INDIRECT LIKELIHOOD INFERENCE
MICHAEL CREEL AND DENNIS KRISTENSEN
ABSTRACT. Gi en a sample om a ully speci ied pa ame ic model, le Znbe a gi en
ini e-dimensional s a is ic - o example, an ini ial es ima o o a se o sample momen s.
We p opose o ( e-)es ima e he pa ame e s o he model by maximizing he likelihood o
Zn. We call his he maximum indi ec likelihood (MIL) es ima o . We also p opose a com-
pu a ionally ac able Bayesian e sion o he es ima o which we e e o as a Bayesian
Indi ec Likelihood (BIL) es ima o . In mos cases, he densi y o he s a is ic will be o
unknown o m, and we de elop simula ed e sions o he MIL and BIL es ima o s. We
show ha he indi ec likelihood es ima o s a e consis en and asymp o ically no mally
dis ibu ed, wi h he same asymp o ic a iance as ha o he co esponding e icien wo-
s ep GMM es ima o based on he same s a is ic. Howe e , ou likelihood-based es ima-
o s, by aking in o accoun he ull ini e-sample dis ibu ion o he s a is ic, a e highe
o de e icien ela i e o GMM- ype es ima o s. Fu he mo e, in many cases hey enjoy
a bias educ ion p ope y simila o ha o he indi ec in e ence es ima o . Mon e Ca lo
esul s o a numbe o applica ions including dynamic and nonlinea panel da a mod-
els, a s uc u al auc ion model and wo DSGE models show ha he p oposed es ima o s
indeed ha e a ac i e ini e sample p ope ies.
Keywo ds: indi ec in e ence; maximum-likelihood; simula ion-based me hods; bias
co ec ion; Bayesian es ima ion.
JEL codes: C13, C14, C15, C33.
Da e: May 2011.
We wish o hank M. A ellano, S. Bonhomme, C. Bos, F. C udu, U. Mülle , P.C.B. Phillips, E. Sen ana and
pa icipan s a semina s a Columbia Uni e si y, G oningen Uni e si y, Singapo e Managemen Uni e si y
and a he G ea e New Yo k A ea Econome ics Colloquium 2010 a NYU o help ul commen s and sug-
ges ions. This wo k was suppo ed by g an s MICINN-ECO2009-11857, SGR2009-578, and NSF g an no.
SES-0961596.
1
INDIRECT LIKELIHOOD INFERENCE 2
1. INTRODUCTION
Suppose we ha e a ully speci ied and hus simulable model, indexed by a pa ame e
θ∈Θ⊂Rk. We ha e obse ed a sample Yn=(y1, ..., yn)gene a ed a he unknown
ue pa ame e alue θ0abou which we wish o lea n. A na u al ool o his end is he
likelihood unc ion, (Yn|θ), and he associa ed maximum likelihood es ima o (MLE),
which has a numbe o a ac i e la ge sample op imali y p ope ies. Howe e , he MLE
is in some si ua ions di icul o compu e due o he complexi y o he model, and i may
equi e nume ical app oxima ions ha can de e io a e he pe o mance o he esul ing
app oxima e MLE. Fo example, i he model in ol es la en a iables, hey mus be in e-
g a ed ou in o de o ob ain he likelihood in e ms o obse ables. Mo eo e , e en i he
MLE is easily compu ed, i may su e om signi ican biases in ini e samples wi h he
esul ing p ecision being a he poo , which complica es ini e-sample in e ence. Well-
known examples a e he biases o leas -squa es es ima o s in au o eg essi e models (An-
d ews, 1993) and in dynamic and nonlinea panel da a models (Hahn and Kue s eine ,
2002; Hahn and Newey, 2004).
To deal wi h he issue o compu a ional complexi y, esea che s o en eso o GMM-
ype me hods whe e a s a is ic Zn=Zn(Yn)is used o d aw in e ence ega ding he
pa ame e o in e es . Suppose o example, ha Znis a se o sample momen s: Then
a na u al way o es ima e pa ame e s is o minimize he dis ance be ween sample and
model-implied momen s. When he o m o he popula ion momen s a e unknown, sim-
ula ions may be used, and one ob ains he simula ed me hod o momen s (SMM; McFad-
den, 1989; Du ie and Single on, 1993). The indi ec in e ence es ima o (II; Gou ié oux,
Mon o , Renaul , 1993; Smi h, 1993) p oposes an al e na i e choice o Zn, namely as
an ex emum es ima o based on an auxilia y model. The e icien me hod o momen s
(EMM; Gallan and Tauchen, 1996) se s Zn o be he sco e ec o o an auxilia y model.
Simila ly, he e exis nume ous me hods designed o educe biases in es ima o s such
as boo s ap (E e ae and Pozzi, 2007; Hall and Ho owi z, 1996), jackkni e (Hahn and
Newey, 2004; Kezdi, Hahn, and Solon, 2001), analy ical me hods (Hahn and Kue s eine ,
2002; Hahn and Newey, 2004) and II (Gou ié oux, Phillips and Yu, 2010; Gou ié oux,
Renaul and Touzi, 2000). Al e na i ely, one can adjus he es ima o o ob ain median-
unbiased es ima o s; see e.g. And ews (1993). Wi h Znchosen as he ini ial es ima o ,
one can hink o hese me hods as a ype o GMM p ocedu e whe e he sample s a is ic
is ma ched agains i s model implied e sion, e.g. i s ini e-sample mean o median o
ob ain a new, imp o ed es ima o . This is in pa icula he case wi h he II es ima o when
he auxilia y model is chosen as he ac ual model.
We he e p opose a me hod ha o e ini e-sample imp o emen s o e he a o emen-
ioned es ima ion me hods. As wi h all he abo e GMM- ype es ima o s1, we ake as
s a ing poin some s a is ic Zn, which, o example, could be an ini ial es ima o o θ0, a
se o sample momen s, o an auxilia y model s a is ic as used in II. Howe e , a he han
minimizing some L2-dis ance, we p opose o ( e-)es ima e he pa ame e s o in e es by
1We use he e m “GMM- ype es ima o s” o e e o GMM, MSM, II o EMM es ima o s based upon a
s a is ic Zn, as desc ibed in he ex .
INDIRECT LIKELIHOOD INFERENCE 3
maximizing he likelihood implied by Zn. This leads o a maximum-likelihood ype es i-
ma o which we call he maximum indi ec likelihood es ima o (MIL), since we ope a e
hough he s a is ic a he han on he sample di ec ly. As a compu a ionally a ac i e al-
e na i e o he MIL, we also p opose a Bayesian e sion o ou es ima o which is e med
a Bayesian indi ec likelihood (BIL) es ima o . These IL es ima o s o e ini e-sample im-
p o emen s o e he co esponding GMM- ype es ima o s based on he same s a is ic as
we will a gue in he ollowing.
We de i e he asymp o ic dis ibu ions o he IL es ima o s and ind ha hey a e i s -
o de equi alen o he GMM es ima o ha is based on he same auxilia y s a is ic and
uses an op imal weigh ing ma ix. Howe e , he o me will in gene al enjoy be e small
sample pe o mance compa ed o he la e o wo easons: Fi s , while GMM es ima-
o s only u ilize he i s and second momen o he s a is ic, IL es ima o s a e based on
a ull desc ip ion o i s ini e sample dis ibu ional cha ac e is ics. As such, we expec
hem o be supe io o he GMM es ima o in e ms o highe -o de op imali y c i e ia
such as he “la ge de ia ions” p inciple (Bahadu , Zabell and Gup a, 1980), highe -o de
e iciency (P anzagl and We elmeye , 1978), and la ge de ia ion p obabili ies o ype II
e o s (Zei ouni and Gu man, 1991).
Second, he i s -o de equi alence esul s ely on he GMM es ima o being compu ed
using he op imal weigh ing ma ix. Since his in gene al is unknown, i has o be es i-
ma ed in o de o he e icien GMM es ima o o be easible. This is pa icula ly di icul
in ime se ies models whe e HAC- ype es ima o s ha e o be employed. In con as , o
ou es ima o s he e is no need o es ima e an op imal weigh ing ma ix since he likeli-
hood unc ion al eady embodies he in o ma ion inhe en in he op imal weigh ma ix.
This elimina es an impo an sou ce o imp ecision ha can ad e sely a ec he small
sample pe o mance o o e iden i ied GMM- ype es ima o s (Al onji and Segal, 1996;
Do an and Schmid , 2006; Hansen, Hea on and Ya on, 1996).
To jus i y he abo e claims o highe -o de op imali y o he MIL o e he co espond-
ing GMM es ima o , we p o ide a highe -o de asymp o ic analysis o bo h es ima o s.
In pa icula , we demons a e ha while he compe ing es ima o s ha e same leading
a iance componen s, and so a e i s -o de equi alen , he MIL es ima o is hi d-o de
e icien in he sense ha i has a smalle highe -o de a iance ela i e o he GMM es i-
ma o s.
The implemen a ion o he indi ec likelihood es ima o s depends on he likelihood
unc ion o he s a is ic being a ailable on closed o m, which will no mally no be he
case. Howe e , i he model is ully speci ied, ou abili y o lea n abou he likelihood
o he s a is ic is limi ed only by willingness o do simula ions. In pa icula , we o -
mula e easible e sions o he MIL and BIL es ima o s by combining simula ions wi h
nonpa ame ic densi y and eg ession echniques espec i ely as in, o example, C eel
and K is ensen (2009), Fe manian and Salanié (2004), and K is ensen and Shin (2008). The
simula ed e sions a e shown o be asymp o ically i s -o de equi alen o he in easible
MIL and BIL es ima o s as he numbe o simula ions inc eases.
The abo e men ioned heo e ical a gumen s o imp o ed ini e-sample pe o mance
o ou indi ec likelihood es ima o s o e GMM- ype es ima o s a e suppo ed by Mon e
INDIRECT LIKELIHOOD INFERENCE 4
Ca lo esul s. We in es iga e he pe o mance o he p oposed es ima o s using a wide
ange o models, including ime se ies, dynamic and nonlinea panel da a, s uc u al auc-
ion, and dynamic s ochas ic gene al equilib ium models. In e ms o oo mean squa ed
e o and bias, we ind ha he simula ed e sion o he BIL es ima o exhibi s pe o -
mance ha is almos always as good, and in mos cases be e , han he co esponding
GMM- ype es ima o s. In pa icula , BIL is ound o inhe i he au oma ed bias-co ec ion
ea u e o he s anda d Indi ec In e ence (II hence o h) es ima o s discussed abo e.
When his pape was nea ly comple ed, we became awa e o so-called App oxima e
Bayesian Compu a ion (ABC) o likelihood- ee Bayesian in e ence (see, e.g., Ta a é e al.,
1997; Ma jo am e al., 2003; Sisson, Fan and Tanaka, 2007) which a e used in he biological
sciences, including gene ics, epidemiology and popula ion biology. One o m o ABC
(Beaumon , Zhang and Balding, 2002) di ec ly implemen s wha we call he simula ed
BIL (SBIL) es ima o . While he ABC li e a u e is qui e ma u e om an empi ical poin o
iew, no heo e ical esul s a e a ailable o ABC es ima o s and hei simula ed e sions,
and so his pape o e s a numbe o con ibu ions in his di ec ion. Mo eo e , he ABC
li e a u e only con ains a he limi ed esul s on he BIL’s ini e-sample pe o mance; we
p o ide ex ensi e Mon e Ca lo examples in es iga ing his. As such, his pape p o ides
an asymp o ic heo y and ini e-sample analysis ha has been missing o his li e a u e.
