Kleibe gen, F ank; Zhan, Zhaoguo
A icle
Double obus in e ence o con inuous upda ing GMM
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: Kleibe gen, F ank; Zhan, Zhaoguo (2025) : Double obus in e ence o
con inuous upda ing GMM, Quan i a i e Economics, ISSN 1759-7331, The Econome ic Socie y, New
Ha en, CT, Vol. 16, Iss. 1, pp. 295-327,
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Quan i a i e Economics 16 (2025), 295–327 1759-7331/20250295
Double obus in e ence o con inuous upda ing GMM
F ank Kleibe gen
Ams e dam School o Economics, Uni e si y o Ams e dam
Zhaoguo Zhan
Coles College o Business, Kennesaw S a e Uni e si y
We p opose he double obus Lag ange mul iplie (DRLM) s a is ic o es ing hy-
po heses speci ied on he minimize o he popula ion con inuous upda ing ob-
jec i e unc ion. The (bounding) χ2limi ing dis ibu ion o he DRLM s a is ic is
obus o bo h misspeci ica ion and weak iden i ica ion, hence i s name. The min-
imize is he so-called pseudo- ue alue, which equals he ue alue o he s uc-
u al pa ame e unde co ec speci ica ion. To emphasize i s impo ance o ap-
plied wo k whe e misspeci ica ion and weak iden i ica ion a e common, we use
he DRLM es o analyze: he isk p emia in Ad ian e al. (2014) and He e al.
(2017); he s uc u al pa ame e s in a nonlinea asse p icing model wi h cons an
ela i e isk a e sion.
Keywo ds. Weak iden i ica ion, misspeci ica ion, obus in e ence, Lag ange
mul iplie .
JEL classi ica ion. C12, C18, G12.
1. In oduc ion
A li le mo e han 20 yea s ago, in e ence p ocedu es o analyzing possibly weakly
iden i ied s uc u al pa ame e s using he gene alized me hod o momen s (GMM) o
Hansen (1982) we e mos ly lacking. Since hen, huge p og ess has been made o de elop
such p ocedu es; see, o example, S aige and S ock (1997), Du ou (1997), S ock and
W igh (2000), Kleibe gen (2002,2005,2009), Mo ei a (2003), And ews and Cheng (2012),
And ews and Mikushe a (2016a,2016b), and Han and McCloskey (2019). A p esen , we
he e o e ha e a a ie y o so-called weak iden i ica ion obus in e ence me hods. Gi en
he p e alence o weak iden i ica ion in applied wo k, a lo o emphasis has also been
pu in aising awa eness among p ac i ione s; see, o example, Kleibe gen and Ma oei-
dis (2009), Beaulieu, Du ou , and Khala (2013), Ma oeidis, Plagbo g-Molle , and S ock
(2014), And ews, S ock, and Sun (2019), and Kleibe gen and Zhan (2020).
When he e is no so-called ue alue o he s uc u al pa ame e s whe e he GMM
momen condi ions exac ly hold, he s uc u al model is ende ed misspeci ied, and
GMM es ima o s p o ide inconsis en es ima es o he ue alue. Ea ly esea ch on
F ank Kleibe gen: [email p o ec ed]
Zhaoguo Zhan: [email p o ec ed]
The esea ch o F ank Kleibe gen has been unded pa ly by he NWO (Du ch Resea ch Council) G an
401.21.EB.002: “Double obus in e ence o s uc u al economic models (DRISEM)”.
©2025 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/QE2347
296 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
misspeci ica ion ocused on in e ence on he ue alue by cha ac e izing exp essions
o he bias and s anda d e o s o inconsis en es ima o s; see, o example, Maasoumi
and Phillips (1982)andMaasoumi (1990). Ins ead o ocusing on he una ainable ue
alue, ecen wo k on misspeci ica ion analyzes he so-called pseudo- ue alue, which
is he minimize o he popula ion objec i e unc ion; see, o example, Whi e (1994). Fo
p ac ical ele ance o he pseudo- ue alue see, o example, Kan, Robo i, and Shanken
(2013) who use he pseudo- ue alue o e alua ing asse p icing models. The pseudo-
ue alue depends on he popula ion objec i e unc ion a hand, so di e en objec i e
unc ions can lead o dis inc pseudo- ue alues. Hall and Inoue (2003), o example,
de elop in e ence me hods o he pseudo- ue alue o he wo-s ep GMM es ima o ,
while Hansen and Lee (2021)do so o an i e a ed GMM es ima o . We use he minimize
o he popula ion con inuous upda ing es ima o (CUE) objec i e unc ion o Hansen,
Hea on, and Ya on (1996) as he pseudo- ue alue because o i s in a iance p ope ies
and since weak iden i ica ion obus in e ence p ocedu es lead o in e ence ha is cen-
e eda oundi .
In case o misspeci ica ion, weak iden i ica ion obus in e ence p ocedu es o es -
ing hypo heses speci ied on he pseudo- ue alue howe e become size dis o ed o
jus small amoun s o misspeci ica ion. This would no sound as much o a p oblem i
i was possible o e icien ly de ec such misspeci ica ion. This is no so since misspeci-
ica ion es s, like he Sa gan–Hansen es (Sa gan (1958)andHansen (1982)), a e i u-
ally powe less in se ings o join misspeci ica ion and weak iden i ica ion; see Gospodi-
no , Kan, and Robo i (2017). Weak iden i ica ion obus in e ence p ocedu es hus came
abou o o e come he gene al c i ique o non obus ness o adi ional in e ence p oce-
du es o a ying iden i ica ion s eng hs (see, e.g., S aige and S ock (1997)andDu ou
(1997)), bu a e simila ly non obus o misspeci ica ion.
While weak iden i ica ion obus in e ence p ocedu es a e size dis o ed when mis-
speci ica ion is p esen , he misspeci ica ion obus in e ence p ocedu es p oposed by,
o example, Hall and Inoue (2003), Kan, Robo i, and Shanken (2013), Lee (2018), and
Hansen and Lee (2021), p o ide misspeci ica ion obus co a iance ma ix es ima o s
o conduc Wald-based in e ence. Because he co a iance ma ix es ima o s a e ini e
by cons uc ion, he esul ing Wald-based es s canno lead o unbounded con idence
se s, which as shown by Du ou (1997), is a necessa y condi ion o size co ec es ing
in se ings wi h po en ial iden i ica ion ailu e. Hence, excep o ce ain speci ica ions
o he s uc u al pa ame e (see, e.g., Gospodino , Kan, and Robo i (2014)),1 hese mis-
speci ica ion obus es s a e po en ially size dis o ed unde weak iden i ica ion.
