∈sa is ying he sys em o linea equa ions Rπ = can be exp essed as
π +μ,whe eis a known lπ×(lπ−lR)ma ix, π is he ee pa ame e ec o o
dimension lπ−lR,andμis a known lπ×1 ec o . Compu a ion o boo s ap weigh ing
ma ix Wb
n(c)includes he ollowing s eps.
S ep 1: Fo any gi en c∈Cn,i a g minπ Gn,c(π +μ)2is no unique, hen se
Wb
n(c)as some known posi i e de ini e ma ix WPsuch as he iden i y ma ix. O he wise,
le π (c)=a gminπ Gn,c(π +μ)2and con inue o S ep 2.
S ep 2: Compu e he boo s ap a iance
b
nc,π (c)=n
B
B

b=1G(b)
n,cπ (c)+μ−Gn,cG(b)
n,cπ (c)+μ−Gn,c,
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 923
A compu a ionally easible es s a is ic de ined as
min
c∈C1
n∪···∪C|z1|
n,Rπ= √nGn,c(π)2
Wn(c)
can be applied and i beha es simila o Tnde ined in (5.4).
6. Mon e Ca lo simula ion
In his sec ion, we conduc a Mon e Ca lo simula ion s udy o examine he pe o -
mance o ou es ima ion and in e ence me hods in ini e samples. We use wo a i-
an s o he Simple Game and wo a ian s o he Gene al Game in his s udy o il-
lus a e he applicabili y o he p oposed me hods beyond he Simple Game and he
Gene al Game. The wo a ian s o he Simple Game a e called Game 1 and Game
2; and he wo a ian s o he Gene al Game a e called Game 3 and Game 4, espec-
i ely. In all ou games, we ully pa ame ize he payo unc ions. Game 1 and Game
2 a e games wi h only unobse ed he e ogenei y bu no mul iple equilib ia. Game 3
and Game 4 a e games wi h bo h unobse ed he e ogenei y and mul iple equilib ia.
Game 1 has been in oduced in Sec ion 2.4. We use i o s udy he e ec o he uning
pa ame e s on unning ime and co ec selec ion a e (CSR) o he MMS p ocedu e,
whe e he CSR is compu ed as he numbe o imes ha 
c=c0di ided by he num-
be o epe i ions. Based on he simula ion esul , we ecommend a ule-o - humb o
choosing he uning pa ame e s. Game 2 adds a common obse ed s a e o he Sim-
ple Game and a s a egic e ec ha a ies wi h s a e a iables. I is used as a obus -
ness check o he e ec i eness o he ule-o - humb. We in es iga e he ini e sam-
ple pe o mance o he es ima o and es . Game 3 and Game 4 a e used o e alua e
he pe o mance o he MMS p ocedu e oge he wi h he consis en g ouping me hod
ha sepa a e mul iple equilib ia om unobse ed he e ogenei y. All he esul s on he
unning ime in his sec ion a e ob ained om a compu e o 2.4GHz CPU and 1TB
RAM.
In Game 1, we le he no malized p i a e in o ma ion i ollow a logis ic dis ibu ion.
The s uc u es o Games 2–4 a e p esen ed below.21 The pa ame e alues a e p o ided
in Supplemen al Appendix D.2.
Game 2 We conside a game wi h a common obse ed s a e a iable x∈X ha akes
disc e e alues. Le he payo unc ion o playe ichoosing di=1be
πi(1, d−i,zi,x,k)=βikx+δikzi
j=i
dj,
whe e (βik,δik)a e he pa ame e s o in e es , βik cha ac e izes he e ec o x,andδikzi
cap u es he s a egic e ec ha a ies wi h zi. Bo h e ec s change wi h k.Theno mal-
21O he models wi h simila pa ame e s o in e es include Example 1 in Kasaha a and Shimo su (2009),
in which a single agen dynamic disc e e choice model has wo pa ame e s ha depend on a common
la en s a e wi h ini e suppo .

