Specification testing for conditional moment restrictions under local identification failure

Do onon, P ospe ; Gospodino , Nikolaj
A icle
Speci ica ion es ing o condi ional momen es ic ions
unde local iden i ica ion ailu e
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: Do onon, P ospe ; Gospodino , Nikolaj (2024) : Speci ica ion es ing o
condi ional momen es ic ions unde local iden i ica ion ailu e, Quan i a i e Economics, ISSN
1759-7331, The Econome ic Socie y, New Ha en, CT, Vol. 15, Iss. 3, pp. 849-891,
h ps://doi.o g/10.3982/QE2242
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Quan i a i e Economics 15 (2024), 849–891 1759-7331/20240849
Speci ica ion es ing o condi ional momen es ic ions unde
local iden i ica ion ailu e
P ospe Do onon
Depa men o Economics, Conco dia Uni e si y, CIREQ, and CIRANO
Nikolay Gospodino
Resea ch Depa men , Fede al Rese e Bank o A lan a
In his pape , we s udy he asymp o ic beha io o speci ica ion es s in con-
di ional momen es ic ion models unde i s -o de local iden i ica ion ailu e
wi h dependen da a. Mo e speci ically, we ob ain condi ions unde which he
con en ional speci ica ion es o condi ional momen es ic ions e ains i s
s anda d no mal limi when i s -o de local iden i ica ion ails bu global iden i i-
ca ion is s ill a ainable. In he p ocess, we de i e some no el in e media e esul s
ha include ex ending he i s - and second-o de local iden i ica ion amewo k
o models de ined by condi ional momen es ic ions, es ablishing he a e o
con e gence o he GMM es ima o and cha ac e izing he asymp o ic ep esen-
a ion o degene a e U-s a is ics unde s ong mixing dependence. Impo an ly,
he speci ica ion es is obus o i s -o de local iden i ica ion ailu e ega dless
o he numbe o di ec ions in which he Jacobian o he condi ional momen e-
s ic ions is degene a e and emains alid e en i he model is i s -o de iden i-
ied.
Keywo ds. GMM, condi ional momen es ic ions, es o o e iden i ying e-
s ic ions, local and global iden i ica ion, i s -o de local iden i ica ion ailu e,
second-o de local iden i ica ion, U-s a is ics, s ong mixing dependence, obus -
ness.
JEL classi ica ion. C01, C1, C14, G12.
1. In oduc ion
While economic models a e designed o be only pa ial and incomple e ep esen a ions
o eal economic phenomena, i is s ill highly desi able o quan i y he deg ee o model
misspeci ica ion and he di ec ions along which he model pe o mance is unsa is ac-
o y. E en i he model is ejec ed by he da a, i can s ill be use ul o policy analysis bu
P ospe Do onon: [email p o ec ed]
Nikolay Gospodino : [email p o ec ed]
We hank ou e e ees and semina and con e ence pa icipan s a LSE, P ince on Uni e si y, UCL, he 2022
mee ing o he Socié é Canadienne de Science économique (Mon eal, Canada), and he 2023 A ica Mee -
ing o he Econome ic Socie y (Nai obi, Kenya) o insigh ul commen s and sugges ions. The i s au ho
acknowledges inancial suppo om he Social Sciences and Humani ies Resea ch Council o Canada. The
iews exp essed he e a e hose o he au ho s and do no necessa ily e lec hose o he Fede al Rese e
Bank o A lan a o he Fede al Rese e Sys em.
©2024 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h p://qeconomics.o g.h ps://doi.o g/10.3982/QE2242
850 Do onon and Gospodino Quan i a i e Economics 15 (2024)
he in e ence and model compa ison p ocedu es should be adjus ed o he unde lying
model unce ain y. Fo hese easons, i is now common p ac ice o i s subjec he can-
dida e models o speci ica ion es ing be o e commi ing o a pa icula analy ical and
in e ence amewo k.
The e a e a leas wo cha ac e is ics o economic models ha make he de elop-
men o ully obus and eliable speci ica ion es ing p ocedu es mo e challenging.
Fi s , economic models a e ypically de ined by a se o condi ional momen es ic ions.
The s anda d app oach is o eso o he law o i e a ed expec a ions and educe he
condi ional es ic ions o uncondi ional momen es ic ions ha a e hen used o de-
sign he p ope es ima ion and es ing amewo k. When his is done in ad hoc manne ,
his app oach could esul in loss o e iciency and e en in inconsis ency o he es ima-
o (see, e.g., Dominguez and Loba o (2004)). On he o he hand, a ans o ma ion ha
p ese es he in o ma ion in he condi ional momen es ic ions leads o modi ied es s
based on a con inuum o momen condi ions (see Bie ens (1982), Bie ens and Plobe ge
(1987), de Jong and Bie ens (1994), Ca asco and Flo ens (2000), Ki amu a, T ipa hi, and
Ahn (2004) among o he s). A common ea u e o all hese es s is ha hey ely on oo -n
consis en es ima o s, which a e eadily a ailable in models ha a e i s -o de locally
iden i ied.
Second, i is o en he case ha he momen es ic ion model is locally unde iden-
i ied. In linea models, o example, he lack o i s -o de local iden i ica ion— ank de-
iciency o he Jacobian ma ix o he momen condi ions—implies global iden i ica ion
ailu e, which ypically ende s he s anda d speci ica ion es s in alid unde bo h he
null and al e na i e hypo heses as he powe o he es , in ce ain con ex s, is bounded
by i s size (Gospodino , Kan, and Robo i (2017)). The in ui ion behind his esul is ha
i is some imes possible o ecas he op imal speci ica ion es as a educed ank es ,
which highligh s he di icul y o de e mining i he educed ank is induced by co ec
speci ica ion o iden i ica ion ailu e. In nonlinea models, howe e , i s -o de iden i i-
ca ion is no longe a necessa y condi ion o global iden i ica ion.
This pape builds on hese wo s ands o li e a u e o ob ain condi ions unde which
he con en ional speci ica ion es s o condi ional momen condi ions emain alid un-
de i s -o de local iden i ica ion ailu e. I should be no ed ha his is no he case in
uncondi ional momen es ic ion models, whe e he second-o de local iden i ica ion
leads o o e ejec ion o he s anda d speci ica ion es (Do onon and Renaul (2013)).
By con as , ou p oposed es is cha ac e ized by obus p ope ies as i p ese es i s
s anda d asymp o ic limi i espec i e o whe he he model is i s -o de o second-
o de iden i ied. Mo e speci ically, he p oposed condi ional speci ica ion es e ains i s
alidi y in he p esence o possible unce ain y abou he pa ame e alues o momen
condi ions ha de e mine i s -o de local iden i ica ion. These obus ness p ope ies
cons i u e he main ad an age o ou app oach and wa an some addi ional ema ks.
Impo an ly, he p oposed es is agnos ic o he p ecise o m o he local iden i ica ion
ailu e, ha is, he deg ee o ank de iciency o he expec ed Jacobian ma ix o he mo-
men condi ions, he alues o he model pa ame e s ha gi e ise o his ank de iciency
o i he Jacobian is exac ly ze o. In he la e case, and i p io knowledge o he ze o Ja-
cobian is a ailable o he esea che , Lee and Liao (2018) p opose o augmen he se
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 851
o momen condi ions wi h his addi ional es ic ion ha es o es he i s -o de local
iden i ica ion and s anda d in e ence. Howe e , i he sou ce o he ank de iciency is
di e en o i s -o de local iden i ica ion holds, imposing hese Jacobian es ic ions
would be in alid and lead o e oneous in e ence. By con as , he implemen a ion and
alidi y o ou p oposed es ob ia es he need o ake a s and on he o m o he ank de-
iciency o he Jacobian ma ix o , mo e gene ally, whe he he model is i s -o de locally
iden i ied o no . When he model is indeed i s -o de locally iden i ied, he condi ional
speci ica ion es con inues o be cha ac e ized by he s anda d no mal limi .