The emains o he pape is o ganized as ollows: Sec ion 2 p esen s he indi ec like-
lihood es ima o s, and Sec ion 3 discusses hei implemen a ion. Fi s - and highe -o de
heo y o he es ima o s a e de eloped in Sec ions 4 and 5 espec i ely. Sec ion 6 con ains
he simula ion s udies, while Sec ion 7 concludes. All p oo s ha e been elega ed o he
Appendix.
2. INDIRECT LIKELIHOOD INFERENCE
We conside he se ing desc ibed in he in oduc ion, whe e we wish o lea n abou
a pa ame e θ∈Θ⊂Rkdesc ibing a model. Gi en a sample Yn=(y1, ..., yn) om he
model, we choose o make in e ence on θ h ough a d-dimensional s a is ic o he sample,
Zn=Zn(Yn)∈Rd. We can hink o Ynas a ( andom) mapping aking a pa ame e
alue in o he co esponding obse ed sample, Yn=Yn(θ). This in u n implies ha he
s a is ic also implici ly is a unc ion o θ h ough he da a, and w i e
Zn(θ)≡Zn(Yn(θ)).
In pa icula , he obse ed s a is ic is his andom mapping e alua ed a he ue pa am-
e e alue which we deno e θ0,Zn=Zn(θ0). Le n(Zn|θ)be he likelihood o he s a is ic
o a gi en alue o he pa ame e . Suppose o now ha he likelihood o he s a is ic is
known on closed o m.2We hen p opose o es ima e he pa ame e s by maximizing he
indi ec likelihood de ined h ough Zn:
(1) ˆ
θMIL =a g sup
θ∈Θ
log n(Zn|θ).
2In gene al, his will no be he case; in he nex sec ion we he e o e de elop a simula ed e sion o i .
INDIRECT LIKELIHOOD INFERENCE 5
The p oposed es ima o is indi ec , because he sample da a is il e ed h ough a s a is ic,
and we e e o he es ima o as a maximum-indi ec likelihood (MIL) es ima o .
Compa ed o he ac ual MLE based on he ull sample, he MIL es ima o will in gen-
e al su e om an in o ma ion loss and will only ob ain ull maximum-likelihood e i-
ciency i he s a is ic is su icien in he sense ha i spans he sco e o he ull sample
log-likelihood. On he o he hand, he compu a ion o he indi ec likelihood is a lowe -
dimensional p oblem compa ed o he ull likelihood (dim(Zn)<dim (Yn)). Mo eo e ,
e en when he ull MLE is compu a ionally easible, he IL es ima o can be used o ad-
jus o ini e-sample biases as a gued below. Finally, we no e ha he IL es ima o in
gene al will be mo e obus compa ed o he ull MLE in ha i can handle misspeci ied
models and emains consis en as long as Zniden i ies he pa ame e o in e es . These
a e o some ex en sha ed by GMM es ima o s based on he same s a is ic. Howe e , in
ini e samples he wo es ima o s will pe o m di e en ly, and he MIL will in gene al ex-
hibi highe -o de imp o emen s ela i e o he GMM es ima o . Two leading examples
illus a ing his gene al phenomenon a e he ollowing:
In he i s example, suppose ha we ha e a ailable some ini ial es ima o , say ˆ
θ. Un-
de sui able egula i y condi ions, his es ima o will be asymp o ically no mally dis-
ibu ed cen e ed a ound he ue pa ame e alue θ0. Howe e , in ini e samples he
es ima o will in gene al no be no mally dis ibu ed and no be cen e ed a ound θ0. I
he e o e appea s sensible o y o lea n abou he es ima o ’s ini e-sample dis ibu-
ion, and u ilize his in o ma ion o ob ain a be e es ima e. By choosing ou s a is ic
as Zn=ˆ
θ, he MIL es ima o is an upda ed e sion o he ini ial es ima o ha akes
in o accoun he ini e-sample cha ac e is ics o ˆ
θ. In pa icula , we expec ha he MIL
es ima o au oma ically adjus s o po en ial biases in he ini ial es ima o . As such i is
simila o he II bias adjus men mechanism epo ed in Gou ié oux, Renaul and Touzi
(2000) and Gou ié oux, Phillips and Yu (2010). Howe e , since MIL es ima o a he same
ime akes in o accoun ea u es o he dis ibu ion o ˆ
θbeyond i s i s momen , i should
be expec ed ha i will in gene al domina e he II es ima o .
As a second example, suppose Znhas been chosen as a se o sample momen s; his
is o example he case wi h simula ed me hod o momen s. These a e mean-unbiased
es ima o s o he co esponding popula ion means and so he e is no need o bias ad-
jus men . As such i would seem ha a ( wo-s ep) GMM es ima o based on Znwould
su ice. Howe e , he s a is ic may in ini e samples s ill be non-No mally dis ibu ed
and aking in o accoun hese ea u es will imp o e he es ima o . Fu he mo e, in he
o e iden i ied case whe e d>k, he e icien GMM equi es ei he knowledge o a p e-
limina y es ima o o he e icien weigh ing ma ix. In con as , he MIL au oma ically
inco po a es in o ma ion abou he e icien weigh and as such is simila o he (gene al-
ized) empi ical likelihood (GEL) es ima o in ha i u ilizes he ull dis ibu ional cha ac-
e is ics o he chosen s a is ic in he es ima ion o he pa ame e s. As a consequence, he
MIL es ima o will sha e he highe -o de op imali y p ope ies o he GEL (see Newey
and Smi h, 2004) and domina e he co esponding GMM es ima o .
In ce ain si ua ions, he op imiza ion p oblem de ining he MIL es ima o may be di -
icul o sol e nume ically. The likelihood unc ion θ7→ n(Zn|θ)may be non con ex,

INDIRECT LIKELIHOOD INFERENCE 6
ha e mul iple local maxima, la spo s, o discon inui ies, in which case he global max-
imize , ˆ
θMIL, can be di icul o compu e in p ac ice. These ea u es may be e en mo e
p onounced when he es ima ion is based on a simula ed e sion o he likelihood unc-
ion. This is pa icula ly an issue when he pa ame e space Θis “la ge” since he sea ch
has o be done o e a la ge-dimensional space. To ci cum en hese po en ial p oblems
in he compu a ion o ˆ
θMIL, we in oduce a Bayesian e sion o i as a compu a ionally
a ac i e al e na i e, since i does no equi e nume ical op imiza ion. In he simula ion
s udies, we ocus on he pos e io mean o θgi en Znde ined as
(2) ˆ
θBIL =ZΘθ n(θ|Zn)dθ,
whe e n(θ|Zn)is he pos e io dis ibu ion gi en by
n(θ|Zn):= n(Zn,θ)
n(Zn)= n(Zn|θ)π(θ)
RΘ n(Zn|θ)π(θ)dθ
o some densi y π(θ)on he pa ame e space Θ. We e e o his pa icula es ima o as
he Bayesian indi ec likelihood (BIL) es ima o . Mo e gene ally, θ0could be es ima ed
by:
(3) ˆ
θBIL =a g in
ζ∈ΘZΘρ√n(θ−ζ) n(θ|Zn)dθ,
o some penal y o loss unc ion ρ(u). This includes he pos e io mean which is ob-
ained by speci ying a quad a ic loss ρ(u)=|u|2, while he τ h quan ile o he pos e io
ollows om choosing he penal y unc ion as he so-called “check” unc ion, ρ(u)=
∑k
i=1(τi−1{ui≤0}), whe e 1{•}deno es he indica o unc ion. The pos e io quan iles
can be used o cons uc asymp o ically alid con idence in e als as shown in he nex
sec ion.
I should be s essed ha we do no gi e he BIL es ima o a Bayesian in e p e a ion
and me ely see i as a compu a ional de ice o ci cum en he nume ical issues ela ed o
he maximiza ion p oblem ha has o be sol ed in o de o compu e ˆ
θMIL. In pa icula ,
we do no in e p e π(θ)as a p io densi y in he Bayesian sense, in ha i does no nec-
essa ily e lec belie s abou he pa ame e . I is simply used o gi e weigh s o di e en
pa s o he pa ame e space, and in ou examples, we alway use a uni o m densi y. As
such, ˆ
θBIL is close in spi i o he class o Laplace ype es ima o s (LTE’s) in oduced in
Che nozhuko and Hong (2003).
3. COMPUTATION OF FEASIBLE ESTIMATORS
In mos si ua ions, i will no be possible o de i e he exac ini e-sample dis ibu ion
o he s a is ic Znon closed o m. Thus he likelihood n(Zn|θ)will no mally no be a ail-
able, and one has o eso o nume ical app oxima ions ins ead. In he compu a ion o
he BIL es ima o , i is in addi ion equi ed o compu e he in eg al RΘρn(θ−ζ) n(θ|Zn)dθ.
Fo he la e p oblem, one could ollow he sugges ions o Che nozhuko and Hong
(2003) and compu e he in eg al using Ma ko chain Mon e Ca lo (MCMC) me hods.
INDIRECT LIKELIHOOD INFERENCE 7
Howe e , we he e op o an al e na i e solu ion which handles he nume ical app oxi-
ma ion o n(Zn|θ)and he in eg al in one s ep; he p oposed me hod which we desc ibe
below is easy o implemen and in gene al qui e obus .
Fi s , o he implemen a ion o he MIL, we ha e o be able o compu e n(Zn|θ)a any
gi en ial alue θ. Since he model is simulable and he mapping Zn(θ)≡Zn(Yn(θ))
is known (as chosen by he econome ician), we p opose o es ima e he densi y using
ke nel densi y me hods: D aw Sindependen samples, Ys
n(θ) o s=1, ..., S, om he
model e alua ed a he ial alue θ, compu e he associa ed s a is ic, Zs
n(θ)≡Zn(Ys
n(θ)),
s=1, ...., S, and hen es ima e he densi y by ke nel me hods (see e.g. Li and Racine, 2007,
Ch. 1 o an in oduc ion):
(4) ˆ
n,S(Zn|θ) =
S
∑
s=1
Kh(Zs
n(θ)−Zn),
whe e Kh(z)=K(z/h)/h,K(z)is a ke nel unc ion and h>0 is a bandwid h. One
hen embeds he app oxima ed densi y inside (1), and uses an op imiza ion algo i hm o
ob ain an es ima o . This yields a simula ed MIL (SMIL) es ima o :
(5) ˆ
θSMIL =a gsup
θ∈Θ
log ˆ
n,S(Zn|θ).
The simula ed e sion is akin o he nonpa ame ic simula ed maximum-likelihood es i-
ma o (NPSMLE) o Fe manian and Salanié (2004) and K is ensen and Shin (2008). The
abo e ke nel densi y es ima o implici ly assumes ha Zn(θ)has a con inuous dis ibu-
ion. Howe e , we show ha e en i his is no he case, he simula ed e sion will s ill
asymp o ically beha e as he MIL es ima o .
Fo he compu a ion o he BIL es ima o , we no only need o e alua e he likelihood
bu also he in eg al o e he quasi-pos e io densi y. Che nozhuko and Hong (2003)
p opose o handle he la e compu a ional p oblem h ough MCMC, bu his can be
qui e a delica e me hod which in some cases has uns able p ope ies (see Ko mil sina
and Nekipelo , 2009). Ins ead, we op o also combine simula ions and nonpa ame ic
echniques in he implemen a ion o he BIL es ima o . Suppose, o illus a e, ha he
penal y unc ion is ρ(u) = |u|2. In his case, he Laplace- ype es ima o is he mean o he
pos e io densi y,
ˆ
θBIL =ZΘθ n(θ|Zn)dθ=E[θ|Zn].
Ou idea is hen o compu e ˆ
θBIL =E[θ|Zn]by combining nonpa ame ic eg ession
me hods and simula ions as ollows: Make i.i.d. d aws θs,s=1, ..., S, om he pseudo-
p io densi y π(θ), o each d aw gene a e a sample Yn(θs) om he model a his pa-
ame e alue, and hen compu e he co esponding s a is ic Zs
n=Z(Yn(θs)),s=1, ..., S.