924 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
ized p i a e in o ma ion i ollows he s anda d no mal dis ibu ion. The condi ional
dis ibu ion o he la en s a e is speci ied as
P (k=A|z,x)=⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
3
4−1
10$|x|+
3

i=1|zi|% o (z,x)=z†,x†,
0.45 o (z,x)=z†,x†,
o some z†∈Zand x†∈X. The alues o (z,x)a e chosen so ha P (k=A|z,x)>
P (k=B|z,x) o all (z,x)= (z†,x†). The anking independence assump ion does no
hold, because he mixing weigh o la en s a e Ais no always s ic ly la ge han ha
o la en s a e B.
Game 3 The game has 5 playe s, 18 obse ed s a es, and 2 la en s a es. The payo
unc ion o playe iwhen choosing di=1isgi enby
πi(1, d−i,zi,k)=xθk+δk
1
N−1
j=i
dj,
whe e (θk,δk)a e he pa ame e s o in e es . The no malized p i a e in o ma ion i ol-
lows he s anda d no mal dis ibu ion.
Game 4 The numbe o playe s, obse ed s a es, and la en s a es a e he same as
Game 3. We le he payo unc ion o playe ichoosing di=1be
πi(1, d−i,zi,k)=xθk+1+x2δk
1
N−1
j=i
(2dj−1),
whe e (θk,δk)a e he pa ame e s o in e es . This (no malized) payo unc ion is a e-
sul o he ollowing wo payo unc ions o di=1anddi=0, espec i ely:
πi(1, d−i,zi,k)=xθk+1+x2δk
1
N−1
j=i
(dj=1)and
πi(0, d−i,zi,k)=1+x2δk
1
N−1
j=i
(dj=0).
The no malized p i a e in o ma ion i ollows he logis ic dis ibu ion.
The iden i ica ion s a egies o Games 2–4 a e simila o he one o Game 1 in Sec-
ion 2.4. We discuss he iden i ica ion and minimum-dis ance c i e ion o Games 2–4 in
Supplemen al Appendix D.1.
6.1 Rule-o - humb choice o uning pa ame e s in he MMS p ocedu e—Game 1
Be o e s udying he e ec o he uning pa ame e s in he MMS p ocedu e, we demon-
s a e ha Assump ion 2.6 holds gene ically. We d aw alues o all p imi i e pa ame e s
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 925
Table 1. E ec o uning pa ame e s in he TMS: unning ime (in seconds) and co ec selec-
ion a e (CSR).
l1=8l1=9l1=10 l1=11 l1=12 l1=13
α1λTime CSR Time CSR Time CSR Time CSR Time CSR Time CSR
0.5% −0.27 0.0482 0.93 0.0417 0.96 0.0432 0.99 0.0598 0.99 0.0891 0.99 0.1529 0.99
−0.09 0.0876 0.93 0.0397 0.96 0.0433 0.99 0.0596 0.99 0.0894 0.99 0.1524 0.99
−0.03 0.1659 0.94 0.0762 0.96 0.0441 0.99 0.0594 0.99 0.0886 0.99 0.1524 0.99
−0.01 0.2198 0.96 0.1082 0.96 0.0445 0.99 0.0607 0.99 0.0901 0.99 0.1537 0.99
1% −0.27 0.0766 0.95 0.0666 0.96 0.0742 0.99 0.0820 0.99 0.1110 0.99 0.1764 0.99
−0.09 0.0949 0.95 0.0689 0.96 0.0704 0.99 0.0816 0.99 0.1117 0.99 0.1770 0.99
−0.03 0.1838 0.95 0.0961 0.96 0.0716 0.99 0.0817 0.99 0.1108 0.99 0.1760 0.99
−0.01 0.2417 0.97 0.1170 0.96 0.0709 0.99 0.0823 0.99 0.1110 0.99 0.1742 0.99
1.5% −0.27 0.0753 0.95 0.0948 0.96 0.0889 0.99 0.1046 0.99 0.1339 0.99 0.1993 0.99
−0.09 0.0950 0.95 0.0900 0.96 0.0885 0.99 0.1057 0.99 0.1343 0.99 0.1992 0.99
−0.03 0.1867 0.95 0.1135 0.96 0.0931 0.99 0.1047 0.99 0.1338 0.99 0.1989 0.99
−0.01 0.2442 0.97 0.1529 0.96 0.0893 0.99 0.1043 0.99 0.1334 0.99 0.1985 0.99
2.5% −0.27 0.1481 0.95 0.1369 0.96 0.1319 0.99 0.1509 0.99 0.1828 0.99 0.2514 0.99
−0.09 0.1598 0.95 0.1329 0.96 0.1323 0.99 0.1507 0.99 0.1800 0.99 0.2511 0.99
−0.03 0.2429 0.95 0.1505 0.96 0.1312 0.99 0.1504 0.99 0.1801 0.99 0.2509 0.99
−0.01 0.2992 0.97 0.1679 0.96 0.1332 0.99 0.1496 0.99 0.1790 0.99 0.2505 0.99
independen ly om uni o m dis ibu ions whose suppo s co e he pa ame e alues
used in he simula ion. Fo all he d awn pa ame e alues, Assump ion 2.6 is sa is ied.