To his end, we o malize he concep s o poin iden i ica ion and i s -o de lo-
cal iden i ica ion ailu e in condi ional momen es ic ion models. Simila o poin -
iden i ied uncondi ional models, he i s -o de local iden i ica ion ailu e allows only
o a educed numbe o di ec ions o he pa ame e ec o o be iden i ied while iden-
i ica ion o he emaining di ec ions is ob ained ia a second-o de expansion o he
momen condi ions. We hen p oceed wi h cha ac e izing he limi ing beha io o he
es ima o and he speci ica ion es in models wi h an expanding se o momen condi-
ions when i s -o de local iden i ica ion ails bu global iden i ica ion is s ill a ainable.
Ou main con ibu ions can be summa ized as ollows. Fi s , we ex end he es o
alidi y o condi ional momen es ic ions (de Jong and Bie ens (1994); Donald, Im-
bens, and Newey (2003)) o momen condi ion models ha a e i s -o de degene a e.
We es ablish ou esul s in a wo-s ep gene alized me hod o momen s (GMM) ame-
wo k wi h gene al o ms o momen condi ion models and dependen da a. While he
es p oposed by de Jong and Bie ens (1994) is ob ained in he con ex o nonlinea e-
g ession models, he es by Donald, Imbens, and Newey (2003) is also de eloped wi hin
he GMM amewo k o a speci ic choice o basis unc ions and c oss-sec ional da a.
Ou esul s he e o e show ha he GMM-based es o Donald, Imbens, and Newey
(2003) e ains i s size con ol wi h dependen da a and unde i s -o de local iden i ica-
ion ailu e so long as second-o de local iden i ica ion holds. We should no e ha he
ex ension o dependen da a and cha ac e izing he limi ing beha io o he GMM es i-
ma o and he speci ica ion es in his con ex is non i ial. We ou line he condi ions
unde which he speci ica ion es wi h an inc easing numbe o uncondi ional momen
es ic ions is obus o he ype o singula i y a ising om i s -o de local iden i ica-
ion ailu e. Mo e speci ically, we ex end he no ion o second-o de local iden i ica ion
o he se ing o models de ined by condi ional momen es ic ions. The limi ing be-
ha io o he GMM es ima o and he speci ica ion es a e s udied in he se up whe e
poin iden i ica ion holds, i s -o de local iden i ica ion ails while local iden i ica ion
is main ained a second o de .
We show ha he GMM es ima o , based on he expanding momen es ic ions,
es ima es he di ec ions o he pa ame e s ha a e locally i s -o de iden i ied a he
s anda d √n- a e while he emaining di ec ions a e es ima ed a a slowe a e. In e es -
ingly, his a e is as e han he n1/4- a e in second-o de iden i ied models wi h a ixed
numbe o momen es ic ions (Do onon and Renaul (2020)). In he condi ional se -
ing, he expanding numbe o momen es ic ions enhances he iden i ica ion signal
and accele a es he a e o con e gence. We also de i e he asymp o ic dis ibu ion o
852 Do onon and Gospodino Quan i a i e Economics 15 (2024)
he GMM es ima o in he scala case, which highligh s he highly nons anda d limi -
ing beha io o he es ima o . Despi e his nons anda d asymp o ic se up, we show ha
he es o alidi y o condi ional momen es ic ions is cha ac e ized by a s anda d
no mal limi e en when he i s -o de local iden i ica ion condi ion is comp omised.
Ano he impo an in e media e esul ha we de elop in he pape is a cen al limi
heo em (CLT) o degene a e U-s a is ics wi h linea ke nels o inc easing dimension
unde s ong mixing dependence. The CLT is no el and o independen in e es . Es ab-
lishing he asymp o ic no mali y o he es o o e iden i ying es ic ions d aws hea ily
on his CLT.
The es o he pape is s uc u ed as ollows. Sec ion 2in oduces he main condi-
ional momen es ic ion se up and he es ing amewo k. I also p esen s he no ions
o i s - and second-o de local iden i ica ion along wi h al e na i e cha ac e iza ions in
he con ex o condi ional momen es ic ions. In o de o enhance he in ui ion behind
he iden i ica ion amewo k, Sec ion 3discusses he model wi h common condi ionally
he e oskedas ic ea u es, which is la e explo ed u he in simula ions and in he em-
pi ical applica ion. This example also allows us o highligh he obus ness o ou es ing
app oach o knowledge abou he p ecise s uc u e o he model. The asymp o ic p op-
e ies o he GMM es ima o a e analyzed in Sec ion 4.Sec ion5p oposes a CLT o U-
s a is ics unde s ong mixing dependence and es ablishes he asymp o ic no mali y o
he speci ica ion es s a is ic unde he null hypo hesis. In addi ion, his sec ion shows
ha his es is consis en agains all al e na i es. Sec ion 6 epo s simula ion esul s o
he p oposed speci ica ion es and p o ides an empi ical applica ion o he p esence
o common condi ionally he e oskedas ic ea u es in po olio bond e u ns. Sec ion 7
concludes. P oo s and addi ional esul s a e p o ided in Appendices A and B, and he
Supplemen al Ma e ial (Do onon and Gospodino (2024)).
Th oughou he pape , we use he ollowing no a ion. Fo any ma ix C,C2=
√λmax(CC)deno es he spec al no m, whe e λmax(·)is he la ges eigen alue unc ion.
I Cis a ec o , his amoun s o i s Euclidean no m as well. Also, le λmin(·)deno e he
smalles eigen alue unc ion, and Z,N,andRmsigni y he se o all in ege s, he se o
na u al numbe s, and he se o eal m×1 ec o s, espec i ely. Fu he mo e, Ca d(S)
deno es he ca dinali y o a ini e se S, de ined o be he numbe o elemen s in he se
S, ec(C)signi ies column ec o iza ion o a ma ix C,a∨bdeno es he maximum o a
and b,Rank(C)is he ank o a ma ix C,andDiag(c11,c22,,cmm )deno es an m×m
diagonal ma ix wi h (c11,c22,,cmm )on i s main diagonal. Con e gence in p obabil-
i y and con e gence in dis ibu ion a e deno ed by P
→and d
→, espec i ely, while he
abb e ia ion a.s. s ands o almos su ely. Le {X : ∈Z}be a sequence o andom a i-
ables and Fb
abe he σ-algeb a gene a ed by {X :−∞≤a≤ ≤b≤∞}.Then{X }is said
o be s ong mixing o α-mixing (And ews (1984)) i
sup
−∞< <∞
sup
A∈F
−∞,B∈F∞
+sP (A∩B)−P (A)P (B)=α(s)→0ass→∞.
Finally, an=oP(1)deno es ha he sequence an ends o ze o in p obabili y and an=
OP(1)signi ies ha anis bounded in p obabili y.

Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 853
2. Model and iden i ica ion
2.1 Condi ional momen es ic ion se up
In his pape , we conside a single condi ional momen es ic ion model:
Eu(y ,θ0)|x =0a.s., (1)
whe e uis a eal- alued unc ion, θ0∈⊂Rpis he pa ame e o in e es , and {(x ,y )}
is a sequence o Rkx×Rky- alued andom ec o s. Many economic equilib ium models
ake his condi ional momen es ic ion o m. A p ominen example o he ole o con-
di ioning is he s ochas ic discoun ac o amewo k in asse p icing (see, o ins ance,
Hansen (2014)).1In his se up, he null hypo hesis o alidi y o he condi ional momen
es ic ion in (1)is
H0:P Eu(y ,θ0)|x =0=1(2)
agains he al e na i e
H1:P Eu(y ,θ)|x =0<1, o any θ∈.(3)
While he single condi ional es ic ion se up co e s a wide ange o p ac ically ele an
models, we ocus on his case me ely o he sake o no a ional simplici y. The main
esul s in his pape ca y o e o highe -dimensional condi ional momen es ic ions
a he cos o mo e cumbe some no a ion.