One o he i s o emphasize he empi ical ele ance o misspeci ica ion in he p es-
ence o weak (o no) iden i ica ion we e Kan and Zhang (1999). Wi h he su ge in applied
wo k on s uc u al es ima ion, awa eness o misspeci ica ion has g own u he . In asse
p icing models, o example, i is now gene ally accep ed ha misspeci ica ion, along-
side weak iden i ica ion, is an impo an empi ical issue; see, o example, Kan, Robo i,
1Gospodino , Kan, and Robo i (2014) show ha he Wald/ es o a ze o isk p emium in a linea asse
p icing model wi h useless pa ialled ou ac o s, which uses hei adjus ed s anda d e o , is bounded by a
χ2dis ibu ion and, he e o e, size co ec o he hypo hesis o in e es .
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 297
and Shanken (2013)andKleibe gen and Zhan (2020). Kan, Robo i, and Shanken (2013)
he e o e de eloped misspeci ica ion obus -s a is ics o he Fama–MacBe h (1973)
wo-pass es ima o , ha is, he mos commonly used es ima o o he isk p emia in
linea asse p icing models. Simila ly, in case o he e ogeneous ea men e ec s, he
local a e age ea men e ec s ha esul o di e en ins umen s can be dis inc (see
Imbens and Ang is (1994)), making he o e iden i ied linea ins umen al a iables e-
g ession model ha uses all ins umen s misspeci ied. E dokimo and Kolesa (2018)
and Lee (2018) he e o e analyze misspeci ica ion obus es s on he ea men e ec
esul ing om using such mul iple ins umen s. These misspeci ica ion obus es s a e
howe e no obus o weak iden i ica ion, so iden ical o he weak iden i ica ion obus
in e ence p ocedu es, hey canno deal wi h he empi ically ele an se ing o bo h mis-
speci ica ion and weak iden i ica ion. Ye he a o emen ioned weak iden i ica ion and
misspeci ica ion li e a u e highligh ha hese wo issues ha e o be aken in o accoun
in o de o conduc alid in e ence on he s uc u al pa ame e s o in e es .
We ex end he weak iden i ica ion obus sco e o Lag ange mul iplie (KLM) es
om Kleibe gen (2002,2005,2009) o a double obus Lag ange mul iplie (DRLM) es ,
which can also be in e p e ed as he misspeci ica ion obus e sion o he KLM es .
The DRLM es is size co ec and obus o bo h misspeci ica ion and weak iden i i-
ca ion, hence i s name. The DRLM s a is ic is a quad a ic o m o he sco e unc ion,
which equals ze o a all s a iona y poin s o he CUE sample objec i e unc ion. This
is also he case o he KLM s a is ic and explains he powe p oblems o he KLM es ;
see, o example, And ews, Mo ei a, and S ock (2006). To o e come he powe p oblems
o he KLM es , he KLM s a is ic can be combined in a condi ional o uncondi ional
manne wi h he Ande son–Rubin (1949, AR) s a is ic; see, o example, Mo ei a (2003)
and And ews (2016). And ews, Mo ei a, and S ock (2006) show ha he condi ional like-
lihood a io (LR) es o Mo ei a (2003) p o ides he op imal manne o combining hese
s a is ics o he homoskedas ic linea ins umen al a iables eg ession model wi h one
included endogenous a iable. In case o misspeci ica ion, i is howe e no ob ious how
o imp o e he powe o he DRLM es by such combina ion a gumen s, since he s a is-
ics wi h which he DRLM s a is ic is o be combined o imp o e powe , ha e non-cen al
limi ing dis ibu ions wi h pa ame e s ha canno be consis en ly es ima ed unde mis-
speci ica ion. We he e o e show ha he powe o he DRLM es can be imp o ed by
exploi ing he de i a i e p ope y o he DRLM s a is ic.
The es o he pape is o ganized as ollows. In Sec ion 2, we discuss con inuous
upda ing GMM wi h misspeci ica ion and p opose he DRLM es . To illus a e he size
co ec ness o he DRLM es , we conduc a simula ion expe imen using linea momen
equa ions in Sec ion 3. A powe s udy o he DRLM es and weak iden i ica ion obus
es s is also p esen ed in Sec ion 3. I shows ha weak iden i ica ion obus es s on he
pseudo- ue alue o he s uc u al pa ame e s a e size dis o ed o jus small amoun s
o misspeci ica ion while he DRLM es is no . I also p oposes powe imp o emen
and shows ha he esul ing es has gene ally good powe . Sec ion 4applies he DRLM
es o isk p emia using asse p icing da a om Ad ian, E ula, and Mui (2014)andHe,
Kelly, and Manela (2017). Sec ion 5ex ends he DRLM es o sub ec o in e ence, s ong
misspeci ica ion, which is deal wi h by an addi ional componen in he weigh ma ix,
298 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
and nonlinea momen equa ions om a nonlinea asse p icing model wi h cons an
ela i e isk a e sion. Sec ion 6concludes. Technical de ails and addi ional esul s a e
elega ed o he Supplemen al Appendix (Kleibe gen and Zhan (2024b)) .
2. DRLM es o possibly misspeci ied GMM
2.1 Se up
We analyze he m×1 pa ame e ec o θ=(θ1θm)whose pa ame e egion is he Rm.
The k ×1 dimensional unc ion (θ,X )is a con inuously di e en iable unc ion o he
pa ame e ec o θand a Bo el measu able unc ion o a da a ec o X ,whichisob-
se ed o ime/indi idual . Since we ocus on misspeci ica ion, he model is o e iden-
i ied, ha is, he e a e mo e momen equa ions han s uc u al pa ame e s so k >m.
The popula ion momen unc ion o (θ,X )equals μ (θ):
EX (θ,X )=μ (θ),(1)
wi h μ (θ)ak -dimensional con inuously di e en iable unc ion. Unlike egula GMM
(see Hansen (1982)) , we do no eques ha he e is a speci ic alue o θ,sayθ0,a which
μ (θ0)=0. Ou analysis hus di e s om a ecen one p oposed by Cheng, Dou, and
Liao (2022), who cons uc a model selec ion p ocedu e o e alua ing po en ially mis-
speci ied models wi h possibly weakly iden i ied s uc u al pa ame e s, which explici ly
uses a se o base momen s con ained in all conside ed models ha a e gua an eed o
hold.
We analyze θusing he con inuous upda ing se ing o Hansen, Hea on, and Ya on
(1996). We use i because o i s in a iance p ope ies and since i leads o in e ence based
on iden i ica ion obus s a is ics in s anda d GMM; see, o example, S ock and W igh
(2000)andKleibe gen (2005). Fo asse p icing s udies, in pa icula , Peña anda and Sen-
ana (2015) ecommend he con inuous upda ing se ing o e wo-s ep o i e a ed GMM
p ocedu es. The accompanying popula ion con inuous upda ing objec i e unc ion is
Qp(θ)=μ (θ)V (θ)−1μ (θ),(2)
wi h V (θ) he co a iance ma ix o he sample momen :2 T(θ,X)=1
TT
=1 (θ,X ),
V (θ)=lim
T→∞ET T(θ,X)−μ (θ) T(θ,X)−μ (θ),(3)
so T(θ,X)is he sample analog o μ (θ) o a da a se o Tobse a ions: X , =1, ,T.