We in e p e his as e idence ha ou iden i ica ion assump ion holds gene ically in he
space o model p imi i es.
Tables 1and 2s udy he e ec o he uning pa ame e s in he TMS and MMS p o-
cedu es on he unning ime and CSR. We use andom samples wi h 500 obse a ions
pe obse ed s a e. The alues in he able a e based on 100 epe i ions. Fo he de-
sign in Table 1,l=18; and o he design in Table 2,l=27. I can be seen om bo h
Table 2. E ec o uning pa ame e s in he MMS: unning ime (in seconds) and co ec selec-
ion a e (CSR).
=1=2=3=4=5=6
l1α1λTime CSR Time CSR Time CSR Time CSR Time CSR Time CSR
10 0.5% −0.03 0.0457 0.99 0.0423 0.99 0.0473 0.99 0.0529 0.99 0.0592 0.99 0.0729 0.99
−0.01 0.0458 0.99 0.0428 0.99 0.0471 0.99 0.0514 0.99 0.0582 0.99 0.0727 0.99
1% −0.03 0.0531 0.99 0.0489 0.99 0.0546 0.99 0.0676 0.99 0.0821 0.99 0.1093 0.99
−0.01 0.0527 0.99 0.0500 0.99 0.0574 0.99 0.0678 0.99 0.0840 0.99 0.1084 0.99
12 0.5% −0.03 0.1555 0.99 0.1564 0.99 0.1588 0.99 0.1953 0.99 0.2312 0.99 0.2639 0.99
−0.01 0.1534 0.99 0.1585 0.99 0.1557 0.99 0.1945 0.99 0.2294 0.99 0.2618 0.99
1% −0.03 0.1771 0.99 0.1767 0.99 0.1922 0.99 0.2349 0.99 0.3078 0.99 0.3687 0.99
−0.01 0.1748 0.99 0.1817 0.99 0.1875 0.99 0.2359 0.99 0.3092 0.99 0.3575 0.99
926 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
ables ha he uning pa ame e l1plays an impo an ole in educing he unning
ime and imp o ing accu acy. As long as l1≥10 =5lπ, he accu acy is highe o equal
o 0.99. As we discussed in Sec ion 3.3, inc easing α1can inc ease he accu acy. Bu
he e ec is ma ginal. A he same ime, he unning ime inc eases wi h α1.When
he accu acy is high, o example, 99%, λdoes no a ec he compu a ion ime. Ta-
ble 2s udies he ole o in he MMS on he unning ime and CSR. The esul s sug-
ges ha when he numbe o momen s is la ge, a mul is ep p ocedu e shall be ap-
plied. The es ima o achie es a high CSR and sho unning ime when =2. When l
inc eases om 18 o 27, he numbe o elemen s in he pa ame e space Cinc eases
mo e han 500 imes. Howe e , because he no el MMS p ocedu e is compu a ionally
e y e icien , i s unning ime only inc eases sligh ly.22 As a compa ison, (
c,π)has an
a e age unning ime a ound 3000 seconds when l=27. E en wi h he leas desi able
choice o he uning pa ame e s, he MMS p ocedu e is housands o imes as e han
(
c,π).