Consis en es ima ion o θ0using (1) equi es poin iden i ica ion, ha is, o all θ∈
,
ρ(x ,θ):=Eu(y ,θ)|x =0, a.s. ⇔θ=θ0.(4)
Mo eo e , in e ence abou θ0hinges on he sha pness o he slope o he unc ion θ→
ρ(x,θ)a θ0. The local beha io o his unc ion de e mines he a e o con e gence o
he es ima o o θ0. The s anda d app oach o in e ence elies on a local iden i ica ion
condi ion, which s a es ha
Eρθ(x,θ0)ρθ(x,θ0)is nonsingula , (5)
whe e ρθ(x,θ):=E(∇θu(y,θ)|x)wi h ∇θu(y,θ0)=∂u(y ,θ)/∂θ|θ=θ0. Following he li -
e a u e on uncondi ional momen es ic ion models (Sa gan (1983); Do onon and Re-
naul (2013); Do onon and Hall (2018); among o he s), we shall e e o his condi ion
as i s -o de local iden i ica ion condi ion o condi ional momen es ic ion models.
This connec ion be ween he iden i ica ion se ups o uncondi ional and condi ional
es ic ion models is o malized in he nex subsec ion. As poin ed ou in he In oduc-
ion, his pape conside s a amewo k whe e he condi ional momen model is poin
iden i ied bu he e is a ailu e o he i s -o de local iden i ica ion condi ion.
1Fo a comp ehensi e ecen discussion o hese issues in he con ex o asse p icing models, we e e
he eade o An oine, P oulx, and Renaul (2020).
854 Do onon and Gospodino Quan i a i e Economics 15 (2024)
The es o ou main analy ical and es ing amewo k can be summa ized as ol-
lows. Conside he sepa able Hilbe space L2(P):=L2(Rkx,B(Rkx),P)o squa e P-
in eg able eal- alued unc ions de ined on Rkx,whe ePis he common p obabili y dis-
ibu ion o x ’s— ha a e assumed o be s a iona y—and B(Rkx)is he Bo el σ-algeb a
o Rkx.WeequipL2(P)wi h he inne p oduc :  (·),h(·)=E( (x )g(x )).Le (gl)l∈N
be a coun able basis (no necessa ily o hono mal) o L2(P)and g(k):=(g1,,gk).2
We in es iga e he null hypo hesis in (2) by p oposing a es o he sequence o uncon-
di ional momen es ic ions
Egl(x )u(y ,θ0)=0, l=1, ,k; =1, ,n,
o w i en mo e compac ly as
Eg(k)(x )u(y ,θ0)=0, (6)
whe e k=k(n)wi h k(n)→∞as n→∞. The GMM es ima o , based on hese es ic-
ions, is de ined as
ˆ
θ=a gmin
θ∈¯
k(θ)ˆ
Wk¯
k(θ),(7)
whe e
¯
k(θ)=1
√n
n

=1
k(x ,y ,θ), k(x ,y ,θ)=g(k)(x )u(y ,θ),
and ˆ
Wkis a sequence o symme ic, posi i e de ini e weigh ing ma ices. The speci ica-
ion es ha we in oduce nex is exp essed as a unc ion o he wo-s ep GMM es ima-
o , which uses he weigh ing ma ix
ˆ
Wk=ˆ
V−1
k,ˆ
Vk=1
n
n

=1
k(x ,y ,˜
θ) k(x ,y ,˜
θ).(8)
The p elimina y ( i s -s ep) GMM es ima o ˜
θused in (8) is ob ained by commonly se -
ing ˆ
Wk=Iko , mo e gene ally, o a non andom ma ix sequence Wk,0. Fu he mo e, we
will de i e ou esul s unde he condi ion ha he sequence ( k(x ,y ,θ0)) is se ially
unco ela ed. This is ensu ed by ou main ained assump ion ha
Eu(y ,θ0)|F =0, whe e F =σx ,u(y −1,θ0),x −1,u(y −2,θ0),.(9)
2To enhance powe , he sequence o unc ions (gl)lis chosen as an enume a ion o some se ies expan-
sion ha does no depend on θ. Fu he mo e, as discussed by de Jong and Bie ens (1994), he condi ioning
andom a iable xcanbeconside ed obeboundedsince
E(Y|x)=EY|(x)
o any one- o-one unc ion  ha maps Rkxin o a compac subse D⊂Rkx.In ha espec ,i x
is no ini ially a bounded andom a iable, one can conside g((x)), ins ead o g(x). Examples o
bounded ans o ma ions (·)include he componen wise a c angen unc ion, ha is, x→ a c an(x)=
(a c an(x1),,a c an(xkx)), while choices o enume a ion o weigh unc ions (gl)linclude polynomial,
igonome ic, and lexible Fou ie o m amilies (de Jong and Bie ens (1994); see also And ews (1991), and
Gallan (1981), o mo e de ails on hese amilies).
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 855
In ac , unde his condi ion and o any k,( k(x ,y ,θ0)) is a ma ingale di e ence se-
quence wi h espec o i s na u al il a ion. The weigh ing ma ix o he wo-s ep GMM
es ima o is hus cons uc ed as he in e se o ˆ
Vk,whe e ˆ
Vkis a sum o ou e p oduc o
k(x ,y ,˜
θ)as de ined by (8).
Ou goal is o de i e he asymp o ic dis ibu ion o he es s a is ic o he null hy-
po hesis in (2)
ˆ
Z=1
√2k¯
k(ˆ
θ)ˆ
V−1
k¯
k(ˆ
θ)−k(10)
unde i s -o de local iden i ica ion ailu e. Cha ac e izing he limi ing dis ibu ion o
ˆ
Z equi es ha we de e mine he limi ing beha io o ˆ
θunde (i) an expanding se o
momen condi ions (k(n)→∞as n→∞) and (ii) second-o de local iden i ica ion.
We show ha e en in his highly nons anda d iden i ica ion se up, he es ˆ
Z e ains
i s N(0, 1)limi unde H0in (2)—which is he asymp o ic dis ibu ion o he es in he
s anda d iden i ica ion se ing (de Jong and Bie ens (1994))—and is consis en unde H1
in (3).
2.2 Iden i ica ion
No e ha , o any kand any ec o o ins umen s z =g(x )∈Rk, a unc ion o x , he
condi ional momen es ic ion in (1) implies he uncondi ional momen es ic ion:
Ez ·u(y ,θ0)=0. (11)
Following Sa gan (1983)andDo onon and Renaul (2013), among o he s, he uncondi-
ional momen es ic ion (11) locally iden i ies θ0a i s o de i
RankEz ·∇θu(y ,θ0)=p, (12)
whe eas lack o i s -o de local iden i ica ion occu s when
RankEz ·∇θu(y ,θ0)<p. (13)
The e o e, i is easonable o conjec u e ha he condi ional momen es ic ion (1)
iden i ies θ0locally a i s o de i and only i he e exis s a se o ins umen s zisuch
ha (12) holds. Rela edly, i s -o de local iden i ica ion ails i and only i (13)holds e-
ga dless o he choice o ins umen s. The ollowing p oposi ion desc ibes his p ope y
in e ms o degene acy o he expec ed Jacobian o u(y ,θ)a θ0.
P oposi ion 2.1. The ollowing wo s a emen s a e equi alen :
(i) Fo any kand any Rk- alued measu able unc ion g,Rank(E(z ·∇θu(y ,θ0))) <p,
whe e z =g(x )and assuming ha he momen exis s.