We de ine he pseudo- ue alue o θ,θ∗as he minimize o he popula ion objec i e
unc ion:
θ∗=a g min
θ∈RmQp(θ).(4)
2Th oughou he pape , we use ecen e ed co a iance ma ices while he con inuous upda ing es ima o
is iden ical unde a ecen e ed o uncen e ed e sion o he same co a iance ma ix es ima o ; see Hansen
and Lee (2021).
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 299
Thus, he pseudo- ue alue θ∗equals he ue alue o θin co ec ly speci ied models.
In misspeci ied models, he in e p e a ion o θ∗depends on he con ex : o example, in
asse p icing s udies, Kan, Robo i, and Shanken (2013) in e p e he pseudo- ue alue
o isk p emia as he alue ha minimizes p icing e o s.
The sample objec i e unc ion o he CUE is
ˆ
Qs(θ)= T(θ,X)ˆ
V (θ)−1 T(θ,X),(5)
wi h ˆ
V (θ)a consis en es ima o o V (θ),ˆ
V (θ)→
pV (θ),so heCUE, ˆ
θis
ˆ
θ=a g min
θ∈Rmˆ
Qs(θ).(6)
A he CUE, he sco e o de i a i e o he CUE objec i e unc ion equals ze o:3
ˆ
s(θ)=1
2
∂
∂θˆ
Qs(θ)= T(θ,X)ˆ
V (θ)−1ˆ
D(θ),(7)
whe e
ˆ
D(θ)=qT(θ,X)−ˆ
Vq1 (θ)ˆ
V (θ)−1 T(θ,X) ˆ
Vqm (θ)ˆ
V (θ)−1 T(θ,X),(8)
wi h qT(θ,X)=∂ T(θ,X)
∂θ|θ=(q1T(θ)qmT (θ)),J(θ)=∂
∂θμ (θ)=(J1(θ)Jm(θ)),
ˆ
Vqi (θ)is a consis en es ima o o Vqi (θ)=limT→∞ E[T(qiT (θ)−Ji(θ))( T(θ,X)−
μ (θ))],i=1, ,m. Co espondingly, he popula ion coun e pa o he sample sco e
ˆ
s(θ)in (7)is
s(θ)=1
2
∂
∂θQp(θ)=μ (θ)V (θ)−1D(θ),(9)
whe e D(θ)is he ecen e ed Jacobian:
D(θ)=J(θ)−Vq1 (θ)V (θ)−1μ (θ)Vqm (θ)V (θ)−1μ (θ).(10)
Ou p oposed double obus Lag ange mul iplie s a is ic on he pseudo- ue alue
equals a quad a ic o m o he sample sco e ˆ
s(θ)in (7), which in ol es he p oduc o
he sample momen and a ecen e ed es ima o o i s Jacobian. While he sample mo-
men T(θ,X)is d i en by he magni ude o misspeci ica ion, he ecen e ed Jacobian
es ima o ˆ
D(θ) e lec s he s eng h o iden i ica ion. Thus, bo h misspeci ica ion and
weak iden i ica ion a e o be accoun ed o when we de elop he explici exp ession o
he double obus Lag ange mul iplie s a is ic based on ˆ
s(θ)in (7).
We ocus on weak (local) misspeci ica ion in he sense ha a he pseudo- ue alue
θ∗, he eexis s
˜μ θ∗=lim
T→∞
√Tμ θ∗,(11)
3The cons uc ion o he de i a i e is in Kleibe gen (2005) and also in Lemmas 3 and 4 in he Supple-
men al Appendix.
300 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
o ˜μ (θ∗)a ini e alued k -dimensional ec o , which could equal ze o. Thus, μ (θ∗)
is allowed o depend on Twi h he dependence kep implici o he pu pose o no-
a ional simplici y. This ea men hus di e s om a s ong misspeci ica ion case o
which μ (θ∗)is conside ed a ixed nonze o ec o . We discuss he s ong misspeci i-
ca ion case la e in Sec ion 5.2 as an ex ension. The p oposed double obus Lag ange
mul iplie es ( o a p ope ly speci ied weigh ma ix) applies o bo h o hese cases.
Analogous o he ea men o μ (θ∗)in (11), he weak iden i ica ion li e a u e widely
uses he so-called weak ins umen asymp o ics (see, e.g., S aige and S ock (1997)),
which in ou con ex amoun s o using a local o ze o sequence o D(θ∗):
˜
Dθ∗=lim
T→∞
√TDθ∗,(12)
o ˜
D(θ∗)a ini e alued k ×mdimensional ma ix, which could be o lowe ank. Al-
hough we ini ially use (12) o ease o exposi ion, i is wo h no ing ha he double
obus Lag ange mul iplie es is equally applicable unde s ong iden i ica ion as we
show when simul aneously discussing s ong misspeci ica ion in Sec ion 5.2. Pu di e -
en ly, whe he D(θ∗)is local o ze o o ixed makes no di e ence o he double obus
Lag ange mul iplie s a is ic, so we allow o all s eng hs o iden i ica ion.
2.2 Example: The linea IV eg ession model
Fo illus a ion, we use a linea ins umen al a iables (IV) eg ession model:4
y =βx +ε ,
x =z + ,(13)
whe e βand a e m×1andk ×mdimensional ma ices con aining unknown pa-
ame e s, y and x a e he scala and m×1 dimensional endogenous a iables, z is he
k ×1 dimensional ins umen al a iables, ε and a e he scala and m×1dimen-
sional e o s.
I s popula ion momen unc ion eads
μ (β)=σzy −zxβ
=μ (0)+J(0)θ, (14)
whe e, assuming ha he obse a ions o e he indi iduals a e i.i.d., σzy =E((z −
μz)(y −μy)),zx =E((z −μz)(x −μx))=Qzz,Qzz =E((z −μz)(z −μz)),μy=
E(y ),μx=E(x ),μz=E(z ). The las line o (14)uses heGMMno a ion om(1)and
(10)soμ (0)=σzy ,J(0)=−zx,andθ=β. Misspeci ica ion occu s when he s uc u al
e o ε is co ela ed wi h he ins umen s z . The e is hen no alue o βa which he
4Fo exposi o y pu poses, we only discuss a simpli ied e sion o he linea IV eg ession model wi hou
so-called, included exogenous a iables. We speci y he linea IV eg ession model, and also he linea asse
p icing model discussed la e , using he no a ion gene ically used in he li e a u e, and also show hei
connec ion o ou ea lie gene ic GMM no a ion.