Based on Tables 1and 2, we ecommend he ollowing ule-o - humb o se ing he
uning pa ame e s: l1=5lπ,α1=0.5%, λ=−0.01, and =2. In he subsequen simula-
ion s udy, we adop his ule o all he games and designs.
To p o ide addi ional insigh on he compu a ional sa ings o he MMS p o-
cedu e, we elabo a e on i s in e media e s eps. Compu ing (
c,π) equi es sol ing
a quad a ic op imiza ion p oblem |C| imes; while calcula ing he MMS es ima o
(
c,π)in ol es he same op imiza ion p oblem "S
s=1|SCs|+|Cn| imes. When l=
18, he ca dinali y o Cis 2l−1=131,072. On he o he hand, when implemen ing
he MMS p ocedu e wi h he ule-o - humb p oposed abo e, we ha e S=4wi h
|SC1|=512 and |SCs|=|Cn|=12 o s=2, 3, 4 based on one simula ion un.
As a esul , (
c,π) equi es only abou 1/250 numbe o op imiza ions compa ed o
(
c,π).
Al hough SC1con ains 512 subselec ion ec o s, many a e ce ainly inco ec .
In consequence, he ou pu o S ep 1, SC1
n, con ains only 3 elemen s. Mo e impo -
an ly, we do no jus elimina e hese 512 −3=509 numbe o subselec ion ec o s,
bu all selec ion ec o s ha sha e he same i s 2 ×l1elemen s as he elimina ed
sub-selec ion ec o s. In o al, we elimina e 509 ×2l−l1=130,304 selec ion ec o s
a e S ep 1, lea ing only a ew selec ion ec o s o be conside ed in he ollowing
s eps.
6.2 The MMS p ocedu e, es ima o , and es based on he ule-o - humb
Applying he sugges ed ule-o - humb o choosing he uning pa ame e s, we in es i-
ga e he ini e sample pe o mance o he es ima o and es in his sec ion. Fo Game
1 and Game 2, we in oduce i e designs o each game co esponding o di e en
numbe s o obse ed s a es. The la ges numbe o obse ed s a es conside ed in he
simula ion is se o be la ge han he sizes o he obse ed s a es in s udies such as
22The unning ime o l=27 can e en be sho e han ha o l=18 because MMS a he han TMS is
used when l=27.
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 927
Swee ing (2009), De Paula and Tang (2012), G ieco (2014), Igami and Yang (2016), and
Xiao (2018).23
Table 3 epo s he unning ime, CSR, and mean squa ed e o (MSE) o he MMS
p ocedu e o di e en sample sizes and di e en designs cons uc ed om Games 1
and 2.24 Fo each game, di e en designs co espond o di e en alues o l.nsde-
no es he numbe o obse a ions pe obse ed s a e, and MSE epo ed in he able
ha i is calcula ed as he sum o MSEs o each pa ame e . The alues in he able a e
compu ed om 1000 epe i ions. Table 3shows ha he unning ime o he MMS in-
c eases e y slowly wi h l. When he numbe o elemen s in he pa ame e space C
inc eases millions o imes, o example, om l=27 o l=64 in Game 1, he un-
ning ime o he MMS only inc eases abou wo imes. E en when he e a e mo e han
1024 (l=81) o 1029 (l=100) ma chings, he MMS akes less han one second o com-
pu e. This is consis en wi h Theo em 3.2, which shows ha wi h p obabili y app oach-
ing one, he ime complexi y o he MMS becomes linea in lwhen he sample size
goes o in ini y. The esul s in Table 3also show ha he unning ime ei he s ays he
same o dec eases wi h he sample size. We suspec ha he o me case occu s be-
cause he unning ime is al eady close o being linea in land has li le oom o im-
p o e wi h he sample size. When he sample size inc eases, he CSR inc eases and he
MSE dec eases. I would be ideal o compa e he CSR and MSE o he MMS wi h (
c,π).
Howe e , he ex emely long unning ime o (
c,π)makes he compa ison impossi-
ble.