(ii) The e exis s a leas one linea combina ion o he elemen s o E(∇θu(y ,θ0)|x ) ha
is almos su ely nil.3
3No e ha , when uis a q- ec o , “·” in pa (i) should be eplaced by he K onecke p oduc “⊗” and
“elemen s” in pa (ii) should be eplaced by “columns,” wi h he unde s anding ha ∇θu(y ,θ0)is a (q,p)-
ma ix.
856 Do onon and Gospodino Quan i a i e Economics 15 (2024)
A p oo o an al e na i e o mula ion o his p oposi ion (P oposi ion A.1(ii))isp o-
ided in Appendix B. The cha ac e iza ion in (ii) alida es (5) as he i s -o de local iden-
i ica ion condi ion in model (1). Fu he mo e, his highligh s some simila i ies wi h
he i s -o de local iden i ica ion ailu e in pa ame ic models as s udied by Lee and
Cheshe (1986) and Ro ni zky, Cox, Bo ai, and Robins (2000). In his se ing, i s -o de
local iden i ica ion ailu e amoun s o linea dependence o he elemen s o he sco e
unc ion o he model, e alua ed a he ue pa ame e alue.
Wi h he basis unc ions g(k)(x)as de ined in Sec ion 2.1, P oposi ion A.1 in Ap-
pendix Aes ablishes he connec ion be ween poin iden i ica ion ( esp., i s -o de local
iden i ica ion ailu e) in condi ional momen models such as (1) wi h poin iden i ica-
ion ( esp., i s -o de local iden i ica ion ailu e) in hei co esponding sequences o
uncondi ional momen models gi en by (6). While poin iden i ica ion by (1) amoun s o
poin iden i ica ion by (6) o somek0, he ac ha g(k)(x)is an inc easingly embedded
sequence o ec o s allows us o claim ha (1) ails i s -o de local iden i ica ion i and
only i he e exis s such ha , o any kla ge enough (say k≥k0), Rank(G(k))= <p,
whe e G(k)is he expec ed Jacobian ma ix:
G(k):=Eg(k)(x)·∇θu(y,θ0).
By cons uc ion, he null space o G(k)and he ange o i s anspose a e ixed o any
kla ge enough. This s abili y o ange and null space will be key o second-o de local
iden i ica ion ha will be imposed on he momen es ic ions in o de o cha ac e -
ize he limi ing beha io o es ima o s and speci ica ion es s. Indeed, he main con-
sequence o i s -o de local iden i ica ion ailu e, while global iden i ica ion holds, is
ha only a ce ain numbe ( <p) o di ec ions o he pa ame e ec o a e iden i ied
h ough i s -o de expansions o he momen unc ion.
Nex , ollowing Do onon and Hall (2018)andDo onon and Renaul (2020), we ocus
on con igu a ions ha allow he iden i ica ion o he emaining di ec ions ia a second-
o de expansion. Le k≥k0such ha Rank(G(k))= <p,R1be a (p, )-ma ix wi h
columns spanning he ange o G(k),andR2deno e a (p,p− )-ma ix wi h columns
spanning he null space o G(k). We say ha he momen es ic ion (1) iden i ies θ0a
second o de i , o all a∈R and b∈Rp− ,weha e
4
G(k)R1a+bR
2Eg(k)
l(x)∇θθu(y,θ0)R2b1≤l≤k=0⇔(a,b)=(0, 0), (14)
whe e ∇θθu(y,θ0):=(∂2/∂θ∂θ)u(y,θ), e alua ed a θ0.
Le ing M(k)be he ma ix o o hogonal p ojec ion on he null space o G(k)(o
equi alen ly he o hogonal o he column span o G(k)), Co olla y 2.3 o Do onon and
Renaul (2020) ensu es ha (14) is equi alen o he exis ence o γk>0such ha , o any
b∈Rp− ,

M(k)bR
2Eg(k)
l(x)∇θθu(y,θ0)R2b1≤l≤k
2≥√γkb2
2,
wi h γk=in
b2=1
M(k)bR
2Eg(k)
l(x)∇θθu(y,θ0)R2b1≤l≤k

2
2. (15)
4F om ou discussion abo e, i (14)holds o agi enk,i holds o allk≥k.
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 863
No e ha D1and H(k)a e nonze o ma ices and he condi ion on hei spec al
no ms in Assump ion 4(ii) ollows i each has a leas one column wi h a numbe o
nonze o elemen s ha is p opo ional o k. The magni ude o γk ollows om he ac
ha (a) i is a nondec easing sequence in k,and(b)i iso o de OP(H(k)2
2).The e-
qui emen ha he a io o he ex eme eigen alues o D
1D1be bounded p e en s his
nonsingula ma ix om being ill-condi ioned. The condi ion on he (k,p− )-ma ix
¯
D2is no pa icula ly es ic i e since each componen o his ma ix is OP(1)by i ue
o he cen al limi heo em.
In Assump ion 4(iii), he o de o magni ude o Sk ollows i λmax(W1/2
kVkW1/2
k)≤<
∞, which is he case, o example, i Vkand Wkha e bounded eigen alues o i Wk=V−1
k.
To see his, no e ha i ¯
λkis bounded, hen o any uni ec o c,weha e
cVa (Sk)c=γ−1
kcH(k)W1/2
kM(k)W1/2
kVkW1/2
kM(k)W1/2
kH(k)c
≤γ−1
kcH(k)W1/2
kM(k)W1/2
kH(k)c≤λmax(Wk)γ−1
k
H(k)

2
2=O(1),
which is su icien o claim ha Sk=OP(1)since E(Sk)=0.
Las ly, Assump ion 4(i ) imposes ha he eigen alues o Wka e bounded and ˆ
Wk
is su icien ly close o Wkas ng ows. No e ha hese wo condi ions a e ul illed by
he GMM es ima o wi h non andom ma ix ha ing bounded eigen alues such as Wk,0.
These condi ions a e also sa is ied o he wo-s ep GMM es ima o as we show in Ap-
pendix Bin he con ex o he a e-o -con e gence esul s in he nex subsec ion.
4.2 Limi ing beha io o he GMM es ima o
Gi en he se o assump ions s a ed abo e, we now p oceed o es ablishing he a e o
con e gence o he GMM es ima o which, in u n, will be use ul o cha ac e ize he
asymp o ic dis ibu ion o he speci ica ion es . In he s anda d case o a ixed numbe
o momen es ic ions (i.e., kis ixed), he GMM es ima o is known o con e ge a a
sha p a e o n1/4al hough a as e a e in some egions o he sample space is possible
(Do onon and Renaul (2013)). This mix u e o a es is essen ial o de i ing he asymp-
o ic dis ibu ion o he GMM o e iden i ica ion es s a is ic as a mix u e o chi-squa ed
andom a iables. We show a simila a e beha io o he GMM es ima o in he cu en
con ex unde local iden i ica ion ailu e al hough he o iginal a e needs o be adjus ed
in o de o e lec he inc easing numbe o momen es ic ions.
The nex heo em s a es he a e o con e gence o he pa ame e ec o . Recall ha
R1deno es a (p, )-ma ix wi h columns spanning he ange o G(k)and R2is a (p,p−
)-ma ix wi h columns spanning he null space o G(k),whe eRank
(G(k))= <p.
Theo em 4.1. I Assump ions 1–4hold and k→∞as n→∞wi h k3/n →0, hen
ˆ
θ−θ02=OPγ−1/4
kn−1/4,
R
1(ˆ
θ−θ0)
2=OPn−1/2,and

R
2(ˆ
θ−θ0)
2=OPγ−1/4
kn−1/4.