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 301
popula ion momen unc ion (14) is equal o ze o. On he o he hand, weak iden i ica-
ion occu s when ins umen s z a e only weakly co ela ed wi h he endogenous x ,so
he ull ank condi ion o zx is a isk.
The popula ion objec i e unc ion o he CUE o he linea IV model wi h ho-
moskedas ic e o s is
Qp(β)=1
ωuu −2ωu β+β β(σzy −zxβ)Q−1
zz (σzy −zxβ),(15)
whe e u =(ωuu
ω u
ωu
)=co (u , ), o u =ε +
β. The CUE hen co esponds wi h
he so-called, limi ed in o ma ion maximum likelihood (LIML) es ima o , so we can use
he well-known k-class no a ion o he LIML es ima o (see, e.g., Hausman (1983)and
And ews (2019)) o exp ess he pseudo- ue alue as
β∗=a g min
β∈RmQp(β)=
zxQ−1
zz zx −τmin −1
zxQ−1
zz σzy −τminω u, (16)
wi h τmin =minβ∈RmQp(β).5We no e ha di e en om he usual k-class no a ion o
τmin =0 he pseudo- ue alue co esponds wi h he pseudo- ue alue o he wo-s age
leas squa es es ima o ; simila ly o τmin =−1, he pseudo- ue alue co esponds wi h
ha o he leas squa es es ima o .
Fo he linea IV eg ession model, he double obus Lag ange mul iplie es is de-
signed o es ing hypo heses on β∗, and i add esses weak iden i ica ion and misspec-
i ica ion simul aneously. The es is based on he sco e o de i a i e o Qp(β),which
equals ze o a β∗. The quad a ic o m o he sample sco e cons i u es he es s a is-
ic, whose explici exp ession will be p o ided using he gene ic GMM no a ion in Sec-
ion 2.4.
2.3 Assump ions
We s a e he assump ions needed o cons uc ing he la ge sample beha io o es
s a is ics cen e ed a ound he CUE o he gene ic GMM se ing. These assump ions
conce n he componen s o he sample sco e ˆ
s(θ)= T(θ,X)ˆ
V (θ)−1ˆ
D(θ).
We i s make Assump ion 1as in Kleibe gen (2005) excep ha i conce ns he la ge
sample beha io o he sample momen s and hei de i a i e a he pseudo- ue alue
θ∗ins ead o he ue alue.
Assump ion 1. The k ×1dimensional de i a i e o (θ)= (θ,X )wi h espec o θi,
qi (θ)=∂ (θ)
∂θi
:k ×1, i=1, ,m(17)
is such ha he join limi beha io o he sums o he se ies ¯
(θ)= (θ)−E( (θ)) and
¯
q (θ)=(¯
q1 (θ)¯
qm (θ)),wi h ¯
qi (θ)=qi (θ)−E(qi (θ)),acco ds wi h he cen al limi
5The minimal alue τmin equals he smalles oo o |τu −(σzy
.
.
.zx)Q−1
zz (σzy
.
.
.zx)|=0.
302 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
heo em a θ=θ∗,whe e θ∗is he minimize o he con inuous upda ing popula ion ob-
jec i e unc ion:
1
√T
T
=1¯
θ∗
¯
q θ∗→
dψ θ∗
ψqθ∗∼N0, Vθ∗,(18)
whe e ψ :k ×1, ψq:kθ×1, kθ=mk ,and V(θ∗)is a posi i e semide ini e symme ic
(k +kθ)×(k +kθ)ma ix,
Vθ∗=V θ∗V qθ∗
Vq θ∗Vqqθ∗, (19)
wi h Vq (θ∗)=V q(θ∗)=(Vq1 (θ∗)Vqm (θ∗)),Vqq(θ∗)=(Vqiqj(θ∗)) :i,j=1, ,m;
V (θ∗),Vqi (θ∗),Vqiqj(θ∗)a e k ×k dimensional ma ices o i,j=1, ,m,and
Vθ∗=lim
T→∞ a √T Tθ∗,X
ecqTθ∗,X. (20)
Assump ion 1 eques s a join cen al limi heo em o hold a he pseudo- ue alue
o he sample momen s and hei de i a i e. I is sa is ied unde mild condi ions, which
a e lis ed in Kleibe gen (2005) like, o example, ini e h momen s o >2incaseo
i.i.d. da a, mixing condi ions o he sample momen s in case o ime-se ies da a. Al-
lowing o a posi i e semide ini e co a iance ma ix V(θ∗)is impo an o applica ions
like, o example, dynamic linea panel da a models. We nex also use Assump ion 2 om
Kleibe gen (2005), which conce ns he con e gence o he co a iance ma ix es ima o .
Assump ion 2. The con e gence beha io o he co a iance ma ix es ima o ˆ
V(θ) o-
wa d V(θ)is such ha (i)ˆ
V(θ)→
pV(θ),and (ii)∂ ec(ˆ
V (θ))
∂θ→
p
∂ ec(V (θ))
∂θ.
Assump ion 2(i) eques s a consis en co a iance es ima o o V(θ). Fo Assump-
ion 2(ii), he co a iance ma ix es ima o ˆ
V (θ)has o be such ha he ˆ
Vq (θ),which
esul s om he de i a i e o ˆ
V (θ)wi h espec o θ, is also a consis en es ima o o
Vq (θ). When using he same co a iance ma ix es ima o o all elemen s o V(θ),As-
sump ion 2holds unde he condi ions o consis ency o (he e oskedas ic au oco e-
la ion consis en ) co a iance ma ix es ima o s; see, o example, Whi e (1980), Newey
and Wes (1987). Lemma 11 in he Supplemen al Appendix p o ides he low-le el egu-
la i y condi ions unde which Assump ion 2holds in he leading i.i.d. case.
Thesco eo heCUEobjec i e unc ionin(7) ac o izes as he p oduc o he sample
momen and a ecen e ed es ima o o i s Jacobian (8). Unde Assump ions 1and 2,
he limi beha io s o he sample momen and his ecen e ed Jacobian es ima o a e
independen a he pseudo- ue alue θ∗:
√T Tθ∗,X−μ θ∗→
dψ θ∗∼N0, V θ∗,
√T ecˆ
Dθ∗−Dθ∗→
dψθθ∗∼N0, Vθθθ∗,
(21)
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 309
The mean o he limi beha io o he sample sco e in (31) equals ze o when he
no malized iden i ica ion s eng h measu e based on he Jacobian, ¯
D¯
D, and misspeci-
ica ion s eng h measu e, ¯μ¯μ, a e equal. I shows ha he pseudo- ue alue θ∗is hen
no iden i ied.