To s udy he ini e sample pe o mance o ou in e ence p ocedu e, we ocus on De-
sign 1 o Game 1 and conside wo null hypo heses o he o m H0:Rπ0= o R=I2,
=(θ1A,δ1A);andR=(0, 1), =δ1A. The i s hypo hesis is on he whole payo pa-
ame e ec o and he second one is on he pa ame e o s a egic in e ac ion, which is
o g ea in e es in empi ical esea ch. The numbe o obse ed s a es is l=18. We se
he nominal size as 5% and use he 95% quan ile o χ2
18 and χ2
17 as he c i ical alues
o R=I2and R=(0, 1). In bo h cases, he boo s ap weigh ing ma ices a e calcu-
la ed wi h 1000 boo s ap samples. The esul s a e based on 5000 Mon e Ca lo epe i-
ions.
Table 4 epo s he esul s on he null ejec ion p obabili ies o di e en sample
sizes. The size is well con olled and is ge ing close o 0.05 as he sample size inc eases
o bo h hypo heses. Table 5p o ides he ini e sample powe esul s when he model
23In Swee ing (2009), De Paula and Tang (2012), and Xiao (2018), he majo obse ed s a e is he ank
o he ma ke acco ding o popula ion, and he e a e 144 anks in o al; in G ieco (2014), he ca dinali y o
obse ed s a es o he baseline model is 12 (3 s a us o disc e ized popula ion, 2 s a us o whe he i is
ac i e in 1998 and 2 s a us o whe he he e is a supe cen e wi hin 20 miles); in he s a ic game e sion o
Igami and Yang (2016), he ca dinali y o obse ed s a es is 16 (4 ca ego ies o popula ion and 4 ca ego ies
o a e age income).
24We also un simula ions wi h e en smalle sample sizes. The es ima ed pa ame e s become less accu-
a e, whe eas he unning ime is s ill sho . As a sequen ial es ima o , ou MMS es ima o is a ec ed by he
well-known ini e sample bias documen ed in pape s like Agui egabi ia and Mi a (2007) and Agui egabi ia
and Ma coux (2021). The bias la gely esul s om poo ly es ima ed CCPs a small sample sizes. Al hough
de eloping CCP es ima o s wi h be e ini e sample pe o mance is beyond he scope o his pape , we see
much alue in u u e esea ch in his di ec ion.
928 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
Table 3. Fini e sample pe o mance o he MMS p ocedu e in di e en designs: Running ime (in seconds), co ec selec ion a e (CSR), and
mean squa ed e o (MSE).
Design 1: l=18 Design 2: l=27 Design 3: l=64 Design 4: l=100 Design 5: l=288
nsTime CSR MSE Time CSR MSE Time CSR MSE Time CSR MSE Time CSR MSE
Game 1 250 0.0400 0.944 0.3824 0.0448 0.967 0.0886 0.1206 0.912 0.2142 0.2338 0.895 0.3373 0.4518 0.795 0.5273
500 0.0363 0.996 0.0749 0.0433 0.997 0.0181 0.0811 0.994 0.0355 0.1980 0.990 0.0323 0.4638 0.982 0.0750
750 0.0363 1 0.0159 0.0443 1 0.0121 0.0606 1 0.0046 0.1409 1 0.0030 0.3900 0.998 0.0162
1000 0.0325 1 0.0121 0.0436 1 0.0086 0.0591 1 0.0033 0.1198 1 0.0020 0.3737 1 0.0012
Design 1: l=24 Design 2: l=36 Design 3: l=54 Design 4: l=81 Design 5: l=162
nsTime CSR MSE Time CSR MSE Time CSR MSE Time CSR MSE Time CSR MSE
Game 2 250 0.0368 0.933 0.7600 0.0307 0.916 1.102 0.0622 0.845 2.366 0.1992 0.761 3.286 0.4459 0.740 3.596
500 0.0388 0.996 0.0725 0.0313 0.990 0.1774 0.0652 0.980 0.3602 0.1658 0.956 0.7004 0.3557 0.956 0.7468
750 0.0224 1 0.0237 0.0324 0.998 0.0623 0.0615 0.999 0.0415 0.1102 0.991 0.1656 0.2742 0.994 0.1844
1000 0.0212 1 0.0203 0.0292 0.999 0.0205 0.0531 1 0.0144 0.0839 0.992 0.1646 0.2194 0.996 0.1422

Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 929
Table 4. Fini e sample ejec ion p obabili ies unde H0 o di e en sample sizes.