Gi en he i.i.d. d aws (θs,Zs
n),s=1,...S, we can ob ain a simula ed e sion o he BIL
(SBIL) h ough nonpa ame ic eg ession echniques. One such is he ke nel es ima o
(see Li and Racine, 2007, Ch. 2),
(6) ˆ
θSBIL =∑S
s=1θsKh(Zs
n−Zn)
∑S
s=1Kh(Zs
n−Zn),
INDIRECT LIKELIHOOD INFERENCE 8
while ano he one is he k-nea es neighbo (KNN) es ima o (see Li and Racine, 2007, Ch.
14), whe e he bandwid h is chosen as h=dk(Zn)wi h dk(Zn)deno ing he Euclidean
dis ance be ween Znand he k- h nea es neighbo among he simula ed alues. As such
he KNN es ima o can be hough o as a ke nel eg ession es ima o wi h an adap i e
bandwid h.
A membe in he gene al class o BIL es ima o s gi en in eq. (3) can be exp essed as
minimizing a condi ional momen ,
ˆ
θBIL =a g in
ζ∈ΘEρ√n(θ−ζ)|Zn,
which can be app oxima ed by eplacing he exac momen by a simula ed nonpa ame ic
e sion,
ˆ
θSBIL =a g in
ζ∈Θ
ˆ
ESρ√n(θ−ζ)|Zn,
whe e ˆ
ESρ√n(θ−ζ)|Zn, o example, can be compu ed by ke nel eg ession,
(7) ˆ
ESρ√n(θ−ζ)|Zn=∑S
s=1ρ√n(θ−ζ)Kh(Zs
n−Zn)
∑S
s=1Kh(Zs
n−Zn),
o nea es neighbo es ima ion whe e again h=dk(Zn). When he penal y unc ion is
chosen as he “check”- unc ion, his leads o simula ed e sions o he pos e io quan iles,
which a e used o compu e con idence in e als. In his case, he ke nel smoo hed e sion
becomes he ke nel quan ile eg ession es ima o (Li and Racine, 2007, Sec. 6.4).
Fo bo h he SMIL and SBIL, he e a e wo sou ces o e o in compa ison wi h he
exac MIL and BIL es ima o s (which only su e om he sampling e o in Zn). Fi s ,
andomness is added due o he use o simula ions, and he e is also a bias componen
due o he use o nonpa ame ic es ima o s. We ea he nonpa ame ic i ing s ep as a
compu a ional ool used o ind he alue o he es ima o , in he same way ha Che -
nozhuko and Hong (2003) ea MCMC as a means o compu ing LTEs. As he numbe
o simula ed d aws Sbecomes la ge, nonpa ame ic densi y and eg ession es ima o s
a e consis en . Thus, bo h he andomness due o use o simula ions and he bias due o
use o nonpa ame ic me hods can be con olled o by choosing Ssu icien ly la ge. We
analyze he impac o simula ions and ke nel smoo hing in Sec ion 6.
One may wish o explo e di e en pseudo-p io s. I a la ge body o simula ions ha e
been gene a ed using he pseudo-p io π(θ), hen one can ob ain esul s o a di e -
en pseudo-p io wi hou doing addi ional simula ions by using impo ance sampling.
The SBIL es ima o based on π(θ) p esen ed in equa ion 6can be w i en as ˆ
θSBIL =
∑S
s=1θswS(Zs
n,Zn), whe e wS(Zs
n,Zn)has an ob ious de ini ion. Gi en simula ions {(θs,Zs
n)}S
s=1
based on π(θ), he SBIL es ima o co esponding o he new pseudo-p io , say π∗(θ), can
be compu ed by
ˆ
θSBIL =
S
∑
s=1
θswS(Zs
n,Zn)π∗(θs)
π(θs).
This may be use ul when Sis e y la ge o when i is cos ly o compu e he auxilia y
s a is ic, as in he case o he DSGE models p esen ed la e in his pape .
INDIRECT LIKELIHOOD INFERENCE 9
In he ABC li e a u e o likelihood- ee li e a u e, discussed in he in oduc ion, me h-
ods o compu ing es ima o s using likelihood- ee Ma ko chain Mon e Ca lo and se-
quen ial Mon e Ca lo ha e been s udied in some de ail (Ma jo am e al., 2003; Sisson, Fan
and Tanaka, 2007; Beaumon e al. 2009). One could also employ so-called impo ance
sampling o educe a iances due o simula ions: Fo any condi ional densi y gn(θ|z)
wi h suppo Θ, we can ew i e ˆ
θBIL as ˆ
θBIL =Rθ{ n(θ|Zn)/gn(θ|Zn)}gn(θ|Zn)dθ,
and so a gene alized e sion o ou p oposed simula ed e sion would be
ˆ
θBIL =1
S
S
∑
s=1
θsˆ
n,S(θs|Zn)
gn(θs|Zn)=∑S
s=1θsπ(θs)/gn(θs|Zn)Kh(Zs
n−Zn)
∑S
s=1Kh(Zs
n−Zn),
whe e θs∼i.i.d.gn(θs|Zn). The op imal choice o gn(θ|z)in e ms o a iance educ ion
is
gn(θ|z)=1{θ∈Θ}|θ| n(θ|z)
RΘ|θ| n(θ|z)dθ.
Un o una ely, i is no easible o d aw om his choice since RΘ|θ| n(θ|z)dθis un-
known, bu app oxima e me hods exis ; see, o example, Zhang (1996). These me hods
should in p inciple be compu a ionally mo e e icien compa ed o he basic sampling
me hod in equa ion (6) o ob ain a gi en le el o p ecision. In ou simula ion s udy we
ocus on he basic sample , and lea e he implemen a ion o impo ance sample s o u-
u e esea ch.Applica ion o hese me hods could p o ide sa ings in compu a ional ime
when i is cos ly o sample om he model, bu on he o he hand equi e mo e ca e-
ul implemen a ion. O he examples conside ed in his pape , only he DSGE models
(below) p esen se ious compu a ional bu den.
4. FIRST-ORDER ASYMPTOTICS
As a i s s ep owa ds a comple e asymp o ic analysis o he MIL and BIL es ima o s,
we he e de i e hei i s -o de asymp o ic dis ibu ion. The asymp o ic analysis o he
MIL es ima o p oceeds along he s anda d s eps o pa ame ic ex emum es ima o s,
while he BIL es ima o on he o he hand equi es a bi mo e ca e. Fo una ely, since
he BIL es ima o can be ega ded as a speci ic LTE, we can employ he gene al esul s o
Che nozhuko and Hong (2003) o es ablish √n-consis ency and asymp o ic no mali y
o ou Bayesian es ima o , as well as he equi alence wi h he MIL es ima o when he
penal y unc ion ρis symme ic.
We impose he ollowing condi ions on he pa ame e space and he weigh ing unc-
ion
Assump ion 1. Assume ha : (i) he pa ame e space Θ⊂Rkis compac wi h θ0being an
in e io poin ; (ii) he weigh ing unc ion π(θ)is a con inuous, uni o mly posi i e densi y; and
(iii) he penal y unc ion is con ex and sa is ies ρ(u)=0⇔u=0, ρ(u)≤1+|u|p o some
p≥1, and φ(x)=Rρ(u−x)eu0audu is uniquely minimized a some x∗ o any a >0.
This se o assump ions is comple ely s anda d, and a e iden ical o he condi ions
ound in Che nozhuko and Hong (2003). I should be no ed ha (ii)-(iii) a e only needed
o de elop heo y o he BIL es ima o , and he asymp o ics o he MIL es ima o only
equi e (i).
INDIRECT LIKELIHOOD INFERENCE 16
we demons a e ha he bias p ope ies o he MIL a e e y a o able and a e as good as
he ones o he CU GMM- ype es ima o .
We now u n ou a en ion o he highe -o de e iciency o he GMM and IL es ima-
o s. As was shown in he p e ious sec ion, hei i s -o de asymp o ic a iances a e
iden ical. Howe e , in ini e samples, he IL es ima o s a e expec ed o domina e o a
numbe o easons: Fi s , he GMM es ima o s a e only i s -o de equi alen o he IL
es ima o s i Wn=Ω−1(θ0)+op(1). I no , he IL es ima o s a e asymp o ically mo e
e icien han GMM. Mo eo e , he i s -s ep es ima ion e o con ained in Wnin gen-
e al has an ad e se impac on he pe o mance o he esul ing wo-s ep es ima o which
may pe o m poo ly in small and mode a e samples; see e.g. Al onji and Segal (1996),
Hansen, Hea on and Ya on (1996) and Newey and Smi h (2004). Fu he mo e, while in-
c easing he dimension o he auxilia y s a is ic inc eases he asymp o ic e iciency o he
GMM es ima o , i also inc eases he dimension o he weigh ma ix o be es ima ed and
nume ical singula i ies can appea making he in e sion di icul . In con as , ou es i-
ma o s do no equi e es ima ion o he op imal weigh ing ma ix, and so inc easing he
dimension o he auxilia y s a is ic causes no di icul ies wi h singula ma ices.
I on he o he hand Ω(θ)is known, hen we can es ima e he pa ame e s using he CU
es ima o which will emo e he addi ional es ima ion e o s due o he use o Wn; see
Donald and Newey (2000) and Newey and Smi h (2004). Howe e , in ini e samples, he
CUE s ill only u ilizes in o ma ion con ained in he i s and second momen s o Zn, while
he MIL akes in o accoun all dis ibu ional cha ac e is ics. This di e ence means ha
he indi ec likelihood es ima o s in gene al will ha e be e small sample pe o mance
han bo h wo-s ep e icien GMM and CU based on he same auxilia y s a is ic.
The o mal p oo o highe -o de e iciency can be done by anking he GMM- ype
and MIL es ima o s in e ms o hei highe -o de MSE. I an es ima o ˆ
θsa is ies he
expansion in eq. (14), we ob ain (again igno ing Rn)
MSE √n(ˆ
θ−θ0)≃BnB0
n+Vn,
whe e Bn=√nBias ˆ
θand Vn=nVa ˆ
θ. Fo each o he h ee es ima o s, he a iance
can be decomposed in o
Vn=J−1+Ξ/n+o(1/n),
whe e J−1=J−1(θ0)is he leading a iance componen , while Ξis he highe -o de a i-
ance. The exp ession o Ξ o each o he h ee es ima o s (CUE, GMM, MIL) is s aigh -
o wa d o ob ain om he expansion, bu i is a he complica ed. This makes a di ec
anking o he es ima o s in e ms o hei espec i e Ξ’s di icul .
Ins ead, we i s de elop an Edgewo h expansion o he dis ibu ion o he MIL es i-
ma o . Fo s anda d maximum-likelihood es ima o s whe e he log-likelihood akes he
o m o a sample a e age o e i.i.d. obse a ions, Edgewo h expansions ha e been es-
ablished; see, o example, Bha acha ya and Ghosh (1978). Howe e , we can in gene al
no w i e log n(Zn|θ)as a sample a e age o i.i.d. a iables and so he s anda d p oo
does no di ec ly ca y o e o ou se ing. Howe e , by impo ing some o he a gumen s
o Bha acha ya and Ghosh (1978), we can s ill show ha ˆ
θMIL ≃H(Wn(Zn)) o some
analy ic unc ion Hand wi h Wn(Zn)deno ing he i s de i a i es o log n(Zn|θ)w. . .