864 Do onon and Gospodino Quan i a i e Economics 15 (2024)
Theo em 4.1 es ablishes ha each o he componen s o he GMM es ima o con-
e ges a leas a a nons anda d a e o γ1/4
kn1/4while he s anda d √n- a e o con e -
gence is possible in some di ec ions. Mo e speci ically, he di ec ions o he pa ame e
ec o ha a e iden i ied a i s o de a e √n-con e gen while he di ec ions ha a e
second-o de locally iden i ied con e ge a a slowe , γ1/4
kn1/4∼k1/4n1/4, a e.In e es -
ingly, his a e is as e han he esul in Do onon and Renaul (2020) who ob ain, in a
con igu a ion o ixed numbe o momen es ic ions, a slowe a e n1/4 o he di ec-
ions iden i ied a second o de . The as e a e in ou con ex is essen ially due o o he
inc eased in o ma ion b ough by he g owing numbe o momen es ic ions.
This inding bea s some simila i ies o Han and Phillips (2006) who show, in he con-
ex o weak ins umen s, ha he GMM es ima o may be consis en i he numbe o
ins umen s is allowed o inc ease wi h he sample size (see also Chao and Swanson
(2005), among o he s). The in ui ion behind his esul is ha he expanding numbe o
momen condi ions, i g owing a an app op ia e a e wi h he sample size, enhances
he iden i ica ion signal and ende s a consis en es ima o (in a loca ion model) e en
wi h possibly i ele an ins umen s. In ou amewo k, poin iden i ica ion is main-
ained and consis en es ima ion is he e o e possible e en i he numbe o momen
es ic ions does no g ow. Bu , as Theo em 4.1 shows, he second-o de local iden i-
ica ion also eaps impo an bene i s om he expanding se o momen es ic ions
as he second-o de iden i ied pa ame e s can be es ima ed a a as e a e. I is wo h
men ioning ha since achie ing consis en es ima ion equi es he numbe o momen
es ic ions o g ow a a slowe a e han he sample size, i will no be possible o ac-
cele a e he con e gence a e o second-o de iden i ied di ec ions o he pa ame ic
√n- a e.
Al hough he a es o con e gence ha a e s a ed in Theo em 4.1 a e su icien o de-
i e he asymp o ic dis ibu ion o he speci ica ion es , i is in e es ing o u he in es-
iga e he la ge sample p ope ies o he GMM es ima o s. Un o una ely, cha ac e izing
he asymp o ic dis ibu ion o he GMM es ima o ˆ
θin he gene al case p o es di icul .
Fo his eason, we es ic ou a en ion o he simples case o single pa ame e (p=1)
models wi h a second-o de local iden i ica ion p ope y.
Theo em 4.2. Suppose ha p=1and Assump ions 1–4hold.In addi ion,i k→∞as
n→∞wi h k4/n →0, and γ−1/2
kH(k)Wk¯
k(θ0)d
→Z:=N(0, σ2) o some σ2>0, hen
√γkn(ˆ
θ−θ0)2d
−→ 1{Z≥0}(2Z).
Theo em 4.2 i s demons a es ha he slow a e o con e gence de i ed in The-
o em 4.1 is, in ac , sha p meaning ha wi hin he assumed model and iden i ica ion
amewo k, he es ima o canno con e ge a a as e a e. Fu he mo e, he asymp o ic
dis ibu ion in Theo em 4.2 can be eadily used o conduc in e ence abou he ue
pa ame e alue θ0by eplacing γkwi h i s sample coun e pa . No e ha his nons an-
da d asymp o ic dis ibu ion wi h an a om mass o 1/2 a he o igin is simila o he one
de i ed by Do onon and Hall (2018) o a ixedk. As poin ed ou abo e, he cha ac e -
iza ion o he asymp o ic dis ibu ion in he gene al case o p>1 appea s o be qui e
in ol ed and is beyond he scope o his pape .
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 865
5. Asymp o ic dis ibu ion o he speci ica ion es
The cha ac e iza ion o he asymp o ic dis ibu ion o ou speci ica ion es s a is ic e-
qui es a cen al limi heo em o degene a e U-s a is ics wi h a linea ke nel o he o m
hn(x ,xs):= 
k(x )V−1
k k(xs),whe e(x ) ∈Zis a s a iona y and s ong mixing p ocess
and ( k(x )) ∈Zis a ma ingale di e ence sequence wi h espec o i s na u al il a ion.
Mo e speci ically, we a e in e es ed in he asymp o ic dis ibu ion o U-s a is ics o he
o m:
Un=1
n
=s
k(x )V−1
k k(xs)
√k, (21)
whe e Vk:=Va ( k(x )).Thedegene acyo Una ises om he ac ha
hn(x,y)dF(y)=0 o allx,wi hFdeno ing he ma ginal dis ibu ion o x . While
he asymp o ic heo y o degene a e U-s a is ics has been ex ensi ely s udied in he
li e a u e (see he Supplemen al Ma e ial), he a ailable esul s a e no well aligned wi h
ou amewo k, which ea u es an inne p oduc wi h an inc easing dimension. Fo his
eason, we de elop a new CLT ha is adap ed o he o m o he U-s a is ic in (21). Since
his esul may be o independen in e es , we collec he su icien condi ions o es ab-
lishing he CLT in he ollowing assump ions.
Assump ion-cl 1. Assume ha (x ) ∈Zis s a iona y and geome ic s ong mixing p o-
cess, k(x )is an Rk- alued measu able unc ion o x such ha he sequence ( k(x )) ∈Z
is a ma ingale di e ence wi h espec o he σ-algeb a σ( k(xs):s≤ ).
Assump ion-cl 2. Assume ha k∼nα o some α∈(0, 1)and he e exis s >0, such
ha
sup
k∈N
1
k
k

h=1
EV−1/2
k k(x )h
4+<∞,
whe e [a]his he h- h elemen o he ec o a.
Assump ion-cl 3. Fo some β≥0,
Emax
1≤ ≤n
V−1/2
k k(x )
/√k=Ologβnand
Emax
1≤ =s≤n k(x )V−1
k k(xs)/√k=Ologβn.
S a iona i y and mixing o (x ) ∈Zis al eady assumed abo e (see Assump ion 1)and
is es a ed in Assump ion-cl 1 o ensu e ha he esul s in P oposi ion S.2 in he Sup-
plemen al Ma e ial and Theo em 5.1 below, which could be o independen in e es , a e
sel -con ained. Assump ion-cl 2is used o ob ain he limi a iance o Unbecause i s
de i a ion equi es dealing wi h ou h-o de momen s o k(x ). Replacing hese mo-
men s by hei analogues unde independence is a common app oach in he li e a u e.
The emainde is hen con olled by eso ing o Lemma S.1 in he Supplemen al Ma e-
ial, due o Roussas and Ioannides (1987), which can be applied i he condi ion on he
866 Do onon and Gospodino Quan i a i e Economics 15 (2024)
momen s in Assump ion-cl 2is sa is ied. No e ha his condi ion is no pa icula ly
es ic i e. I imposes he exis ence o momen s o o de highe han he ou h o he
no malized componen s o k(x ). The boundedness o he a e age o hese momen s
means ha no componen domina es he o he s in e ms o hese momen s.
The i s bound in Assump ion-cl 3is no es ic i e as i holds wi h β=1p o ided
ha he momen gene a ing unc ion o z :=V−1/2
k k(x )2/√kexis s. This holds e-
ga dless o he dependence s uc u e.6The second bound in Assump ion-cl 3is no
oo es ic i e ei he . I k(x )and k(xs)a e independen , hen E[ k(x )V−1
k k(xs)/
√k]2=1so ha | k(x )V−1
k k(xs)|/√k=OP(1)and, as be o e, we can claim ha he
s a ed bound accommoda es a la ge class o p ocesses.
We a e now eady o s a e he ollowing CLT o he scaled U-s a is ic in (21).
Theo em 5.1. Unde Assump ions-cl 1,2,and 3,
Un
√2
d
−→ N(0, 1).