ns500 625 750 875 1000 1125 1250
R=I20.0336 0.0334 0.0340 0.0394 0.0430 0.0406 0.0472
R=(0, 1)0.0252 0.0262 0.0294 0.0350 0.0392 0.0386 0.0436
de ia es om he null hypo hesis. When he sample size is ixed, Table 5shows ha as
he ue alue de ia es u he om he hypo hesized alue, he p obabili y o ejec ing
he null hypo hesis inc eases. A he same ime, o any ixed de ia ion, he ejec ion
p obabili y inc eases wi h he sample size.
6.3 Sepa a ing unobse ed he e ogenei y om mul iple equilib ia
We use Games 3 and 4 o examine he pe o mance o ou es ima o o he Gene al
Game. Fo bo h games, we conside wo designs: in Design 1, he e a e 2 equilib ia on
la en s a e A o he 8 h obse ed s a e; and in Design 2, he e a e 2 equilib ia on la-
en s a e A o bo h he8 hand he16 hobse eds a e.Fo obse eds a esonwhich
he e a e no mul iple equilib ia, we le P (k=A|x)=0.5. Fo obse ed s a es on which
he e a e mul iple equilib ia, he condi ional dis ibu ion o he composi e la en a i-
able ωis speci ied as ollows: ω=1 wi h p obabili y 0.4, which co esponds o la en
s a e Aand equilib ium 1; ω=2 wi h p obabili y 0.3, which co esponds o la en s a e
Aand equilib ium 2; ω=3 wi h p obabili y 0.3, which co esponds o la en s a e B.
The esul s o he simula ions a e epo ed in he able below. F om 100 epe i ions, we
documen he a e age unning ime and he co ec g ouping a e (CGR), whe e he un-
ning ime is he o al ime o un ou p ocedu es de eloped in Sec ions 5.2.1 and 5.2.2,
and he CGR is p obabili y ha 
S
Kis equal o he co ec pa i ion. When 
S
Kequals o
he co ec pa i ion, 
K=|K|and #
πkis he a e age o consis en es ima o s o πk
0 o
k=1, ,|K|.
F om Table 6, we see ha ou es ima ion p ocedu e de eloped in Sec ion 5.2 o he
Gene al Game is bo h as o un and accu a e. The CGR is e y close o equal o one
ac oss di e en designs and inc eases wi h sample size.
Table 5. Fini e sample ejec ion p obabili ies unde H1 o di e en de ia ions and sample
sizes.
De . −0.15 −0.1 −0.05 −0.025 0.025 0.05 0.1 0.15
R=I2ns=500 0.9586 0.4704 0.0662 0.0338 0.0722 0.2004 0.7822 0.9950
ns=750 0.9996 0.8298 0.1526 0.0422 0.0892 0.3012 0.9440 0.9998
ns=1000 1 0.9608 0.2596 0.0672 0.1126 0.4228 0.9904 1
De . −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2
R=(0, 1)ns=500 0.8850 0.4902 0.1364 0.0330 0.1358 0.4966 0.8708 0.9440
ns=750 0.9908 0.8064 0.2806 0.0470 0.2420 0.7710 0.9870 0.9950
ns=1000 1 0.9380 0.4372 0.0666 0.3618 0.9170 0.9998 1
930 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
Table 6. Pe o mance o he consis en g ouping me hod on op o MMS: unning ime (in sec-
onds) and co ec g ouping a e (CGR).
Game 3 Game 4
Design 1 Design 2 Design 1 Design 2
nsTime CGR Time CGR Time CGR Time CGR
250 4.8851 0.99 5.0575 1 5.4538 0.96 5.9654 0.98
500 4.7580 1 5.0110 1 5.0065 0.98 5.0314 0.99
750 5.0105 1 4.6777 1 4.9731 1 5.0438 1
1000 4.7189 1 4.6639 1 4.8997 1 4.9798 1
7. Conclusion
In his pape , we ha e p oposed a compu a ionally as sequen ial me hod o es ima e
he payo unc ion and o conduc uni o m in e ence in s a ic games o incomple e in-
o ma ion wi h unobse ed he e ogenei y and mul iple equilib ia. I builds on a no el
cha ac e iza ion o he ma ching- ypes p oblem as a minimum-dis ance p oblem wi h
bo h co ec and inco ec momen s. Based on his cha ac e iza ion, we de elop a new
MMS p ocedu e ha is ex emely as o implemen . Fo in e ence, we cons uc an
asymp o ically uni o mly alid es o linea hypo heses on he payo unc ion. The
es is easy o implemen wi h known c i ical alues om he chi-squa ed dis ibu ion.