INDIRECT LIKELIHOOD INFERENCE 17
θ. Since ( he no malized e sion o ) Znsa is ies an Edgewo h expansion, we can hen
apply he gene al esul s o Phillips (1977) on Edgewo h expansions o ans o ma ions
o andom sequences o ob ain he desi ed esul :
P oposi ion 5. Unde Assump ions 1-4wi h E |Zn|p<∞ o all n,p≥1, and
P|Zn−Z(θ0)|>c1qlog (n)/n=on− /2,
he MIL sa is ies an h o de Edgewo h expansion:
sup
yP√nˆ
θMIL −θ0J(θ0)≤y−
y
Z
−∞
φ(x)"1+
∑
i=1
n−i/2 ˜
πi(x)#dx=on− /2,
whe e ˜
πi(x)is a polynomial o o de 3i, i =1, ..., .
The assump ion ha Znhas momen s o all o de s is somewha es ic i e and ules
ou hea y ails. We conjec u e ha his assump ion is no s ic ly necessa y o he abo e
esul o holds. In pa icula , one migh be able o show P oposi ion 5by using he esul s
o Sko gaa d (1981) whe e weake momen es ic ions a e equi ed. This would on he
o he hand complica e he p oo and so o cla i y we main ain he assump ion o all
momen s exis ing. The ail p obabili y condi ion is sa is ied o mos egula s a is ics;
see, e.g., Bha acha ya and Ghosh (1978, Theo em 3).
Once we ha e shown ha he dis ibu ion o he MIL es ima o can be app oxima ed
by an Edgewo h expansion, i now ollows by s anda d esul s o maximum-likelihood
es ima o s (see e.g. Ghosh, 1994 and Bickel, Gö ze, and an Zwe , 1985), ha he bias-
adjus ed MIL es ima o is hi d-o de e icien amongs all es ima o s elying on he
s a is ic Zn. In pa icula , ΞGMM ≥ΞMIL and ΞCUE ≥ΞMIL.
6. PROPERTIES OF SIMULATED VERSIONS
We analyze he impac o he use o simula ions and nonpa ame ic es ima ion in he
implemen a ion o he MIL and BIL es ima o s. Fo he simula ed e sion o he MIL
es ima o , we combine he gene al esul s o K is ensen (2009) and K is ensen and Shin
(2008) o show ha i is i s -o de asymp o ically equi alen o he in easible MIL es i-
ma o . The analysis o he simula ed e sion o he BIL es ima o can be done di ec ly
since we can w i e i up on closed o m.
To u ilize exis ing esul s on con e gence a es o ke nel es ima o s, we make he ol-
lowing assump ions ega ding he ke nel unc ion used in he compu a ion o he SMIL
and SBIL de ined in Sec ion 3:
Assump ion 5. The ke nel K sa is ies: The e exis C,L<∞such ha ei he (i) K(u) = 0 o
kuk>L and |K(u)−K(u´)| ≤ Cku−u0k,o (ii) K(u)is di e en iable wi h supu|K0(u)|<
∞. Fo some a >1,|K(u)| ≤ Ckuk−a o ||kuk>L„ and RK(z)dz =1,RzK (z)dz =0,
Rz2K(z)dz <∞.
The abo e assump ions imposed on he ke nel a e qui e s anda d and a e o example
sa is ied by he Gaussian ke nel. We i s es ic ou sel es o he case whe e he likeli-
hood is a densi y:
INDIRECT LIKELIHOOD INFERENCE 18
Assump ion 6. The indi ec likelihood n(z|θ)is a densi y wi h espec o he Lebesgue measu e
and is wice con inuously di e en iable in z.
Unde his assump ion on he ke nel and he likelihood, he ollowing esul holds:
P oposi ion 6. Assume ha Assump ions 1-6hold. Then he SMIL and SBIL es ima o s a e
asymp o ically i s -o de equi alen o he ac ual ones unde he ollowing condi ions:
Fo he ke nel-smoo hed e sions, nh2→0and nlog (S)/Shd→0.
Fo he nea es -neighbo e sions, n [k/S]2/d→0, and nlog (S)/k→0.
The es ic ions on Sand ha e ai ly s anda d and equi e he numbe o simula ions
o g ow a a sligh ly as e a e han he numbe o obse a ions. In pa icula , s anda d
bandwid h selec o s will sa is y he abo e a es and so hese can be used in he imple-
men a ion o he simula ed e sions.
We also no e ha he simula ed e sions o ou es ima o s su e om a cu se o di-
mensionali y. This appea s explici ly in he condi ions on Sand hgi en in P oposi ion 6
whe e we equi e nlog (S)/Shd→0. Thus, he la ge d=dim (Zn)(which mus be a
leas ha o θ, and which is la ge in mos o he applica ions below), he mo e simula-
ions a e equi ed o he simula ions o ha e a negligible impac on he es ima o . This
is a well-known issue which is sha ed by mos o he simula ion-based es ima o s: The
la ge he dimension o he space o e which we need o in eg a e, he la ge he numbe
o simula ions should be chosen o con ol he simula ion e o .
The abo e esul equi es he likelihood o be a densi y. We now demons a e ha he
SMIL and SBIL es ima o s enjoy he same asymp o ic p ope ies e en i his is no he
case. In ac , we will no e en equi e ha Assump ion 4holds and as such allow o
bo h con inuous and disc e e obse a ions. To be mo e speci ic, we eplace Assump ions
4and 6wi h he ollowing one:
Assump ion 7. Fo some N ≥1: supn≥NEhsupθ∈Θ√n(Zn(θ)−Z(θ))2i<∞.
This uni o m in eg abili y assump ion is sa is ied i , o example, Zn(θ)is a sam-
ple a e age wi h second momen . I imposes no smoo hness es ic ions on he ini e-
sample likelihood and holds o bo h con inuous and disc e e unde lying da a. I is used
in conjunc ion wi h Assump ion 2 o ensu e ha √nE [|Zn(θ)−Z∗
n(θ)|]→0, whe e
Z∗
n(θ)∼N(Z(θ),Ω(θ)/n)is i s No mal limi sequence. We use his o show ha he
ke nel smoo he based on simula ions om he dis ibu ion o Zn(θ)con e ges owa ds
he one based on simula ions o Z∗
n(θ). Since Z∗
n(θ)sa is ies Assump ion 4and 6by con-
s uc ion, his in u n implies ha SMIL and SBIL ha e he desi ed asymp o ic p ope ies:
P oposi ion 7. Assume ha Assump ions 1-3,5and 7hold, and he ke nel K is uni o mly
Lipsch iz, |K(u)−K( )|≤D|u− |. Then he SMIL and SBIL ha e he same asymp o ic
p ope ies as hose s a ed in P oposi ion 1unde he bandwid h condi ions s a ed in P oposi ion 6
oge he wi h nh2→∞(ke nel smoo he ) and n/k2→∞(nea es neighbo ).
The in ui ion behind he abo e esul is he ollowing: I he dis ibu ion o Zn(θ)can-
no be desc ibed by a densi y, one can hink o he ke nel smoo hing inhe en in bo h
he SMIL and SBIL as a ype o egula iza ion ha gene a es a smoo h objec i e unc ion
INDIRECT LIKELIHOOD INFERENCE 19
which can be used ins ead o he mo e i egula ly beha ed ue likelihood. As such he
SMIL and SBIL es ima o s a e simila in na u e o he smoo hed maximum sco e es ima-
o p oposed in Ho owi z (1992) whe e a non-smoo h es ima o is egula ized h ough
smoo hing.
In p ac ice, we choose he numbe o simula ions Sso la ge, ha he addi ional a i-
ance due o simula ions is negligible. Howe e , o comple eness, we no e ha he simu-
la ed e sion o he BIL es ima o sa is ies
ˆ
θSBIL =ˆ
θBIL +ES(Zn),
o a s ochas ic unc ion ES(z)which is independen o ˆ
θBIL and sa is ies ei he (in he
case o ke nel-smoo he s),
√ShdES(z)→dN0, kKk2σ2
n(z)
n(z),
o (in he case o nea es -neighbo es ima o s),
√kES(z)→dN0, kKk2σ2
n(z),
whe e d=dim (Zn),kKk2=RK2(z)dz, and σ2
n(z)=Va [θ|Zn=z]. Thus, he a i-
ance es ima o o he ke nel-smoo hed e sion o SBIL could be adjus ed by adding
kKk2σ2
n(z)
n(z)/Shd o J−1(θ0), and simila ly o he nea es -neighbo e sion. A simila
adjus men can be de eloped o he MIL es ima o by using he a gumen s o K is ensen
and Salanié (2010).
7. MONTE CARLO RESULTS
In his sec ion we explo e he pe o mance o he SMIL and SBIL es ima o s, compa ing
hem o o he es ima o s, using a a ie y o econome ic models including simple ime se-
ies models, a dynamic and nonlinea panel da a models, a s uc u al econome ic model
o an auc ion and wo dynamic s ochas ic gene al equilib ium (DSGE) models. We ocus
on se e al issues. Fi s , he SBIL es ima o is conside ably mo e con enien o use han
is he SMIL es ima o , om a compu a ional poin o iew, so we would like o know i
he wo es ima o s pe o m simila ly be o e ocusing ou a en ion on he SBIL es ima o .
Second, P oposi ion 5 ells us ha he exac MIL is highe -o de mo e e icien han he
GMM es ima o ha uses he op imal weigh ma ix. This leads us o hope ha he SMIL
and SBIL es ima o s ha e be e small sample pe o mance han GMM- ype compe i o s.
A ac o ha could unde mine hese po en ial gains is he need o use simula ions and
nonpa ame ic i ing o implemen he easible e sions ( he easible SMIL and SBIL e -
sus he in easible MIL and BIL). This sec ion h ows ligh on he ac ual pe o mance o
he easible e sions. A hi d pu pose o his sec ion is simply o gi e examples o how
he SMIL and SBIL es ima o s may be implemen ed in p ac ice. Examples o p ac ical
issues o deal wi h a e he choice o he auxilia y s a is ic, and he speci ica ion o he
pa ame e space in he case o he SBIL es ima o .
A ou h issue is he accu acy o con idence in e als compu ed using es ima ed quan-
iles o he pseudo-pos e io . We ind mixed esul s o con idence in e al co e age: in
INDIRECT LIKELIHOOD INFERENCE 20
some cases co e age is e y accu a e, while in o he s he con idence in e als a e oo
b oad, so ue size is smalle han he nominal size. Because he indings a e mixed, we
do no p esen abula esul s, and we lea e his issue o u u e esea ch. I is pe ec ly
easible o use o he means (asymp o ic, boo s ap, Mon e Ca lo) o compu ing con i-
dence in e als and s anda d e o s o he SBIL es ima o . Fo example, Li (2010) ound
ha boo s ap con idence in e als a e e y accu a e o he II es ima o o he s uc u al
auc ion model discussed below. The same me hod could be used o he SBIL es ima o .
We do no pu sue he issue u he in his pape .