The p oo o Theo em 5.1 ollows simila a gumen s as in Kim, Luo, and Kim (2011)
and is p o ided in he Supplemen al Ma e ial. We es ablish his CLT by showing ha
he momen s o Uncon e ge o hose o he no mal dis ibu ion. Unde Assump ion-
cl 3, we show ha he summands o Una e essen ially bounded by a slowly inc easing
unc ion o he sample size, which u ns ou o be essen ial o con olling he di e ence
be ween he momen s Unand hose o i s Gaussian limi .
Building on his cen al limi heo em, we now cha ac e ize he asymp o ic dis ibu-
ion o he speci ica ion es s a is ic ˆ
Z=1
√2k(¯
k(x ,y ,ˆ
θ)ˆ
V−1
k¯
k(x ,y ,ˆ
θ)−k)unde he
null hypo hesis ha he condi ional momen es ic ion (1)isco ec lyspeci ied.
Theo em 5.2. Suppose ha Assump ions 1,2,3(i), 4(i,ii), A.1(ii,iii,i ), and
Assump ions-cl 2–3wi h k(x):= k(x,y,θ0),hold.Also,assume ha k=o(n1/5)and
Wk,0 has bounded eigen alues.Then,as n→∞,
ˆ
Zd
−→ N(0, 1).
Se e al ema ks a e wa an ed ega ding he esul in Theo em 5.2.Fi s ,i isim-
po an o unde sco e ha he s anda d no mal limi dis ibu ion in Theo em 5.2 is ob-
ained in a highly non-s anda d se ing. In pa icula , we ha e a lack o i s -o de local
iden i ica ion which, as discussed ea lie , gi es ise o nons anda d limi ing beha io o
he GMM es ima o . The second-o de local iden i ica ion, in conjunc ion wi h he ex-
panding se o momen condi ions, ensu es he consis ency o he es ima o and de e -
mines i s a e o con e gence. The condi ions o he consis ency o he wo-s ep GMM
es ima o a e collec ed in Assump ion A.1 in Appendix Aand a e used in es ablishing he
6Fo ins ance, β=1i z has a Gamma dis ibu ion, and β=1/2i z is Gaussian. I is wo h no ing ha
z =OP(1)since E(z2
)=1. I z ’s a e i.i.d. wi h common dis ibu ion F, i is known ha his bound holds
o a la ge class o Fbu ules ou hose wi h Pa e ian ail (Pe ei a (1983); see also Be man (1964) and Isae ,
Rodiono , Zhang, and Zhuko skii (2020) o simila esul s o ime-dependen p ocesses).
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 867
limi in Theo em 5.2. While he ˆ
Z es s a is ic is based on he wo-s ep GMM es ima o
wi h ˆ
Wk=[1
nn
=1 k(x ,y ,˜
θ) k(x ,y ,˜
θ)]−1, s a ing explici ly ha Wk,0 has bounded
eigen alues allows us o in oke Assump ion 4(iii) in o de o ensu e he desi ed a e o
con e gence o he p elimina y GMM es ima o ˜
θ. (See Rema k 1in Appendix B.) Also,
as discussed ea lie , Assump ion 4(iii) holds p o ided ha λmax(W1/2
kVkW1/2
k)≤<∞,
which is i ially sa is ied by he wo-s ep GMM es ima o ha se s Wk=V−1
k.
In he con en ional amewo k whe e he condi ional model is poin iden i ied, he
p ope ly ecen e ed and s anda dized speci ica ion es wi h an inc easing numbe o
momen condi ions con e ges, unde some egula i y condi ions, o a s anda d no -
mal limi (see, e.g., Ca asco and Flo ens (2000); Donald, Imbens, and Newey (2003);
T ipa hi and Ki amu a (2003); among o he s). Theo em 5.2 es ablishes ha he s an-
da d no mal dis ibu ion con inues o be he co ec limi o he ˆ
Z es s a is ic unde
he null o co ec speci ica ion, p o ided ha k=o(n1/5)as n→∞. Unlike he egu-
la se up, his limi is ob ained wi hin he second-o de local iden i ica ion amewo k
in Assump ion 2, which is cha ac e ized by i s -o de local iden i ica ion ailu e. In u-
i i ely, his is achie ed by combining and balancing he bene i s om he second-o de
local iden i ica ion and he expanding numbe o momen condi ions. Impo an ly, o
heapp op ia echoiceo k(as a unc ion o n), in e ence o he co ec speci ica ion o
he condi ional momen es ic ion model is s aigh o wa d in p ac ice as i is based on
he c i ical alues om he s anda d no mal dis ibu ion. In ou simula ions and empi -
ical applica ion, we se k∝n1/6. One may e en choose k o g ow a bi a ily slowly wi h n
and he esul s in his pape would con inue o hold. Howe e , a k ha g ows oo slowly
may comp omise powe . I is wo h s essing ha he cons uc ion and implemen a ion
o he es is agnos ic abou he p ecise o m o i s -o de local iden i ica ion ailu e.
This obus ness p ope y is u he enhanced by he ac ha he es emains alid e en
i he model happens o be i s -o de iden i ied.
We comple e ou heo e ical analysis by cha ac e izing he limi ing beha io o ˆ
Z
unde he al e na i e hypo hesis H1, speci ied in (3). Wi h app op ia e choices o se ies
unc ions gl(·),H1implies ha in θ∈E(g(k0)(x)u(y,θ))2>0 o a ixedk0so ha he
uncondi ional momen es ic ion E(g(k0)(x)u(y,θ)) =0 is misspeci ied. In his case, i
is known ha he Sa gan–Hansen speci ica ion es o his uncondi ional es ic ion—
albei in easible because k0is unknown—would be consis en . Theo em 5.3 shows ha
his esul ca ies o e o he easible s a is ic ˆ
Z, which makes ou speci ica ion es con-
sis en agains all al e na i es.7
Theo em 5.3. Le ˆ
Vk(θ):=n−1n
=1 k(x ,y ,θ) k(x ,y ,θ).Assume ha k2=o(n), he
gl(·)se ies a e as in Lemma B.4 in Appendix B,and he condi ions o ha lemma a e sa -
is ied wi h u(θ):=u(y,θ).Assume u he ha he e exis s ¯
λ>0such ha ,wi h p oba-
bili y app oaching one,supθ∈λmax(ˆ
Vk(θ)) ≤¯
λk,and supθ∈|(1/n)n
=1gl(x )u(y ,θ)−
7S udying he asymp o ic beha io o he es unde local al e na i es p o es o be e y in ol ed as i
equi es cha ac e izing he limi ing beha io o he GMM es ima o in misspeci ied condi ional es ic-
ion models unde d i ing sequences and i s -o de local iden i ica ion ailu e. This analysis is beyond he
scope o his pape .
868 Do onon and Gospodino Quan i a i e Economics 15 (2024)
E(gl(x )u(y ,θ))|=oP(1) o each l.Then,unde H1,
∃δ>0: lim
n→∞P k3/2n−1|ˆ
Z|>δ
=1.
The condi ions o his heo em a e essen ially a subse o hose o he main Theo em
5.2. The pu pose o he condi ion on he bound o λmax(ˆ
Vk)is o acili a e he p oo as we
can ely on mo e p imi i e condi ions. Theo em 5.3 shows ha |ˆ
Z|di e ges o in ini y
i kis such ha k3/2=o(n). No e ha in Theo em 5.2,whichs udies ˆ
Zunde H0,we
impose k5=o(n). This shows ha he p oposed es is consis en and has powe agains
all al e na i es.
6. Simula ions and empi ical analysis
In his sec ion, we p o ide simula ion e idence on he empi ical size and powe o he
s anda d no mal asymp o ic app oxima ion o he speci ica ion es . We also apply he
p oposed es ing amewo k o s udy he p esence o a common CH ac o in bond po -
olio e u ns.