An ex ensi e Mon e Ca lo s udy is ca ied ou o in es iga e he ini e sample pe o -
mance o ou es ima ion and in e ence p ocedu es.
Ins ead o employing sequen ial es ima o s, esea che s could make use o he equi-
lib ium condi ion o imp o e he ini e sample pe o mance o s uc u al es ima o s.
Agui egabi ia and Mi a (2007) de elop a nes ed pseudo-likelihood algo i hm ha im-
poses he equilib ium condi ion i e a i ely while a oids sol ing he equilib ium condi-
ion exac ly o any pa ame e as in nes ed ixed-poin algo i hm. In a companion pape ,
we aim o ex end he nes ed pseudo-likelihood algo i hm o games wi h bo h mul iple
equilib ia and unobse ed he e ogenei y.25 We shall add ess open ques ions on he s a-
bili y and con e gence o he algo i hm and o combine i wi h ou consis en g ouping
me hod o sepa a e mul iple equilib ia om unobse ed he e ogenei y.
The MMS p ocedu e in oduced in his pape has b oad applicabili y besides he
s udy o games. We a e cu en ly wo king on i s ex ensions o he gene al momen se-
lec ion p oblems discussed in And ews (1999).
Appendix A: Addi ional de ails o he Gene al Game
In his Appendix, we p o ide addi ional de ails o he Gene al Game discussed in Sec-
ion 5. In Appendix A.1, we p o ide he exp essions o πand  o cons uc ing he sys-
em o momen unc ions. Appendix A.2 con ains he de ailed p ocedu e o es ima ing
(ch
0,πh
0) o h=1, ,|z1|by ex ending he MMS p ocedu e de eloped in Sec ion 3.1
o he Simple Game. In Appendix A.3, we p o e ha ou es ima o s a e consis en and
25We hank an anonymous e e ee o sugges ing his u u e line o esea ch.
Quan i a i e Economics 15 (2024) Es ima e games wi h unobse ed he e ogenei y 931
as o compu e. Appendix A.4 shows ha he es is asymp o ically uni o mly alid and
consis en .
A.1 Exp essions o πand 
Fo s=1, ,|z|, we de ine ω(s,z)as he s h alue o he composi e la en a iable on
obse ed s a e ec o z. The coe icien ma ices in he momen unc ion G(π) o he
Gene al Game a e
π=π1z1,,π1zland =1z1,,1zl,
whe e πhas dimension J"l
=1|z |,has dimension J"l
=1|z |by J(J+1)N−1,and
o =1, ,l,
π1z =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
π11, z ,ω1, z 
.
.
.
π1J,z ,ω1, z 
.
.
.
π11, z ,ω|z |,z 
.
.
.
π1J,z ,ω|z |,z 
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
and 1z =⎡
⎢
⎣
ι11, z 
.
.
.
ι1|z |,z ⎤
⎥
⎦.
Fo s={1, ,|z |}, he ma ix ι1(s,z )is a block diagonal ma ix wi h Jiden ical blocks
gi en by
ι1s,z =⎡
⎢
⎣
p−1z ,ωs,z  0
...
0p
−1z ,ωs,z ⎤
⎥
⎦,
whe e p−1(z ,ω(s,z ))is a ow ec o wi h (J+1)N−1elemen s being he p obabili ies o
join ac ions o all o he playe s in he same spi i as he Simple Game in Sec ion 2.2.2.
A.2 MMS p ocedu e o es ima ing πh
0
We ocus on de eloping he MMS p ocedu e o (c1
0,π1
0) o simpli y ou discussion and
no a ion.