To implemen he SMIL and SBIL es ima o s, we use be ween S=106and S=107
simula ed poin s d awn andomly om he pa ame e space, depending on he appli-
ca ion. The auxilia y s a is ics we use a e in mos cases compu a ionally inexpensi e,
so gene a ing a la ge numbe o eplica ions is no bu densome. The excep ions a e he
DSGE models, which equi es app oxima ely wo days o ime on a 32 co e clus e pe
106 eplica ions o he auxilia y s a is ic3. Fo all p oblems, we use a leas 5000 Mon e
Ca lo eplica ions a each design poin . The nonpa ame ic i is done using he knea -
es neighbo s app oach4, using he ANN lib a y (A ya, Malama os and Moun , 2009;
h p://www.cs.umd.edu/~moun /ANN/). Using his C++ lib a y, he KNN nonpa a-
me ic i ing s ep equi es a mos se e al minu es o ime on a single co e. I is also a
simple ma e o swi ch o using app oxima e nea es neighbo s, which can speed up he
nonpa ame ic i ing s ep i one uses an ex emely la ge numbe o simula ions. The
numbe o neighbo s kused o he nonpa ame ic i is chosen (wi h one excep ion) as
k=1.5 ×S0.25, ounded down o he nea es in ege . Mo e ca e ul choice using me hods
such as c oss alida ion migh imp o e he esul s, bu we do no explo e his possibili y
in his pape . We epo he SBIL es ima o compu ed as he pos e io mean. The e -
sion compu ed as he pos e io median gi es e y simila esul s. Fo all applica ions he
pseudo-p io π(θ)is a uni o m dis ibu ion o e he pa ame e space Θ, so he only e-
maining issue is speci ying he bounds o pa ame e space. Fo some o he applica ions
( he MA and dynamic panel da a models), p io belie s such as s a iona i y o in e ibil-
i y lead di ec ly o he speci ica ion o a leas some o he bounds o pa ame e space.
Fo o he s ( he auc ion model and he DSGE models) we ha e less in o ma ion a ail-
able ega ding plausible bounds on a leas some o he pa ame e s. The issue o se ing
he pa ame e space in such cases is add essed in he subsec ion p esen ing he auc ion
model.
7.1. Dynamic panel da a. Gou ié oux, Phillips and Yu (2010; hence o h GPY) in es i-
ga e he pe o mance o he II es ima o using a linea dynamic panel model
(17) yi =αi+φ0yi −1+ei
3Pe o ming Mon e Ca lo on a clus e is qui e s aigh o wa d. We use PelicanHPC (h p://pelicanhpc.
o g/), a amewo k e y simila o ha desc ibed in C eel (2007).
4We also ha e used ke nel eg ession, which gi es e y simila esul s o he KNN esul s epo ed he e.
INDIRECT LIKELIHOOD INFERENCE 21
whe e ei ∼N(0,1),αi∼N(0,1),φ0=0, 0.3, 0.6, 0.9 and αiand eia e independen ly
dis ibu ed. The ini ial condi ion is
yi0|αi∼Nαi
1−φ0,1
1−φ2
0.
GPY use he (inconsis en ) ML “ ixed e ec s” es ima o as he auxilia y s a is ic. They
ind ha he II es ima o ou pe o ms a numbe o al e na i e es ima o s, in e ms o oo
mean squa ed e o (RMSE). While ou asymp o ic esul s do no s aigh o wa dly gen-
e alize o dynamic panel da a models (whe e he heo y no mally equi es he numbe o
ime pe iods, T, o g ow wi h sample size), we conjec u e ha he highe -o de e iciency
esul s also hold in his con ex . We he e in es iga e his claim by compa ing he pe o -
mance o he SMIL and SBIL es ima o s o he II esul s ob ained by GPY. GPY also epo
esul s o o he bias co ec ion me hods such as jackkni e and analy ical bias co ec ion
and ind ha hei II es ima o domina es hose; we he e o e ocus on he II es ima o
and do no ep oduce he esul s o he o he es ima o s. The pa ame e space is se o
he s a iona y egion φ0∈(−1, 1). We conside wo auxilia y s a is ics: he same ML
es ima o as used by GPY, and also he ML es ima o augmen ed wi h he OLS es ima o
o he nai e model yi =δyi −1+νi ha igno es he p esence o indi idual e ec s.
The SMIL es ima o equi es a nonpa ame ic densi y i embedded inside an op i-
miza ion p oblem, while he SBIL es ima o elimina es he op imiza ion. In he p esen
case, he pa ame e o es ima e is a scala , so o his p oblem i is ela i ely easy o apply
bo h he SMIL and SBIL es ima o s. By compa ing he wo in his ela i ely simple case,
we can ge an indica ion o whe he ocusing on he SBIL es ima o in mo e compu a ion-
ally demanding cases is wa an ed by a compa able pe o mance o he wo es ima o s.
To implemen he SMIL, we use a di e en app oach han wha is ou lined in equa ions
(4) and (5). The eason o his o ake ad an age o he la ge se o eplica ions o (θs,Zs
n)
ha a e al eady a ailable a e compu ing he SBIL es ima o . Ins ead o ope a ing on
he condi ional densi y n(Zn|θ), we wo k wi h he join densi y n(Zn,θ). When θsis
d awn om a uni o m densi y, as is he case he e, n(Zn,θ)and n(Zn|θ)a e maximized
a he same alue o θ, because he ma ginal densi y o θdoes no depend upon θ. We o
cou se do no know he join densi y, so i mus be i nonpa ame ically. We use he sim-
ple KNN densi y es ima o gi en in equa ion 14.2 o Li and Racine (2007) o i n(Zn,θ).
This nonpa ame ic i o he join densi y, ˆ
n(Zn,θ)is hen maximized wi h espec o θ
using a g id sea ch, in o de o deal wi h he ough, nondi e en iable na u e o he KNN
densi y es ima o . Because θis a scala in he p esen case, use o g id sea ch does no
p esen a signi ican compu a ional bu den.
Table 1p esen s he bias o he es ima o s, and Table 2p esen s he oo mean squa ed
e o s (RMSEs). In hese Tables, he columns labeled II, SBIL and SMIL all e e o use
o he auxilia y s a is ic Zn=b
φML, while he columns labeled SBIL(OI) and SMIL(OI)
e e o use o he o e iden i ying auxilia y s a is ic Zn=b
φML,b
δOLS. Resul s o he
inconsis en ML es ima o a e also p esen ed, o e e ence. We see ha he II and SBIL
es ima o s ha e e y small biases in almos all cases. Wi h an exac ly iden i ying auxil-
ia y s a is ic, he es ima o s (excep ML) all ha e simila biases and RMSEs, especially o

INDIRECT LIKELIHOOD INFERENCE 22
la ge sample sizes. Fo small sample sizes, he SBIL es ima o pe o ms somewha be -
e han he II es ima o , o e all. When he di e ence a o s he II es ima o , i is small,
bu when i a o s he SBIL es ima o , i is la ge . Fo he SMIL and SBIL es ima o s, i is
easy o use an o e iden i ying auxilia y s a is ic, because no co a iance ma ix need be
es ima ed. Looking a he columns labeled SMIL(OI) and SBIL(OI), we see ha he e a e
gains om doing so: bias is essen ially unchanged, bu RMSE is educed conside ably,
especially o smalle sample sizes. The e seems o be no eason o p e e SMIL o SBIL,
as he RMSEs o he wo a e essen ially he same in he case o he exac ly iden i ying aux-
ilia y s a is ic, while SBIL almos uni o mly domina es SMIL when he o e iden i ying
auxilia y s a is ic is used.
Based on he good pe o mance o SBIL compa ed o SMIL in his example, and he
ac ha he wo es ima o s a e i s o de equi alen , we ocus on SBIL in he emaining
examples. Mos o he emaining examples ha e pa ame e ec o s o highe dimension,
which would make a global maximiza ion s a egy such as g id sea ch o simula ed an-
nealing mo e edious o employ ( ecall ha a nonpa ame ic densi y i mus be done o
each ial pa ame e alue). The SBIL es ima o does no equi e his op imiza ion s ep,
so i a oids his di icul y.
7.2. Mo ing a e age. The p e ious sec ion compa ed he p oposed es ima o s o a jus
iden i ied II es ima o . I is also desi able o compa e o an o e iden i ied II es ima o ,
because his is he si ua ion whe e i is necessa y o es ima e he e icien weigh ma ix
in o de o ob ain an e icien II es ima o , gi en he chosen auxilia y s a is ic. We would
like o see i he SBIL es ima o bene i s om he ac ha i does no equi e es ima ion
o he e icien weigh ma ix. The i s o de mo ing a e age (MA(1)) model has been
widely used o in es iga e he pe o mance o he indi ec in e ence es ima o , and a
p h-o de au o eg essi e model is o en used o gene a e he auxilia y s a is ic (see, o
example, Gou ié oux,Mon o and Renaul , 1993; Chumace o, 2001). In his sec ion we
es ima e he MA(1) model
y =e +ψe −1
e ∼i.i.d.N(0, σ2)
using sample sizes o n=50, 100 and 200 obse a ions. The pa ame e ψis one o he al-
ues {−0.95, −0.9, −0.5, 0, 0.5, 0.9, 0.95}, so he model is always in e ible. The pa ame-
e σis always equal o 1. The pa ame e ec o is θ= (ψ,σ). We se he pa ame e space
o Θ=(−1,1)×(0, 2), which imposes in e ibili y, which is needed o he pa ame e
o be iden i ied. The s a is ic Znis he ec o o es ima ed pa ame e s ρ0,ρ1, ..., ρP,σ2
υo
an AR(P) model y =ρ0+∑P
p=1ρpy −p+υ , i o he da a using o dina y leas squa es.
Fo simplici y, we hold he o de o he AR(P)model cons an a P=10 ac oss he Mon e
Ca lo eplica ions. Thus, he dimension o Znis 12, while he dimension o θis 2, so we
ha e conside able o e iden i ica ion.
We es ima e θusing SBIL and II, whe e bo h a e based on he auxilia y s a is ic de-
ined in he las pa ag aph. The II es ima o is compu ed using con inuously upda ed
INDIRECT LIKELIHOOD INFERENCE 23
GMM (Hanson, Hea on and Ya on, 1996). The momen condi ions ha de ine he con-
inuously upda ed indi ec in e ence (CU-II) es ima o a e mn(θ) = Zn−¯
ZS,n(θ)whe e
¯
ZS,n(θ) = 1
S∑S
s=1Zs
n(θ), and he weigh ma ix a each i e a ion is he in e se o ΩS
n(θ) =
1
S∑S
s=1[Zs
n(θ)−¯
ZS,n(θ)] [Zs
n(θ)−¯
ZS,n(θ)]0, whe e S=100. Fo e e ence, we also es i-
ma e θusing he condi ional maximum likelihood es ima o (Gaussian MLE wi h e0se
o ze o). When a eplica ion o he ML o CU-II es ima o lies in he non-in e ible pa o
he pa ame e space, we use he obse a ionally equi alen in e ible pa ame e alue in
i s place. The need o doing his and he means o doing so a e explained by Chumace o
(2001).
Table 3 epo s he esul s. In his Table, SBIL(AR) e e s o he SBIL es ima o ha
uses he AR(10) auxilia y s a is ic, while SBIL(ML) is he SBIL es ima o ha uses he ML
es ima o as he auxilia y s a is ic. We see ha he SBIL(AR) and CU-II es ima o s ha e
biases ha a e o compa able magni udes, o e all. Compa ing RMSEs, he SBIL(AR)
es ima o pe o ms be e han he CU-II es ima o , almos uni o mly. This esul is no
unexpec ed, gi en he p e ious heo e ical esul s o highe o de e iciency o MIL com-
pa ed o CU-II. These heo e ical g ounds o e iciency plus he a oidance o es ima ion
o he weigh ma ix appea o lead o eal small sample e iciency gains. Compa ing o
he ML es ima o , o he smalle sample size, SBIL(AR) has a la ge RMSE han does
ML, which is no doub an indica ion ha an AR(10) auxilia y model is excessi ely pa-
ame e ized when he sample size is only 50. When he sample size is 200, he SBIL(AR)
es ima o has bias and RMSE compa able o hose o he ML es ima o . When he ML
es ima o is used as he auxilia y s a is ic o SBIL, he e is no bene i in e ms o RMSE
when he sample size is 50, bu o samples o size 100 and 200, he SBIL(ML) es ima o
has an RMSE lowe han ha o he ML es ima o .