3.4 Simula ed powe o DRLM
Nex , we illus a e he powe o 5% signi icance DRLM es s o H0:θ∗=0compa ed o
o he weak iden i ica ion obus es s including GMM-AR, KLM, and LR es s; see, o
example, S ock and W igh (2000), Mo ei a (2003), and Kleibe gen (2005).
We conside a simula ion se ing wi h k =25 momen equa ions, weak misspeci i-
ca ion, ¯μ¯μ=10, and a ying iden i ica ion s eng h. When he e is no misspeci ica ion,
And ews, Mo ei a, and S ock (2006) show ha he LR es is op imal, so we e ain om
using ha se ing. Figu es 3and 4show he powe cu es o KLM (panel 3.1), DRLM
(panel 3.2), LR (panel 4.1), and GMM-AR (panel 4.2) es s o H0:θ∗=0 o a ious iden-
i ica ion s eng hs and a ixed amoun o misspeci ica ion.
The powe cu es o he di e en es s in Figu es 3and 4show ha only he DRLM
es is size co ec o all se ings o he iden i ica ion s eng h. The size dis o ion o
some o he o he weak iden i ica ion obus es s can be qui e p onounced, which es-
pecially holds o he LR and GMM-AR es s. Fo he LR es , he ejec ion equency a
ze o dec eases om 30% o 8% when he iden i ica ion s eng h inc eases, while o he
KLM es , i dec eases om 10% o 5%. The ejec ion equency o he GMM-AR es a
ze o is equal o 36% o all iden i ica ion s eng hs.
Co olla y 2shows ha he pseudo- ue alue is no iden i ied when he iden i ica ion
s eng h equals he amoun o misspeci ica ion, ¯
D¯
D=¯μ¯μ. This holds o a alue o
a ound h ee (≈√10)on he “leng h ¯
D” axis, which explains why he powe cu es o
all es s a e la in he opposi e di ec ion a his alue. Fo he DRLM es , he powe cu e
is la wi h a ejec ion equency, which is p o en o be a mos 5%, while o he o he
es s i exceeds 5%, and o he LR and GMM-AR es s e en by a subs an ial amoun .
Figu e 3. Powe o 5% signi icance KLM and DRLM es s o H0:θ∗=0 wi h misspeci ica ion,
¯μ¯μ=10, k =25, =I2.
310 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
Figu e 4. Powe o 5% signi icance LR and GMM-AR es s o H0:θ∗=0 wi h misspeci ica ion,
¯μ¯μ=10, k =25, =I2.
When ¯
D¯
Dis less han ¯μ¯μ, he minimal alue o he popula ion con inuous up-
da ing objec i e unc ion is no longe a θ∗bu a θ1=−1
θ∗.9TheGMM-ARs a is icis
he sample analog o he popula ion con inuous upda ing objec i e unc ion, which ex-
plains why i s powe is minimal a ze o and maximal a in ini y when “leng h ¯
D” exceeds
h ee, so ¯
D¯
D> ¯μ¯μ, and ice e sa when “leng h ¯
D” is less han h ee, so ¯
D¯
D< ¯μ¯μ.In
he la e case, he pseudo- ue alue is no iden i ied because i is a in ini y. Kleibe gen
and Zhan (2024a) p o e ha he pseudo- ue alue o he CUE is no iden i ied when
he popula ion alue o he adi ional ank s a is ic o he Jacobian, which hey label
IS, equals he minimal alue o he CUE popula ion objec i e unc ion, which hey label
MISS and equals he popula ion alue o he J-s a is ic. They also show ha IS −MISS
is always nonnega i e and p o ides a measu e o he iden i ica ion s eng h in misspec-
i ied linea GMM whose sample analog is a (quasi-) likelihood a io no-iden i ica ion
s a is ic. When ¯
D¯
Dis less han ¯μ¯μ,MISS =IS, which u he indica es ha he pseudo-
ue alue is hen no iden i ied.
Figu e 5shows he dis ibu ion unc ion o he misspeci ica ion J-s a is ic, which
equals he minimal alue o he GMM-AR s a is ic when he null hypo hesis holds, so
o alues o θ∗equal o ze o. I shows he dis ibu ion unc ion o h ee di e en alues
o he iden i ica ion s eng h ¯
D¯
D: 0, 10, and 100. Recognizing ha he 95% c i ical alue
o he χ2(24)dis ibu ion, since k −1=24, is abou 36.42, Figu e 5shows ha we ne e
ejec no misspeci ica ion a he 5% signi icance le el when ¯
D¯
Dequals 0, 7% o he
imes when ¯
D¯
D=10, and 33% when ¯
D¯
Dequals 100. This indica es he di icul y o
de ec ing misspeci ica ion; see also Gospodino , Kan, and Robo i (2017).
The powe cu es o he LR and GMM-AR es s in Figu e 4inc ease when mo -
ing away om he hypo hesized alue and when he iden i ica ion s eng h exceeds
9The i s elemen o helimi beha io o hesamplesco ein(31) equals ze o a he wo s a iona y
poin s o he CUE popula ion objec i e unc ion: θ∗and −1
θ∗. The sign o he Hessian also esul s om i
and is indica ed by ¯μ¯μ−¯
D¯
D.Hence,when ¯μ¯μ−¯
D¯
D<0, he s a iona y poin s a θ∗and −1
θ∗a e he min-
imum and maximum, espec i ely, and ice e sa when ¯μ¯μ−¯
D¯
D>0; see Lemma 2(i) in he Supplemen al
Appendix.
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 311
Figu e 5. Dis ibu ion unc ion o he J-s a is ic o misspeci ica ion when θ∗=0 holds, solid
line: ¯
D¯
D=0, dash-do : ¯
D¯
D=10 =s eng h o misspeci ica ion, dashed: ¯
D¯
D=100.
he measu e o misspeci ica ion, ¯
D¯
D> ¯μ¯μ. This howe e does no hold o he powe
cu es o he wo LM es s in Figu e 3, which a e e en ually dec easing. To imp o e he
powe o he DRLM es , we p opose o use a p ope y o i s de i a i e as discussed nex .
3.5 Powe imp o emen wi h m=1
The sco e is equal o ze o a all s a iona y poin s o he CUE sample objec i e unc ion,
so he same holds o es s based on a quad a ic o m o i like, o example, he DRLM
and KLM es s, as well. I explains he powe dec easing away om he null hypo hesis
o he KLM and DRLM es s in Figu e 3. Tes s wi h be e powe p ope ies he e o e ex-
is in co ec ly speci ied GMM ha implici ly o explici ly combine he KLM es wi h an
asymp o ically independen J- es in ei he a condi ional o uncondi ional manne ; see
Mo ei a (2003), Kleibe gen (2005), And ews, Mo ei a, and S ock (2006), And ews (2016),
and And ews and Mikushe a (2016a,2016b). In ou misspeci ied GMM se ing, his is
howe e no possible since he limi ing dis ibu ion o he J-s a is ic is a noncen al χ2
dis ibu ion wi h an unknown noncen ali y pa ame e . Hence, we canno combine his
limi ing dis ibu ion wi h ha o he DRLM s a is ic o ob ain he (condi ional) c i ical
alues o a combina ion es .