Le sc deno e he subselec ion ec o o dimension J"l
=1|z | ha consis s o e1
and e0. Following De ini ion 5.1,deno esc o =1, ,las he h sub ec o o sc such
ha sc ≡[sc1,,scl]. De ine Jn(sc)≡minπ∈Gn,sc(π)2.
S ep 0: Se l1∈{lπ,lπ+1, ,l},α1∈(0, 1],λ∈(−1, 0),and∈{1, ,l−l1}.Le
S=l−l1
and α=(2|z1|−1)−. The alue o αis chosen acco ding o he same spi i
as he one in he MMS p ocedu e o he Simple Game: he mo e momen s we add in
each s ep, he smalle p opo ion o ma chings we end o keep. See Sec ion 3.3 o mo e
de ail. Relabel {z1,,zl}such ha |z| o z∈{z1,,zl}a e in an ascending o de .
932 Fan, Jiang, and Shi Quan i a i e Economics 15 (2024)
S ep 1: The inpu o S ep 1 is
SC1≡⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
[sc1,,scl]:sc1=[sc1,1,,sc1,|z1|]wi h sc1,1 =e1
and sc1,j∈{e1,e0} o j=1;
o =2, ,l1,sc =[sc ,1,,sc ,|z |]wi h sc ,w ∈{e1,e0},whe e
w∈1, ,|z |and sc =[e0,,e0]; o =l1+1, ,l,sc =[e0,,e0]
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
.
So Jn(sc1) o all sc1∈SC1, and deno e Jα1
nas he alue o he 100α1% smalles . The
ou pu o S ep 1 is
SC1
n≡sc1∈SC1:Jnsc1≤maxJα1
n,nλ.
S eps2,3,...S:Fo s=2, ,S, de ine ls≡ls−1+. The inpu o S ep sis he collec-
ion o subselec ion ec o s de ined as
SCs≡⎧
⎪
⎨
⎪
⎩
[sc1,,scl]:[sc1,,scls−1]=scs−1
1,,scs−1
ls−1 o some scs−1∈SCs−1
n;
o =ls−1+1, ,ls,sc =[sc ,1,,sc ,|z |]wi h sc ,w ∈{e1,e0},whe e
w∈1, ,|z |and sc =[e0,,e0]; o =ls+1, ,l,sc =[e0,,e0]
⎫
⎪
⎬
⎪
⎭.
The ou pu o S ep sis
SCs
n≡scs∈SCs:Jnscs≤maxJα
n,nλ.
S ep (S+1):The e ec i e pa ame e space o c1
0is
C1
n≡⎧
⎪
⎨
⎪
⎩
[c1,,cl]:[c1,,clS]=scS
1,,scS
lS o some scS∈SCS
n;
o =lS+1, ,l,c =[c ,1,,c ,|z |]wi h c ,w ∈{e1,e0},
whe e w ∈1, ,|z |and c =[e0,,e0]
⎫
⎪
⎬
⎪
⎭.
Gi en he e ec i e pa ame e space, he MMS es ima o o (c1
0,π1
0)is de ined as

c1,π1≡a g min
c∈C1
n,π∈Gn,c(π)2
Wn(c)−ρ1c0κ1,n/n.
By elimina ing inco ec ma chings in mul iple s eps, he size o he e ec i e pa-
ame e space C1
nis much smalle han ha o C1. As a esul , he mul is ep es ima o
(
c1,π1)is much as e o compu e han (
c1,π1). We epea he abo e p ocedu e o ob-
ain es ima o s (
ch,πh) o h=2, ,|z1|.
A.3 Asymp o ic p ope ies o he es ima o s
In his sec ion, we p esen esul s on he consis ency o ou es ima o s and he ime and
space complexi ies o ou MMS p ocedu e. We p o ide su icien condi ions ha pa allel
hose s a ed in he Simple Game. The p oo s o he esul s in his sec ion ollow he sim-
ila a gumen s o he esul s in Sec ions 3.2. All p oo s a e collec ed in he Supplemen al
Appendix.
We i s p o ide assump ions o he consis ency o he p oposed es ima o (
ch,πh)
o h=1, ,|z1|.