7.3. Nonlinea panel model. Sec ion 7.1 explo es a linea panel da a model wi h no -
mally dis ibu ed e o s. One migh expec ha a nonlinea model could lead o a la ge
di e ence be ween he SBIL and II es ima o s, especially o smalle sample sizes, as in
such a case he small sample dis ibu ion o he auxilia y s a is ic, which cha ac e izes
he objec i e unc ions o he IL es ima o s, could be less well app oxima ed by he co e-
sponding no mal limi ing dis ibu ion, which cha ac e izes he objec i e unc ion o he
II es ima o . To in es iga e his conjec u e, we use he s a ic logi panel model ha A el-
lano and Bonhomme (2009) used in some o hei Mon e Ca lo wo k o compa e a se o
semi-pa ame ic nonlinea panel da a es ima o s. Thei s a ic logi Mon e Ca lo design
(see hei Sec ion 7.1) is used he e o compa e he SBIL and CU-II es ima o s. The design
o he expe imen is
yi =1[xi φ0+αi0+ei >0]
whe e xi ∼N(0,1)and he indi idual e ec s αi0∼N(¯
xi,1), whe e ¯
xi=1
T∑T
=1xi .
The ei a e independen d aws om he logis ic CDF. The ue alue o φ0=1. We se
N∈{30,100}and T=5. The i s componen o he auxilia y s a is ic is he es ima o o
he misspeci ied logi model ha esul s om he abo e model, wi h he excep ion ha ,
e oneously, i is assumed ha he indi idual e ec s a e all iden ical. To be p ecise, i is
he quasi-ML es ima o esul ing om logi es ima ion o he misspeci ied model yi =
INDIRECT LIKELIHOOD INFERENCE 24
1[α+xi φ+ei >0]. The second componen o he auxilia y s a is ic is he OLS es ima o
o he linea p obabili y model yi =α+xi φ+ηi . The logi and OLS es ima o s o αand
φ oge he yield an auxilia y s a is ic o dimension 4, so we ha e o e iden i ica ion o
he es ima o o he scala φ0. We use SBIL and CU-II o es ima e φ0, using his auxilia y
s a is ic. SBIL uses 2 ×106simula ions, and CU-II was implemen ed as desc ibed in he
p e ious sec ion. Fo bo h SBIL and CU-II, he pa ame e space o φis se o [0,2]and
he pseudo p io o SBIL is a uni o m dis ibu ion o e he pa ame e space.
Table 4p esen s he esul s o bias, RMSE and mean absolu e e o (MAE). Fo bo h
sample sizes, he SBIL es ima o is less biased and has smalle RMSE and MAE han he II
es ima o . Fo he smalle sample size, he RMSE o he SBIL es ima o is 88.4% ha o he
CU-II es ima o , while o he la ge sample size he pe cen age is 91.6%. This esul sup-
po s he conjec u e ha he SBIL es ima o will ha e be e small sample pe o mance
han ha o GMM- ype es ima o s based on he same auxilia y s a is ic. Compa ing hese
esul s o hose o he linea dynamic panel da a model, i seems ha he nonlinea i y
o he model also con ibu es o accen ua e he di e ence in pe o mance o he SBIL
and GMM- ype es ima o s. Fo he sample size N=100, T=5, he MAE and bias e-
sul s may be compa ed wi h he i s panel o Table I in A ellano and Bonhomme (2009).
Bo h SBIL and CU-II ha e less bias and lowe MAE han any o he es ima o s consid-
e ed by A ellano and Bonhomme. This is o be expec ed, because hose es ima o s a e
semi-pa ame ic, in ha he dis ibu ion o he indi idual e ec s is unknown. The SBIL
and CU-II es ima o s, in con as , a e based on simula ions ha equi e knowledge o he
dis ibu ion o he ixed e ec s. The assump ion ha he dis ibu ion o he indi idual
e ec s be known is qui e implausible in his example. Ne e heless, he example se es
o illus a e how he SBIL and II es ima o s can achie e a good bias educ ion in small
samples, h ough use o a simple nai e auxilia y model, when one is able o w i e a ully
simulable model.
7.4. S uc u al model o an auc ion. Li (2010) p oposes o use indi ec in e ence o es i-
ma ion o s uc u al econome ic models, and illus a es wi h a Mon e Ca lo example o
es ima ion o he pa ame e s o a Du ch auc ion, whe e only he winning bid is obse ed.
The numbe o bidde s is ixed a N=6, and he sample size is n=100, meaning ha he
ou comes o 100 auc ions a e obse ed. A each auc ion i=1,2,..., 100, he quali y, xi, o
he i em being auc ioned is he squa e o a uni o m (0,2) andom a iable, o in oduce
he e ogenei y in he alues o he objec s ac oss he auc ions. The 6 bidde s d aw hei
independen p i a e alues om a common exponen ial dis ibu ion wi h densi y
( |xi) = 1
exp(θ0+θ1xi)exp −
exp(θ0+θ1xi)
so ha exp(θ0+θ1xi)is he mean alua ion o he i em, o e he bidde s. The equilib ium
s a egy o he winning bid is
b∗
i= ∗
i−1
FN−1( ∗
i|xi)Z ∗
i
0FN−1(u|xi)du
INDIRECT LIKELIHOOD INFERENCE 25
whe e ∗
iis he highes p i a e alua ion, and F(·|xi)is he exponen ial dis ibu ion unc-
ion. Fo a gi en alue o N(6 in his case), symbolic compu a ion so wa e can be used o
ob ain an analy ic solu ion o he winning bid, which acili a es simula ion o he model.
The obse ed da a a e he 100 alues o {xi,b∗
i}, and we seek o es ima e θ0and θ1. The
ue alues a e se o θ0=1 and θ1=0.5. Li p esen s esul s o indi ec in e ence us-
ing wo auxilia y s a is ics: he i ed coe icien s o a pseudo ML es ima o , and he OLS
eg ession coe icien s (b
β0,b
β1)ob ained by i ing he model b∗
i=β0+β1xi+σei.
To apply he SBIL es ima o , we mus speci y he pa ame e space. The p esen appli-
ca ion is in e es ing, because we ha e no clea a p io i bounds o he wo pa ame e s θ0
and θ1. Ou side o he Mon e Ca lo con ex , one would only ha e he sample da a, bu
would no know he ue pa ame e alue. We discuss he issue o how he pa ame e
space may be speci ied a some leng h, because i is a necessa y s ep o apply he SBIL es-
ima o . Ou p oposal is o s a wi h a pa ame e space ha seems conse a i ely la ge,
and o check ha i in ac con ains elemen s ha can gene a e simula ed s a is ics Zs
n ha
di e in impo an espec s om he Zngene a ed by he sample da a. To do his, one can
gene a e a p elimina y se o Zs
nse ing Ssmall enough o be con enien . Then one may
compu e he dis ance be ween each simula ed s a is ic and he s a is ic using he sample
da a, gi ing he Sdis ances ds. Then one can so he S eplica ions o (θs,Zs
n,ds)by ds
and check ha he θs ha gene a e ela i ely small dis ances a e always com o ably a
away om he bounds o he p oposed pa ame e space. I his is no he case, he pa-
ame e space can be expanded, and he p ocedu e epea ed again. Con e sely, one may
ind e idence ha he p oposed pa ame e space is excessi ely b oad, in ha egions o
he pa ame e space ne e gene a e s a is ics close o Zn. Such simula ions will no con-
ibu e o he nea es neighbo s e sion o SBIL, and as such a e was ed. This could be
a oided by using impo ance sampling, bu we he e o simplici y ake a b u e o ce ap-
p oach and simply choose ini ially a la ge pa ame e space and a mode a e numbe o
simula ions, S o an in ial explo a ion o he dis ibu ion o he s a is ic ac oss di e en
pa ame e alues. We hen sh ink he pa ame e space emo ing pa s whi h li le o no
con ibu ion o he pos e io dis ibu ion.
We ini ially se he pa ame e space o Θ=(−5,5)×(0, 5). We gene a e a single sam-
ple a he ue pa ame e alue, and a ai ly small numbe (105)simula ed samples om
he p oposed pa ame e space. Inspec ion o he dis ibu ion o he auxilia y s a is ic
used by Li e eals ha he auxilia y s a is ic when sampling om he p oposed pa am-
e e space p esen s some ex eme ou lie s. This is a p oblem ha may no be de ec ed
when using he II es ima o wi h a limi ed numbe o eplica ions o he auxilia y s a is-
ics (Li uses only one d aw), because he II es ima o main ains he unde lying andom
d aws ixed o e he i e a ions, o a oid he phenomenon o “cha e ” when doing he
minimiza ion o compu e he es ima o . The chances o encoun e ing an ou lying alue
o he auxilia y s a is ic a e small, because only a e andom d aws gene a e ou lie s, by
de ini ion, and a ai ly small numbe o d aws a e used. Howe e , when a la ge numbe
o auxilia y s a is ics a e gene a ed, as is he case wi h he SBIL es ima o , ou lie s will
e en ually appea i he dis ibu ion o he auxilia y s a is ic has ou lie s in i s suppo .
INDIRECT LIKELIHOOD INFERENCE 32
achie ed by wo king wi h a ini e dimensional s a is ic a he han wi h he ull sample
con e s po en ially in ini e dimensional p oblems (as he sample g ows) in o ac able
ini e dimensional p oblems. This is an impo an simpli ica ion when nonpa ame ic
es ima ion me hods a e used. Mo eo e , wi h a ca e ul choice o auxilia y s a is ic, once
can hope o app oxima e su iciency. As we ha e seen in he DSGE examples, he SBIL
es ima o may be compu ed e en when he auxilia y s a is ic is o ai ly high dimen-
sion, a he cos o equi ing mo e simula ions. The possibili y o using a ai ly high (bu
ini e) dimensional auxilia y s a is ic makes i easonably hope ul ha he s a is ic ap-
p oxima ely spans he space o he e icien sco e, in which case he SBIL es ima o will
be app oxima ely ully asymp o ically e icien . Ou Mon e Ca lo esul s o he dynamic
panel and nonlinea panel examples can be compa ed o he esul s o o he au ho s o
o he es ima o s, gi ing suppo o he good ela i e e iciency o he SBIL es ima o . Ad-
di ional suppo comes om ou MA example, whe e he SBIL es ima o o en exhibi s
an RMSE smalle han ha o he ML es ima o .
The ac ha he SBIL es ima o may ha e be e small sample pe o mance han he
ML es ima o may be ele an when one seeks o es ima e complex DSGE models. The
combina ion o pa icle il e ing and MCMC discussed abo e seeks o compu e he ML
es ima o o ela ed Bayesian likelihood-based es ima o s. The il e ing/MCMC echnol-
ogy is ela i ely complica ed o implemen , and is compu a ionally ex emely demand-
ing. In compa ison, he SBIL es ima o is simple o implemen . In addi ion, i is ce ainly
possible ha he SBIL es ima o could ha e be e small sample pe o mance han he ML
es ima o o such complex and o en nonlinea models. An in e es ing a enue o explo e
would be o compa e ou es ima o wi h he he MLE based on pa icle il e ing/MCMC
al e na i es
In ou implemen a ion, we ha e ocused on he basic sample as gi en in equa ion (6)
choosing he numbe o neighbo s k h ough he simple ule k=1.5 ×S0.25. The e is
ce ainly scope o use o mo e sophis ica ed ules, such as c oss- alida ion, o di e en
ke nels, which could lead o be e pe o mance. Simila ly, mo e complica ed sample s
using impo ance sampling me hods could be used o imp o e on he compu a ion ime.
We lea e hese nume ical issues o u u e esea ch.