6.1 Simula ions
The simula ion design o assessing he ini e-sample p ope ies o he speci ica ion es
ˆ
Zis ailo ed o he common CH ac o example discussed in Sec ion 3and used in he
subsequen empi ical applica ion. Mo e speci ically, he da a gene a ing p ocess has he
o m:
Y +1=D τ+ +1+e +1, (22)
whe e Y +1and e +1∼iidN(0, κIm)a e m×1 ec o s,and +1is an m×1 ec o o
unobse ed CH ac o s. The i h componen i, +1o +1 ollows a GARCH(1,1) p ocess:
i, +1=σi, εi, +1,σ2
i, =ωi,0 +ωi,1 2
i, +ωi,2σ2
i, −1, (23)
wi h ωi,0,ωi,1,ωi,2 >0andεi, +1∼iidN(0, 1).8Finally, is an m×mma ix o ac o
loadings, D :=Diag(σ2
1, ,,σ2
m, )and τis an m×1 ec o o ma ke p ices o isk (see,
e.g., King, Sen ana, and Wadhwani (1994)). In all cases conside ed below, we se κ=0.1
and ωi,0 =1−ωi,1 −ωi,2 o i=1, 2, 3, whe e (ω1,1,ω1,2 )=(0.2, 0.6),(ω2,1,ω2,2)=
(0.4, 0.4)and (ω3,1,ω3,2)=(0.1, 0.8).
The pa ame e iza ion o he ac o loading ma ix de e mines i he model is unde
he null (common CH ea u es) o unde he al e na i e. In e alua ing he size p ope ies
o he speci ica ion es s, we conside wo cases: (i) m=2, bi a ia e Y +1wi h a single
common CH ac o , and (ii) m=3, i a ia e Y +1wi h wo common CH ac o s. Fo
case (i), =10
0.5 0 , and o case (ii), =100
110
0.5 0.5 0 . In assessing he powe p ope ies o
he es s, we se  o be he iden i y ma ix.
8Bouge ol and Pica d (1992) de i e he condi ions o s ic s a iona i y o GARCH p ocesses.

Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 869
As demons a ed in Sec ion 3, he p esence o common CH ea u es amoun s o es -
ing he condi ional momen es ic ion E(u1, +1(θ)|F )) =0. In case (i), u1, +1(θ)is pa-
ame e ized as u1, +1(θ):=(β1Y1, +1+(1−β1)Y2, +1)2−cwi h θ=(β1,c)and o case
(ii), u1, +1(θ):=(β1Y1, +1+β2Y2, +1+(1−β1−β2)Y3, +1)2−cwi h θ=(β1,β2,c).9
Fo some choice o ins umen s z ∈F , he pa ame e ec o θis es ima ed by he wo-
s ep GMM based on he momen condi ions E[z u1, +1(θ)] =0. Fo he uncondi ional
GMM app oach and he co esponding J(Sa gan–Hansen) es , we use z =(Y
,Y2
),
wi h Y2
:=(Y2
1, ,,Y2
m, ),whichgi es ise ok1−po e iden i ying es ic ions: wi h
k1=4andp=2 o case (i), and k1=6andp=3 o case (ii). We epo wo e sions
o he J es : one based on c i ical alues om χ2(k1−p)and one based on c i ical al-
ues om χ2(k1). As a gued abo e, he o me es is no alid when he model is no
i s -o de locally iden i ied while he la e emains alid bu is conse a i e.
The ˆ
Z es uses x:=Y as condi ioning a iables in cons uc ing he unc ions
gl(·),l=1, ,k, so ha he GMM es ima ion is based on E(z u1, +1(θ)) =E(gl(Y )×
u1, +1(θ)) =0. In he case whe e kx:=size(x)=1, he uning pa ame e s o he es
a e (·):R→[−π,π],x→ 2a c an(x), se ies o bounded unc ions gl(·):[−π,π]→
[−1, +1],x→cos(lx) o l=1, k,andk=n1/6.10 In hecasewhe ekx>1, (·)and
gl(·)a e applied componen wise o xleading o k=kx·n1/6momen es ic ions.11 We
epo esul s o he one-sided es ˆ
Za nominal le el α,ˆ
Z>q
1−α,whe eq1−αdeno es
he (1−α)quan ile o he N(0, 1)dis ibu ion.
The ou h speci ica ion ha we conside is also a J es bu i isbasedon heaug-
men ed se o momen condi ions E(z u1, +1(θ)
u2, +1(β))=0, whe e u2, +1(β):=β1Y1, +1+
(1−β1)Y2, +1 o case (i), and u2, +1(β):=β1Y1, +1+β2Y2, +1+(1−β1−β2)Y3, +1
o case (ii). The alue o he pa ame e τin model (23) de e mines i he addi ional
es ic ion E(z u2, +1(β)) =0 es o es he i s -o de local iden i ica ion (τ= 0) o no
(τ=0). Fo case (i), we se τ=(0, 0)o τ=(0.1, 0.1), and o case (ii), τ=(0, 0, 0)o
τ=(0.1, 0.1, 0.1).TheJs a is ic in his augmen ed model is compa ed o c i ical alues
om χ2(k2−p),whe ek2=8andp=2 o case(i)andk2=12 and p=3 o case (ii).
The empi ical ejec ion p obabili ies o he ou speci ica ion es s a e based on
n=(2000, 5000, 10,000)and 10,000 Mon e Ca lo eplica ions. Tables 1and 2p esen
he esul s o case (i), m=2, and case (ii), m=3, espec i ely, wi h he op panel in
9We conside in his sec ion a linea combina ion o asse s wi h coe icien s adding o one o mimic
po olio o ma ion. This yields he same iden i ying p ope ies o he esul ing momen es ic ions as
hose ob ained using he weigh (1, β), conside ed in Sec ion 3, so long as he ma ix o ac o loadings
does no con ain a ele an column wi h equal elemen s.
10E en o la ge sample sizes, he choice k=n1/6may esul in a ela i ely small numbe o ins umen s.
I appea s ha u he imp o emen s can be ob ained i we se k=cons ·n1/6, whe e he cons an “cons ”is
calib a ed o he pa icula se up ( o he alues o nand m). Ideally, i seems desi able o a ge a mo e da a-
d i en choice o k ia subsampling o esampling me hods. Bu , as a gued in he concluding sec ion, such
me hods may be di icul o implemen due o he highly challenging na u e o ou se up: a condi ional mo-
men es ic ions model wi h local i s -o de local iden i ica ion ailu e, second-o de local iden i ica ion,
and dependen da a. Fo una ely, ou simula ion and empi ical esul s sugges ha he p ope ies o he
es a e no pa icula ly sensi i e o he alues o “cons ”ink=cons ·n1/6 o cons ≥1.
11We expe imen ed wi h he cons uc ion o he basis unc ions as in Bie ens (1990) bu he esul s a e
b oadly simila .
870 Do onon and Gospodino Quan i a i e Economics 15 (2024)
Table 1. Empi ical ejec ion a es o speci ica ion es s unde he null (size) and al e na i e
(powe ): case m=2.