To imp o e he powe o he DRLM es , we aim o ejec hypo hesized pseudo- ue
alues o θ, which a e close o a s a iona y poin o he CUE sample objec i e unc ion
o he han he CUE. This would be simila o he condi ional o uncondi ional iden i i-
ca ion obus combina ion es s in egula GMM, which use ha while he KLM es does
no ejec a such alues o θ,J, and/o GMM-AR es s (see Ande son and Rubin (1949)
and S ock and W igh (2000)) likely do. Fo hypo hesized alues o θclose o he CUE,
hese combina ion es s pu mos weigh on he KLM es , bu shi he weigh owa d
he Jand GMM-AR es s when θis close o o he s a iona y poin s; see And ews (2016)
and Kleibe gen (2007). Since he limi ing dis ibu ions o he Jand GMM-AR s a is ics
depend on unknown nuisance pa ame e s in ou misspeci ied GMM se ing, i is no
clea how we can use hese s a is ics o imp o e powe .
312 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
S a iona y poin s o he sample CUE objec i e unc ion lead o ze o alues o he
DRLM s a is ic. The CUE is one o hese s a iona y poin s and leads o he smalles alue
o he objec i e unc ion. To imp o e he powe o he DRLM es , we u he ejec al-
ues o θ ha lead o nonsigni ican alues o he DRLM s a is ic, which esul om s a-
iona y poin s di e en om he CUE. We can do so by ejec ing alues o θin a closed
se o non ejec ed alues ha does no con ain he CUE, o which an illus a i e appli-
ca ion is p esen ed la e in Sec ion 4(see Figu e 7; a closed se o non ejec ed alues is
on he le o Figu e 7as discussed o ha igu e).
To p o e why he abo e powe imp o emen ule leads o a size co ec es p oce-
du e, we use he de i a i e o he DRLM s a is ic (see Lemma 12 in he Supplemen al
Appendix). In line wi h he p e ious Sec ion 3.3, we also ocus on m=1 when discussing
powe imp o emen in Theo em 3below.
Theo em 3. When Assump ions 1and 2hold,m=1, T(θ,X)is linea in θ,and es ing
H0:θ∗=θ∗
0,wi h θ∗ he pseudo- ue alue,a he 100 ×α%signi icance le el, he powe
imp o ed DRLM es ing p ocedu e ha ejec s a pseudo- ue alue o θbo h i :
1. he DRLM s a is ic is signi ican a he 100 ×α%signi icance le el o
2. he DRLM s a is ic is no signi ican bu he hypo hesized pseudo- ue alue o θlies
in a closed se o nonsigni ican alues,which does no con ain he CUE wi h some
signi ican alues in be ween he closed se and he CUE,
is a es p ocedu e o H0:θ∗=θ∗
0wi h size 100 ×α%.
P oo . See he Supplemen al Appendix.
While he gene ic speci ica ion o he DRLM es is o a s a iona y poin o he pop-
ula ion con inuous upda ing objec i e unc ion, he powe imp o ed DRLM es om
Theo em 3explici ly es s o he minimize . La e , Sec ion 4con ains an empi ical ap-
plica ion (see Figu e 7) ha illus a es he powe imp o emen ule om Theo em 3,in
pa icula , i s S ep 2. When compu ing he size o he es a he hypo hesized alue o ,
say, ze o, we he e o e ha e o asce ain ha i is he minimize o he popula ion objec-
i e unc ion. Fo he se up in Figu es 1–2, which uses he limi exp ession o he DRLM
s a is ic in (29), he popula ion minimize is a ze o i he amoun o misspeci ica ion is
less han he s eng h o iden i ica ion so he leng h o ¯μis less han ha o ¯
D. When he
leng h o ¯μexceeds ha o ¯
D, he minimize o he popula ion objec i e unc ion is a
±∞. In s anda d GMM, he e is no misspeci ica ion, so he minimal alue o he popu-
la ion objec i e unc ion is equal o ze o. The amoun o misspeci ica ion is hen always
less han o equal o he iden i ica ion s eng h, so he hypo hesized alue au oma ically
co esponds wi h he minimize o he popula ion objec i e unc ion.
3.6 Simula ed powe o powe imp o ed DRLM
Figu e 6shows he ejec ion equency and powe o he powe imp o ed DRLM es .
Panel 6.1 con ains he ejec ion equency when he minimize o he popula ion con in-
uous upda ing objec i e unc ion equals he hypo hesized alue, which is ze o. I he e-
o e does no show he ejec ion equency o alues whe e he leng h o ¯μexceeds ha
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 313
Figu e 6. Powe imp o ed 5% signi icance DRLM es s o H0:θ∗=0 using a condi ional 95%
c i ical alue as a unc ion o he leng hs o ¯μand ¯
D(panel 6.1), ¯μ¯μ=10 (panel 6.2), k =25,
=I2.
o ¯
D, since he hypo hesized alue does hen no co espond wi h he minimize o he
popula ion objec i e unc ion, which is a ±∞. The ejec ion equencies in panel 6.1
a e compu ed using he calib a ed condi ional c i ical alues. In line wi h Theo em 3,
panel 6.1 shows ha he powe imp o emen does no a ec he size o he DRLM es
when he hypo hesized alue equals he minimize o he con inuous upda ing popula-
ion objec i e unc ion.
Panel 6.2 o Figu e 6shows powe cu es o he powe imp o ed DRLM es . I uses
he same se up as o he powe cu es in Figu es 3and 4.A i s glance, hepowe
imp o ed DRLM es in panel 6.2 seems mino ly-size dis o ed because i s ejec ion e-
quency can each 8% when he leng h o ¯
Dis below 3. A hese alues, he minimize o
he popula ion CUE objec i e unc ion is howe e a ±∞since he misspeci ica ion ex-
ceeds he iden i ica ion s eng h, so he 8% is indica i e o powe and no o size dis o -
ion. When he minimize is a ze o, so he iden i ica ion s eng h exceeds he amoun o
misspeci ica ion, ha is, he leng h o ¯
Dexceeds 3, he ejec ion equency o he DRLM
es is a mos 5% and e eals no size dis o ion. Fo he la e se ing, panel 6.2 also
shows ha he powe cu es a e (almos ) mono onic and he dec ease in powe ha we
obse ed o he DRLM es in panel 3.2 is no longe p esen .