INDIRECT LIKELIHOOD INFERENCE 33
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INDIRECT LIKELIHOOD INFERENCE 36
APPENDIX A: PROOFS
P oo . [P oposi ion 1]We i s in es iga e he MIL es ima o : To his end, i s no e ha by
Lemma 1 he log-likelihood sa is ies
1
nlog (Zn|θ) = 1
nlogφ∗
n(Zn|θ)+LRn(θ)=1
nlogφ∗
n(Zn|θ)+oP1/√n
uni o mly in θ. Thus, o he i s -o de analysis, we can ea Ln(θ):=logφ∗
n(Zn|θ)as
he ac ual log-likelihood. To show consis ency, no e ha uni o mly in θ∈Θ:
1
nLn(θ) = −1
2nlog (|Ω(θ)|)−Tn(θ)0Tn(θ)
2n+oP(1)
=−1
2(Z(θ0)−Z(θ))0Ω−1(θ) (Z(θ0)−Z(θ)) +oP(1)
(21)
=:L(θ)+oP(1),
whe e L(θ)is a con inuous unc ion wi h a unique minimum a θ=θ0by Assump ion 3.
I now ollows by s anda d esul s (see e.g. Newey and McFadden, 1994, Theo em 2.1),
ha he MLE is consis en .
Nex , we show asymp o ic no mali y: Wi h ˙
Z(i)(θ)=∂Z(θ)/(∂θi)and ˙
Ω(i)(θ)=
∂Ω(θ)/(∂θi),
∆n,i(θ):=∂Ln(θ)
∂θi
=−1
2Ω−1(θ)˙
Ω(i)(θ)−√nTn(θ)0Ω−1/2 (θ)˙
Z(i)(θ)+1
2Tn(θ)0˙
Ω(i)(θ)Tn(θ)
=−√nTn(θ)0Ω−1/2 (θ)˙
Z(i)(θ)+oP√n
and wi h ¨
Z(i,j)(θ)=∂2Z(θ)/∂θi∂θjand ¨
Ω(i,j)(θ)=∂2Ω(θ)/∂θi∂θj,
Jn,ij (θ):=1
n
∂2Ln(θ)
∂θi∂θj
=1
nΩ−2(θ)˙
Ω(i)(θ)˙
Ω(j)(θ)−1
nΩ−1(θ)¨
Ω(i,j)(θ)
+˙
Z(i)(θ)0Ω−1(θ)˙
Z(j)(θ)+Tn(θ)0Ω−1/2 (θ)¨
Z(i,j)(θ)/√n+oP(1),
Wi h J(θ)de ined in Assump ion 3, i now holds ha
(22) 1
√n∆n(θ0)=−Tn(θ0)0Ω−1/2 (θ0)˙
Z(θ0)+oP(1)→dN(0, J(θ0)) ,
and, uni o mly in θ,Jn(θ)=J(θ)+oP(1). Since he sco e o he log-likelihood con e ges
weakly owa ds a no mal dis ibu ion while he Hessian con e ges uni o mly owa ds a
non-singula limi in p obabili y, i now ollows by a s anda d Taylo expansion o he
sco e ha he MILE is √n-asymp o ically no mally dis ibu ed wi h asymp o ic a iance
J−1(θ0).
Nex , he p ope ies o he BIL a e es ablished by e i ying Assump ions 1-4 in Che -
nozhuko and Hong (2003), CH hence o h, wi h Ln(θ)chosen as abo e. Fi s no e ha
CH’s Assump ions 1-2 a e sa is ied by ou Assump ion 1. Wha emains is o e i y hei
Assump ion 3-4. Bu by combining hei Lemmas 1-2 wi h he abo e de i a ions, hese
INDIRECT LIKELIHOOD INFERENCE 37
a e easily e i ied. We can now appeal o CH’s Theo em 2 which yields he desi ed e-
sul . 
P oo . [P oposi ion 2]This ollows di ec ly om Che nozhuko and Hong (2003, Theo-
em 3) since eqs. (22) and Jn(θ)=J(θ)+oP(1)imply ha he gene alized in o ma ion
equali y holds. 
P oo . [P oposi ion 3]By assump ion, ¯
Zn(θ)=Z(θ)+o1/√n, while Zn=Zn(θ0)→P
Z(θ0). Thus, Dn(θ) = D(θ) + op(1), whe e he limi is gi en by
D(θ) = 1
2(Z(θ0)−Z(θ))0Ω−1(θ0) (Z(θ0)−Z(θ)) .
By Assump ion 3in conjunc ion wi h s anda d a gumen s, i now ollows ha ˆ
θGMM is
consis en . To de i e i s asymp o ic dis ibu ion, i s no e ha ˆ
θGMM sol es
0=∂Dn(θ)
∂θ0=−∂Zn(θ)
∂θ
0Wn(Zn−¯
Zn(θ)) =−˙
Z(θ)0Wn(Zn−Z(θ)) +oP1/√n,
whe e, by Assump ion 2,
Zn−Z(θ)=Zn−Z(θ0)−˙
Zθ(θ−θ0),
whe e θlies on he line be ween θand θ0. Combining hese wo equa ions,
0=−˙
Zˆ
θGMM0WnZn−Zˆ
θGMM+oP1/√n
=−˙
Zˆ
θGMM0Wn{Zn−Z(θ0)}+˙
Zˆ
θGMM0Wn˙
Zθ(ˆ
θGMM −θ0) + oP1/√n.
The esul now ollows by Assump ion 2 oge he wi h Wn→PΩ−1(θ0).
P oo . [P oposi ion 4]Fi s , conside he wo-s ep GMM es ima o , ˆ
θGMM. Wi h
mn(θ)=(Zn−¯
Zn(θ))0Wn
∂¯
Zn(θ)
∂θ ,
we can apply Lemma 2. The i s and second o de de i a i es a e gi en by
∂mn(θ0)
∂θ =−∂¯
Zn(θ0)0
∂θ Wn
∂¯
Zn(θ0)
∂θ +(Zn−¯
Zn(θ))0Wn
∂2¯
Zn(θ0)
∂θ2,
and ∂2mn(θ0)
∂θ2=−3∂2¯
Zn(θ0)0
∂θ2Wn
∂¯
Zn(θ0)
∂θ +(Zn−¯
Zn(θ))0Wn
∂3¯
Zn(θ0)
∂θ3
Wi h
D¯
mn=−∂¯
Zn(θ0)0
∂θ Ω−1
n(θ0)∂¯
Zn(θ0)
∂θ ,
D2¯
mn=−3∂2¯
Zn(θ0)
∂θ2Ω−1
n(θ0)∂¯
Zn(θ0)
∂θ ,

INDIRECT LIKELIHOOD INFERENCE 38
whe e Ω−1
n(θ0)deno es he a iance o Zn−¯
Zn(θ), and ∆nde ined in he p oposi ion,
An:=∂mn(θ0)
∂θ −D¯
mn
=∂2¯
Zn(θ)0
∂θ2Ω−1
n(θ0) (Zn−¯
Zn(θ)) −∂¯
Zn(θ0)0
∂θ ∆n
∂¯
Zn(θ0)
∂θ
+∂2¯
Zn(θ)0
∂θ2∆n(Zn−¯
Zn(θ))
=∂2¯
Zn(θ)0
∂θ2Ω−1
n(θ0) (Zn−¯
Zn(θ)) −∂¯
Zn(θ0)0
∂θ ∆n
∂¯
Zn(θ0)
∂θ +OP(1/n).
Thus,
E[Anmn(θ0)] =∂2¯
Zn(θ)0
∂θ2Ω−1
n(θ0)Eh(Zn−¯
Zn(θ)) (Zn−¯
Zn(θ))0iΩ−1
n(θ0)∂¯
Zn(θ)
∂θ
−∂¯
Zn(θ0)0
∂θ E∆n
∂¯
Zn(θ0)
∂θ (Zn−¯
Zn(θ))0Ω−1
n(θ0)∂¯
Zn(θ)
∂θ
=1
n
∂2¯
Zn(θ)0
∂θ2Ω−1
n(θ0)∂¯
Zn(θ)
∂θ
−1
n
∂¯
Zn(θ0)0
∂θ E∆n
∂¯
Zn(θ0)
∂θ (Zn−¯
Zn(θ))0Ω−1
n(θ0)∂¯
Zn(θ)
∂θ
≃1
n1
3D2¯
m+BW,n
wi h BW,nde ined in he p oposi ion. The o he bias componen can be w i en as:
Em2
n(θ0)=∂¯
Zn(θ)0
∂θ Ω−1
n(θ0)Eh(Zn−¯
Zn(θ)) (Zn−¯
Zn(θ))0iΩ−1
n(θ0)∂¯
Zn(θ)
∂θ
=1
n
∂¯
Zn(θ)0
∂θ Ω−1
n(θ0)∂¯
Zn(θ)
∂θ ≃ −1
nJ(θ0).
Thus, by Lemma 2,
Eˆ
θGMM−θ0≃ −J−2(θ0)E[Anmn(θ0)] −1
2
D2¯
mn
D¯
mn
Em2
n(θ0).
≃1
nJ−2(θ0)1
6D2¯
m+BW,n
Nex , conside he CU es ima o : I is easily checked ha he expansion goes h ough
wi h
mn(θ):=2∂¯
Zn(θ)0
∂θ Ω−1
n(θ) (Zn−¯
Zn(θ)) +(Zn−¯
Zn(θ))0∂Ω−1
n(θ)
∂θ (Zn−¯
Zn(θ)) ,
and D¯
mnand D2¯
mngi en as be o e. Howe e , in he case o CU,
An:=∂mn(θ0)
∂θ −D¯
mn=∂2¯
Zn(θ)0
∂θ2Ω−1
n(θ0) (Zn−¯
Zn(θ)) +OP(1/n)
and so he bias e m due o he i s -s ep es ima ion o he weigh ing ma ix anishes and
we ob ain he claimed esul .
INDIRECT LIKELIHOOD INFERENCE 39
Finally, conside he MIL es ima o : Since LR (θ)=oP1/n2, we can choose mn(θ)=
n−1∂log ∗
n(Zn|θ)/(∂θ)such ha
∂mn(θ)
∂θ =1
n
∂2log ∗
n(Zn|θ)
∂θ2,∂2mn(θ)
∂θ2=1
n
∂3log ∗
n(Zn|θ)
∂θ3.
F om he de ini ion o ∗
n(Zn|θ),mn(θ)=mn,1 (θ)+mn,2 (θ), whe e he i s e m is he
Gaussian componen ,
mn,1 (θ)≃˙
Z(θ)0Ω−1(θ) (Zn−Z(θ)) ,
while he second one is due o he highe -o de componen ,
mn,2 (θ)≃1
n3/2
∂π1(Tn(θ)|θ)/∂θ
1+π1(Tn(θ)|θ)/√n
≃1
n3/2
∂π1(Tn(θ)|θ)
∂θ
≃ −1
nπ(1)
1(Tn(θ)|θ)Ω−1/2 (θ)˙
Z(θ).
The de i a i es sa is y
∂m1,n(θ)
∂θ ≃1
2¨
Z(θ)0Ω−1(θ) (Zn−Z(θ)) −˙
Z(θ)0Ω−1(θ)˙
Z(θ),
∂m2,n(θ)
∂θ ≃ −1
nπ(1)
1(Tn(θ)|θ)Ω−1/2 (θ)¨
Z(θ)
+1
√n˙
Z(θ)0Ω−1/2 (θ)π(2)
1(Tn(θ)|θ)Ω−1/2 (θ)˙
Z(θ),
and ∂2m1,n(θ)
∂θ2≃...
Z(θ)0Ω−1(θ) (Zn−Z(θ)) −3¨
Z(θ)0Ω−1(θ)˙
Z(θ),
∂2m2,n(θ)
∂θ2≃ −1
nπ(1)
1(Tn(θ)|θ)Ω−1/2 (θ)...
Z(θ)
+2
√n˙
Z(θ)0Ω−1/2 (θ)π(2)
1(Tn(θ)|θ)Ω−1/2 (θ)˙
Z(θ)
+∑
i
˙
Z(θ)0Ω−1/2 (θ)˜
π(3)
1,i(Tn(θ)|θ)Ω−1/2 (θ)˙
Z(θ),
whe e ˜
π(3)
1,i(Tn(θ)|θ)=∂π(2)
1(Tn(θ)|θ)/(∂ i)˙
Tn,i(θ)/√n. Since ¯
Tn(θ0)=O1/√n, we
can choose D¯
mnand D¯
m2
nas o he GMM and CU es ima o s excep ha Z(θ) eplaces
¯
Zn(θ). Nex , in o de o ob ain an exp ession o he bias, we Taylo -expanding w. . . he
s a is ic: Wi h ¯
Z0,n:=¯
Zn(θ0),¯
∗
n:=¯
∗
n(¯
Z0,n|θ0)and ¯
Tn(θ):=√nΩ−1/2 (¯
Z0,n−Z(θ)),
∂imn(θ)
∂θi≃1
n
∂i+1log ¯
∗
n(θ)
∂θi+1
n
∂i+2log ¯
∗
n(θ)
∂θi∂z(Zn−¯
Z0,n),
o i=0,1,2, whe e
1
n
∂2log ¯
∗
n(θ)
∂θ∂z≃˙
Z(θ)0Ω−1(θ)+1
√nΩ−1/2 (θ)π(2)
1(¯
Tn(θ)|θ)Ω−1/2 (θ)˙
Z(θ),
INDIRECT LIKELIHOOD INFERENCE 40
1
n
∂3log ¯
∗
n(θ)
∂θ2∂z≃¨
Z(θ)0Ω−1(θ)−1
√nΩ−1/2 (θ)π(2)
1(¯
Tn(θ)|θ)Ω−1/2 (θ)¨
Z(θ)
+∑
i
˙
Z(θ)0Ω−1/2 (θ)¯
π(3)
1,i(¯
Tn(θ)|θ)Ω−1/2 (θ)˙
Z(θ),
whe e ¯
π(3)
1,i( )is de ined in he p oposi ion, and
1
n
∂4log ¯
∗
n(θ)
∂θ3∂z≃...
Z(θ)0Ω−1(θ)+1
√nΩ−1/2 (θ)π(2)
3(¯
Tn(θ)|θ)Ω−1/2 (θ)...
Z(θ)
+2∑
i
˙
Z(θ)0Ω−1/2 (θ)¯
π(3)
3,i(¯
Tn(θ)|θ)Ω−1/2 (θ)˙
Z(θ).
We no e ha ¯
Tn(θ0)=O1/√nsuch ha π(i)
1(¯
Tn(θ)) ≃π(i)
1(0). Thus,
E[Anmn(θ0)] ≃1
n2
∂2log ¯
∗
n(θ0)
∂θ∂z0Eh(Zn−¯
Zn(θ0)) (Zn−¯
Zn(θ0))0i∂3log ¯
∗
n(θ0)
∂θ2∂z
≃1
3nD2¯
m+1
nBπ,
and
Em2
n(θ0)≃1
n2
∂2log ¯
∗
n(θ0)
∂θ∂z0Eh(Zn−¯
Zn(θ0)) (Zn−¯
Zn(θ0))0i∂2log ¯
∗
n(θ0)
∂θ∂z≃1
nJ(θ0).
Lemma 2now yields he claimed esul . 
P oo . [P oposi ion 5]As usual, we can ea ∗(Zn|θ)as he ac ual likelihood due o
Lemma 1. By an h o de Taylo expansion o he co esponding sco e equa ion w. . .
θ,
(23) 0 =Wn,1 (Zn)+
∑
i=1
1
i!Wn,i(Zn)ˆ
θMIL −θ0i+Rn,=:AWn(Zn),ˆ
θMIL+Rn,
whe e Wn(z)=(Wn,1 (z), ...,Wn, (z)) wi h Wn,i(z)=n−1∂ilog ∗
n(z|θ0)/∂θi
0, and
Rn=n−1|∂ log ∗
n(z|θ)/(∂θ )|θ=θˆ
θMIL −θ0 .
Fi s , igno e Rnand ede ine ˆ
θMIL as he solu ion o AWn(Zn),ˆ
θMIL=0. F om he
exp ession o ∗(Zn|θ), i is easily seen ha
Wn(Z(θ0)) =W∞(Z(θ0)) +
∑
i=1
1
ni/2 Mi+on− /2,
whe e Mia e cons an s depending on de i a i es o he polynomials π1, ..., π and W∞,i(Z(θ0))
is he leading e m o n−1∂ilog ∗(Z(θ0)|θ0)/∂θi
0. In pa icula , he limi ing sco e and
Hessian sa is y ¯
W∞,1 (Z(θ0)) =0 and ¯
W∞,2 (Z(θ0)) =−J(θ0). Thus, A(W∞(Z(θ0)) ,θ0)=
0, and ∂A(W∞(Z(θ0)) ,θ)/∂θ|θ=θ0=−J(θ0)has ull ank. Hence, by he implici unc-
ion heo em, he e exis s an analy ic unc ion H(w)in a neighbo hood o W∞(Z(θ0))
such ha θ0=H(W∞(Z(θ0))). Mo eo e , o all nla ge enough, he solu ion θ0,n o
A(Wn(Z(θ0)) ,θ0,n)=0, can be exp essed as θ0,n=H(Wn(Z(θ0))) since Wn(Z(θ0))
INDIRECT LIKELIHOOD INFERENCE 41
lies in a neighbo hood o W∞(Z(θ0)) o all nla ge enough. The sequence θ0,nsa is ies
θ0,n−θ0=H(Wn(Z(θ0))) −H(W∞(Z(θ0)))
=
∑
i=1
∂iH(W∞(Z(θ0)))
∂wi[Wn(Z(θ0)) −W∞(Z(θ0))]i+on− /2
=:
∑
j=1
1
nj/2 ˜
Mj+on− /2,
whe e ˜
Mjis a cons an depending on M1,..., M and he i s de i a i es o H(W∞(Z(θ0))),
j=1,..., .
We ob ain an Edgewo h expansion o ˆ
θMIL −θ0,n=H(Wn(Zn)) −H(Wn(Z(θ0))) by
applying he gene al esul o Phillips (1977) o Edgewo h expansions o ans o ma-
ions o andom sequences: We de ine he ollowing sequence o unc ions
en(q):=H(Wn(q+Z(θ0))) −H(Wn(Z(θ0))) ,
such ha en:=ˆ
θMIL −θ0,n=en(qn), whe e qn:=Zn−Z(θ0), and e i y Phillips (1977,
Assump ions 3-5): Fi s , since he dis ibu ion o he no malized s a is ic Tn(θ0)=√nqn
sa is ies an Edgewo h expansion by Assump ion 4, Phillips (1977, Assump ion 3) holds.
Nex , he wo unc ion Hand Wna e bo h imes con inuously di e en iable and he
de i a i es o Wn(z)con e ges owa ds hose o W∞(z). Thus, en(q)is imes di e -
en iable wi h i s de i a i es uni o mly bounded in a neighbo hood a ound 0. Finally,
we know om he implici unc ion heo em ha ∂H(W∞(Z(θ0))) /(∂w)has ull ank
while i is easily checked ha ∂W∞,1 (Z(θ0)) /(∂z)=Ω−1/2 (θ0)˙
Z(θ0). Hence, by he
chain ule, |∂en(q)/∂q|is bounded away om ze o as n→∞. This shows ha Phillips
(1977, Assump ions 4-5) hold.
We ha e shown ha √nenadmi s an Edgewo h expansion, say
∗
en(x)=φ(x)"1+
∑
i=1
n−i/2 ¯
πi(x)#.
This in u n implies ha he dis ibu ion o en:=√nˆ
θMIL −θ0=√nen+bn, whe e
bn=√n(θ0,n−θ0)=∑
j=1n−j/2Mj+on− /2, can be app oxima ed by
∗
en(x)=φ(x−bn)"1+
∑
i=1
n−i/2 ¯
πi(x−bn)#.
Expanding a ound ∗
en(x)and ea anging e ms, we hen ob ain he desi ed esul whe e
he coe icien s o he polynomial ˜
πi(x)depend on he ones o ¯
πj(x)and he coe icien s
Mj,j=1,..., .
Finally, we ha e o e i y ha we a e allowed o igno e he emainde e m Rnin he
Taylo expansion. By he a gumen s in Ro henbe g (1984, p. 898), his will ollow i
P(|Rn|>logcn)=on− /2. This will in u n hold i
P|Wn(Zn)−Wn(Z(θ0))|>c1qlog (n)/n=on− /2,
INDIRECT LIKELIHOOD INFERENCE 48
TABLE 6. Fully obse ed DSGE model wi h monopolis ic compe i ion
Bias RMSE
Pa ame e Lowe Bound Uppe bound T ue alues P io mean SBIL P io mean SBIL
α0.15 0.4 0.33 -0.055 -0.002 0.091 0.006
β0.95 0.999 0.99 -0.016 -0.000 0.021 0.001
δ0.005 0.06 0.023 0.010 0.001 0.019 0.001
ψ1 3 1.75 0.250 0.004 0.629 0.014
ρ0.85 0.99 0.95 -0.030 -0.017 0.050 0.024
σ0.005 0.04 0.01 0.012 0.000 0.016 0.001
e9 13 10 1.000 0.002 1.529 0.033
TABLE 7. Pa ially obse ed DSGE model wi h habi o ma ion, i s design
Bias RMSE
Pa ame e Lowe Bound Uppe bound T ue alues P io mean SBIL P io mean SBIL
α0.25 0.4 0.36 -0.035 -0.003 0.056 0.008
β0.93 0.99 0.95 0.010 0.001 0.020 0.004
δ0.02 0.04 0.025 0.005 0.000 0.008 0.001
η0 0.5 0.2 0.050 -0.024 0.153 0.044
γ1 4 2 0.500 0.220 1.000 0.283
ρ0.8 0.99 0.85 0.045 -0.005 0.071 0.017
σ0.01 0.08 0.04 0.005 0.001 0.021 0.004
ψNA NA 3.197 9.854 0.356 22.529 0.530
TABLE 8. Pa ially obse ed DSGE model wi h habi o ma ion, second design
Bias RMSE
Pa ame e Lowe Bound Uppe bound T ue alues P io mean SBIL P io mean SBIL
α0.25 0.4 0.36 -0.035 0.001 0.056 0.006
β0.93 0.99 0.95 0.010 -0.003 0.020 0.004
δ0.02 0.04 0.025 0.005 0.001 0.008 0.002
η0 0.5 0.4 -0.150 -0.033 0.208 0.056
γ1 4 3 -0.500 0.018 1.000 0.185
ρ0.8 0.99 0.85 0.045 -0.003 0.071 0.016
σ0.01 0.08 0.04 0.005 0.000 0.021 0.003
ψNA NA 13.562 -0.511 0.792 22.266 3.052

INDIRECT LIKELIHOOD INFERENCE 49
FIGURE 1. Fully obse ed DSGE model. Pseudo-p io s, ue pa ame e
alues, and densi y o SBIL
(A)α(B)β(C)δ
(D)ψ(E)ρ(F)σ
(G)e
FIGURES
UNIVERSITAT AUTÒNOMA DE BARCELONA AND MOVE
COLUMBIA UNIVERSITY AND CREATES (CENTER FOR RESEARCH IN ECONOMETRIC ANALYSIS OF TIME
SERIES, UNIVERSITY OF AARHUS).