4. Applica ions
To show he impo ance and ele ance o he DRLM es o applied esea ch whe e lin-
ea momen equa ions a e common, we b ie ly e isi he linea models conside ed in
Ad ian, E ula, and Mui (2014)andHe, Kelly, and Manela (2017) using ou DRLM es
and he iden i ica ion obus GMM-AR, KLM, and LR es s; see also Kleibe gen (2009)
and Kleibe gen and Zhan (2020). Fo he linea IV eg ession model, he Supplemen al
Appendix con ains an empi ical s udy using he da a o Ca d (1995).
314 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
Ad ian, E ula, and Mui (2014) p opose a le e age isk ac o (“Le Fac”) o asse
p icing. The le e age le el is he a io o o al asse s o e he di e ence be ween o al
asse s and liabili ies, and he le e age isk ac o equals i s log change. The empi ical
s udy o Ad ian, E ula, and Mui (2014) uses qua e ly da a be ween 1968Q1 and 2009Q4.
Following Le au, Lud igson, and Ma (2019), we ex end he ime pe iod o 1963Q3–
2013Q4 and use N=25 size and book- o-ma ke so ed po olios as es asse s. Ad ian,
E ula, and Mui (2014) show ha he le e age ac o p ices he c oss-sec ion o many es
po olios, as e lec ed by he signi ican Fama–MacBe h (FM) (1973) and Kan–Robo i–
Shanken (KRS) -s a is ics on he isk p emium epo ed in Table 1.TheKRS -s a is ic is
obus o misspeci ica ion bu no o weak iden i ica ion; see Kan, Robo i, and Shanken
(2013).
He, Kelly, and Manela (2017) p opose he banking equi y-capi al a io ac o
(“EqFac”) o asse p icing. We conside one o hei speci ica ions wi h “EqFac”and
he ma ke e u n “Rm” as he wo ac o s. The signi ican FM and KRS -s a is ics o
he isk p emium on “EqFac”inTable1show ha his ac o is conside ed o be p iced
by he es asse s.
DRLM: Ad ian, E ula, and Mui (2014)
Using he same da a as o Table 1, Figu e 7shows he p- alues o es ing he isk p e-
mium on he le e age ac o (ho izon al line) using he DRLM, GMM-AR, KLM, and LR
es s. We also apply he LR no-iden i ica ion es o Kleibe gen and Zhan (2024a), whose
es s a is ic 3.55 is conside ably below i s 95% condi ional c i ical alue o 83.3 ( o con-
di ioning s a is ic equal o 119), so we canno ejec he hypo hesis ha he pseudo-
ue alue is no iden i ied a he 5% (and much la ge ) signi icance le el. Mos o he
p- alues in Figu e 7a e he e o e abo e he 5% le el, which implies ha none o he
DRLM, GMM-AR, KLM, and LR es s leads o igh 95% con idence in e als o he isk
p emium on he le e age ac o as shown in Table 1. Gi en he smallish p- alue o he
J- es , 0.20, and he likely weak iden i ica ion o he isk p emium on he le e age ac o
e lec ed by he unbounded 95% con idence se s, misspeci ica ion could be p esen , so
i would be app op ia e o use he DRLM es .
The p- alues o he DRLM es in Figu e 7a e equal o one a wo di e en poin s.
The p- alues o he GMM-AR es show ha one o hese wo poin s ela es o he min-
imal alue o he GMM-AR es and he o he one o he maximal alue o he GMM-AR
es . Using he powe enhancemen ule o DRLM s a ed in Theo em 3,wecan ejec
nonsigni ican alues esul ing om he DRLM es ha lie wi hin he closed in e al
indica ed by he signi ican maximize o he GMM-AR s a is ic (a ound he le peak in
Figu e 7, which does no con ain he CUE), so he nonsigni ican p- alues o he DRLM
es , which occu a ound he maximize o he GMM-AR es can all be ca ego ized as
signi ican ones acco ding o he powe enhancemen ule. The esul ing 95% con i-
dence se o he DRLM es ejec s a ze o alue o he isk p emium o he le e age
ac o and is epo ed in Table 1alongside he one which esul s om jus applying he
DRLM es . The FM and KRS -s a is ics epo ed in Table 1also ejec a ze o alue o he
isk p emium, bu hese es s a e no eliable because o he po en ially weak iden i ica-
ion o he isk p emium o he le e age ac o and he likely misspeci ica ion e lec ed
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 315
Table 1. In e ence on isk p emia λFin Ad ian, E ula, and Mui (2014)andHe, Kelly, and Manela (2017). The es asse s a e he N=25 size and
book- o-ma ke po olios om 1963Q3 o 2013Q4 aken om Le au, Lud igson, and Ma (2019). “Le Fac” is he le e age ac o o Ad ian, E ula,
and Mui (2014). “EqFac” is he banking equi y-capi al a io ac o o He, Kelly, and Manela (2017). “Rm” is he ma ke e u n. The es ima e o λF
and he FM -s a is ic esul om he Fama–MacBe h (1973) wo-pass p ocedu e. The KRS -s a is ic is based on he KRS - es o Kan, Robo i,
and Shanken (2013). The poin es ima es o λFa e iden ical o hose epo ed in Le au, Lud igson, and Ma (2019). The F-s a is ic ank es is
om Kleibe gen and Zhan (2020) o es ing H0: ank(β)=m−1, wi h m he numbe o isk ac o s. The LR no-iden i ica ion es and 95%
condi ional c i ical alue a e om Kleibe gen and Zhan (2024a).
Ad ian, E ula, and Mui (2014)He, Kelly, and Manela (2017)
Le Fac RmEqFac
Es ima e o λF13.91 1.19 6.88
FM 3.58 0.81 2.14
KRS 2.55 0.77 2.10
CUE o λF51.77 23.22 94.02
95% con idence se
FM (6.29, 21.54) (−1.67, 4.05) (0.57, 13.19)
KRS (3.22, 24.60) (−1.84, 4.22) (0.46, 13.30)
DRLM (−∞,−91.4)∪(−9.2, 1.7)∪(17.8, +∞)(
−∞,+∞)(
−∞,+∞)
DRLM (powe enh.) (−∞,−91.4)∪(17.8, +∞)(
−∞,+∞)(
−∞,+∞)
GMM-AR (−∞,−101.3)∪(18.3, +∞)(
−∞,−64.6)∪(8.1, +∞)(
−∞,−244.1)∪(37.7, +∞)
KLM (−∞,−185.5)∪(−5.7, −0.9)∪(20.8, +∞)(
−∞,−7.2)∪(−4.7, −0.3)∪(1.0, +∞)(
−∞,−23.8)∪(−8.1, 1.7)∪(11.8, +∞)
LR (−∞,−274.2)∪(21.8, +∞)(
−∞,−9.7)∪(2.2, +∞)(
−∞,−33.8)∪(16.2, +∞)
IS χ2-s a is ic (p- alue) 31.97 (0.13) 35.88 (0.04)
Rank F-s a is ic (p- alue) 1.17 (0.28) 1.33 (0.16)
LR no-iden i ica ion (95%) 3.55 (83.3) 0.57 (56.3)
316 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
Figu e 7. Ad ian, E ula, and Mui (2014). p- alue om he DRLM (dashed ed), GMM-AR
(dashed blue), KLM (solid black), LR (dash-do ed g een), and he 5% le el (do ed black).
J-s a is ic (=minimum GMM-AR) equals 28.42, wi h p- alue o 0.20 esul ing om χ2(N−2),
IS =31.97.
by he smallish p- alue o he J- es . Since he ze o isk p emium is ejec ed by DRLM,
he le e age ac o does appea o ha e i s p icing abili y; on he o he hand, since he
con idence se o he isk p emium is wide, we canno p ecisely in e he le e age isk
p emium by using he adop ed da a.
DRLM: He, Kelly, and Manela (2017)
Figu e 8shows he join 95% con idence se s (shaded a eas) o he isk p emia on he
banking equi y-capi al a io ac o “EqFac” and he ma ke e u n “Rm,” om using he
DRLM, GMM-AR, KLM, and LR es s. The p- alue o he J- es shows ha misspeci i-
ca ion is likely p esen , so i is app op ia e o use he DRLM es o he con idence se
o he minimize o he popula ion con inuous upda ing objec i e unc ion. The LR no-
iden i ica ion s a is ic o Kleibe gen and Zhan (2024a), 0.57 is a below i s 95% condi-
ional c i ical alue, 56.3 ( o condi ioning s a is ic equal o 114), so we canno ejec he
hypo hesis ha he pseudo- ue alue is no iden i ied. The 95% con idence se s o he
DRLM and KLM es s consis o wo o h ee a he disjoin se s. The powe enhance-
men ule o he DRLM es shows ha he smalle disjoin closed se can be disca ded
o he join 95% con idence se ha esul s om he DRLM es . The esul ing 95% con i-
dencese om heDRLM es includesaze o alue o he iskp emiumon“EqFac”and
is also unbounded, which indica es ha he p icing abili y o “EqFac”isunde doub .
Gospodino and Robo i (2021) also c i icize he wo- ac o model o He, Kelly, and
Manela (2017). They wa n ha “EqFac”and“Rm” a e closely ela ed, so join ly using
hem could lead o a educed ank o he be a ma ix. To compa e wi h Figu e 8,we e-
place he “EqFac” isk ac o wi h he “SMB” (small minus big) ac o om Fama and
F ench (1993) and simila ly cons uc Figu e 9. The GMM-AR es now indica es mis-
speci ica ion, since i ejec s e e y hypo hesized isk p emia as shown in panel 9.2,
so he 95% con idence se ha esul s om he GMM-AR es is emp y. The LR no-
iden i ica ion s a is ic is now 69 wi h a 5% condi ional c i ical alue o 56.4 ( o con-
di ioning s a is ic equal o 214, see Kleibe gen and Zhan (2024a)), so we ejec he hy-
po hesis ha he pseudo- ue alue is no iden i ied a he 5% signi icance le el. Ou
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 317
Figu e 8. He, Kelly, and Manela (2017). 95% con idence se s om DRLM, GMM-AR, KLM,
and LR. J-s a is ic (minimum o GMM-AR) equals 35.32, wi h p- alue o 0.036 esul ing om
χ2(N−3),IS =35.88.
DRLM es , which allows o misspeci ica ion he e o e yields a igh con idence se in
panel 9.1. This igh con idence se , in con as wi h he wide one in panel 8.1, indica es
ha he p icing abili y o “EqFac” di e s subs an ially om “SMB,” ha is, “SMB” leads
o s onge iden i ica ion o isk p emia han “EqFac.” Because o he misspeci ica ion,
he 95% con idence se s esul ing om he KLM and LR es s a e no ep esen a i e o
he minimize o he popula ion objec i e unc ion.
5. Ex ensions
We nex b ie ly discuss how he DRLM es can be used o sub ec o in e ence. Follow-
ing up, we s a e he DRLM es ha also allows o s ong misspeci ica ion. The ea e ,
we illus a e he size p ope ies o he DRLM es o a se ing wi h nonlinea momen
equa ions and discuss an applica ion o i .
5.1 Sub ec o in e ence
The exp ession o he DRLM s a is ic applies o se ings whe e he s uc u al pa ame e
ec o po en ially has mul iple elemen s, so he pseudo- ue alue θ∗=(θ∗
1θ∗
m)wi h
m≥1. Many imes, p ac i ione s a e in e es ed in cons uc ing con idence se s on a sub-
318 Kleibe gen and Zhan Quan i a i e Economics 16 (2025)
Figu e 9. Rmand SMB. 95% con idence se s om DRLM, GMM-AR, KLM, and LR. J-s a is ic
(minimum o GMM-AR) equals 59.34, wi h p- alue o 0.00 esul ing om χ2(N−3),IS =128.35.
ec o o indi idual elemen o he s uc u al pa ame e ec o . We discuss how o use
he DRLM es o cons uc ing such a con idence se in he Supplemen al Appendix.
5.2 DRLM es o s ong misspeci ica ion
Fo he s ong misspeci ica ion case, ha is, μ (θ∗)is conside ed as a ixed nonze o
ec o ins ead o a d i ing sequence o ze o, he weigh ma ix in ol ed in he DRLM
s a is ic needs o be modi ied. Fo he gene al obus ness o he DRLM es , we ex end
Assump ion 1 o Assump ion 1∗, which conce ns he join limi beha io o he sample
momen , i s de i a i e, and he co a iance ma ix es ima o .
Assump ion 1∗.Fo a alue o θequal o he minimize o he con inuous upda ing
popula ion objec i e unc ion,θ∗,we assume ha he join limi beha io o he sample
momen ,i s de i a i e,and he co a iance ma ix es ima o acco ds wi h he cen al limi
heo em:
√T⎛
⎜
⎝
Tθ∗,X−μ θ∗
ecqTθ∗,X−Jθ∗
echˆ
V θ∗−V θ∗⎞
⎟
⎠→
d⎛
⎜
⎝
ψ θ∗
ψqθ∗
ψ θ∗⎞
⎟
⎠∼N0, Vθ∗, (32)
Quan i a i e Economics 16 (2025) Double obus in e ence o GMM 325
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The eplica ion package o his pape is a ailable a h ps://doi.o g/10.5281/zenodo.14025673.
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