Panel A: τ=(0, 0)Panel B: τ=(0.1, 0.1)
JTes JTes JAugmen ˆ
ZTes JTes JTes JAugmen ˆ
ZTes
nχ
2(k1-p)χ2(k1)χ2(k2-p)N(0, 1)χ2(k1-p)χ2(k1)χ2(k2-p)N(0, 1)
size 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5%
2000 8.1 4.0 1.6 0.7 5.3 2.2 6.6 4.0 8.2 4.3 1.7 0.7 5.2 2.3 6.7 4.2
5000 10.7 5.7 2.4 1.1 6.6 3.1 9.0 5.6 11.3 6.0 2.6 1.2 5.4 2.3 8.8 5.7
10,000 13.3 7.7 3.6 1.7 8.9 4.5 9.4 5.9 13.6 7.6 3.7 1.8 6.2 2.7 9.7 6.2
powe 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5%
2000 10.8 5.5 2.3 0.9 10.7 5.6 99.1 98.4 14.9 8.0 3.3 1.4 89.4 82.6 99.0 98.5
5000 10.8 5.4 2.2 0.9 10.7 5.4 100 100 24.8 14.3 6.9 3.3 99.8 99.5 100 100
10,000 11.2 5.9 2.4 1.1 11.5 5.9 100 100 40.2 27.4 15.8 8.6 100 100 100 100
No e: In his simula ion design (m=2), “size” co esponds o a bi a ia e Y +1wi h a single common CH ac o , and “powe ”
co esponds o a bi a ia e Y +1wi h wo CH ac o s. The able p esen s he empi ical size and powe a 5% and 10% nominal
le el o h ee J es s o o e iden i ying es ic ions (p=2,k1=4,andk2=8)and he ˆ
Z es , p oposed in his pape . ‘‘J
augmen ” s ands o he J es , augmen ed wi h an addi ional condi ional momen es ic ion. The alue o τ(τ= 0o τ=
0) de e mines i he augmen ed model is i s -o de locally iden i ied o no . The esul s a e based on 10,000 Mon e Ca lo
eplica ions.
each able epo ing he empi ical size o he es s and he bo om panel epo ing hei
empi ical powe .
In he se up whe e he model is no i s -o de locally iden i ied bu globally iden-
i ied, he s anda d J es o o e iden i ying es ic ions is known o o e ejec unde
he null (Do onon and Renaul (2013)). These o e ejec ions a e con i med in Table 1
Table 2. Empi ical ejec ion a es o speci ica ion es s unde he null (size) and al e na i e
(powe ): case m=3.
Panel A: τ=(0, 0, 0)Panel B: τ=(0.1, 0.1, 0.1)
JTes JTes JAugmen ˆ
ZTes JTes JTes JAugmen ˆ
ZTes
nχ
2(k1-p)χ2(k1)χ2(k2-p)N(0, 1)χ2(k1-p)χ2(k1)χ2(k2-p)N(0, 1)
size 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5%
2000 4.3 1.8 0.5 0.1 2.4 1.0 3.6 2.0 4.8 2.1 0.7 0.2 3.9 1.6 3.7 2.2
5000 6.5 2.8 0.7 0.3 3.3 1.5 5.2 2.9 7.0 3.3 0.9 0.4 5.3 2.5 5.4 3.2
10,000 8.1 3.7 1.1 0.5 4.3 1.8 6.1 3.9 9.4 4.7 1.3 0.4 6.7 2.9 6.1 3.5
powe 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5%
2000 2.5 1.2 0.4 0.1 4.7 2.1 50.8 41.9 3.3 1.6 0.5 0.1 55.6 46.2 52.9 43.4
5000 1.8 0.6 0.1 0.0 2.4 1.0 97.3 95.7 3.0 0.9 0.3 0.1 95.2 91.0 97.8 96.4
10,000 5.2 1.9 0.3 0.1 4.6 2.3 100 100 9.7 3.9 0.8 0.3 100 99.9 100 100
No e: In his simula ion design (m=3), “size” co esponds o a i a ia e Y +1wi h wo common CH ac o s, and “powe ”
co esponds o a i a ia e Y +1wi h h ee CH ac o s. The able p esen s he empi ical size and powe a 5% and 10% nominal
le el o h ee J es s o o e iden i ying es ic ions (p=3,k1=6,andk2=12)and he ˆ
Z es , p oposed in his pape . “J
augmen ” s ands o he J es , augmen ed wi h an addi ional condi ional momen es ic ion. The alue o τ(τ= 0o τ=
0) de e mines i he augmen ed model is i s -o de locally iden i ied o no . The esul s a e based on 10,000 Mon e Ca lo
eplica ions.
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 871
and hey con inue o ge la ge as he sample size inc eases.12 While he J es based
on χ2(k1)c i ical alues is alid, i is e y conse a i e, which esul s in loss o powe .
Fo τ=0, he s anda d J es also exhibi s lack o powe due o he ac ha only he
ins umen s Y2
ca y in o ma ion abou he model pa ame e s while he es ic ions
based on he ins umen s Y a e unin o ma i e as hese es ic ions a e co ec un-
de bo h he null and he al e na i e hypo heses and ende he model jus -iden i ied.
When τ=0, he powe o he s anda d J es is somewha imp o ed (in he case m=2)
bu i emains low because his es does no exploi explici ly he momen es ic ion
E(z u2, +1(β)) =0. This es ic ion is used by he J es based on he augmen ed model
(“Jaugmen ” in Tables 1and 2) whose powe app oaches 100% when τ= 0. Because
he augmen ed model egains i s i s -o de local iden i iabili y o τ=0, he s anda d
in e ence (based on he χ2(k2−p)dis ibu ion) es o es i s alidi y unde bo h he null
and he al e na i e hypo heses.
Bu when τ=0, he addi ional momen es ic ion E(z u2, +1(β)) =0is edundan
and he model emains i s -o de locally uniden i ied. This should mani es i sel in size
dis o ions unde he null ( o n=50,000, he ejec ion a es o he augmen ed J es
unde he null a e 14.9% and 8.6% a he 10% and 5% nominal le el, espec i ely) and
lack o powe unde he al e na i e. This highligh s he need o a obus es ha does
no equi e p io knowledge o he ue s uc u e o he model and he alue o τ.
Indeed, he ˆ
Z es p oposed in his pape o e s p ecisely his ype o obus ness
as i emains agnos ic abou he i s -o de iden i iabili y o he model. Tables 1and 2
demons a e ha he ˆ
Z es con ols size o bo h τ=0andτ=0. The mino unde ejec-
ions o he es o m=3 a ise om he ac ha e en o n=10,000, he ini e-sample
dis ibu ion o he ˆ
Z es is sligh ly asymme ic and i equi es e en la ge sample sizes
o he N(0, 1)asymp o ics o ully asse i sel . Fu he simula ion e idence and dis-
cussion ega ding he absolu e and ela i e ini e-sample pe o mance o he ˆ
Z es is
p o ided in he Supplemen al Ma e ial.
6.2 Empi ical applica ion
In his subsec ion, we in es iga e he p esence o a common CH ac o in U.S. bond e-
u ns o di e en ma u i ies. A e p esen ing some p elimina y e idence on commonal-
i y in he GARCH-based ola ili y dynamics in bond e u ns, we subjec hese po olio
e u ns o he es o common CH ea u es, which amoun s o es ing he alidi y o a
e sion o he condi ional momen es ic ion E(u(y ,θ0)|x )=0.
Le (j)
+1deno e he holding e u n, be ween pe iods and +1, on a bond wi h j
yea s o ma u i y, in excess o he isk- ee a e. Le Y +1=( (1)
+1,, (m)
+1).Asin he
p e ious subsec ion, we posi ha he m- ec o o excess bond e u ns Y +1,adap ed o
he inc easing il a ion F , admi s he ep esen a ion (22) wi h common CH ea u es.13
12In un epo ed esul s o n=50,000 and m=2, he empi ical ejec ions o he s anda d J es based
on χ2(k1−p)a e 18.2% and 11.3% o τ=0, and 16.9 and 10.1% o τ=0 a 10% and 5% nominal le els,
espec i ely. A simila inc ease in o e ejec ions o he J es a e obse ed o m=3 in sample sizes ha
exceed hose epo ed in Table 2.
13The con en ional e m s uc u e models impose no-a bi age es ic ions on he ac o loading ma ix
. Recen esea ch (Du ee (2011); Joslin, Single on, and Zhu (2011); among o he s) cas s doub on he ole
872 Do onon and Gospodino Quan i a i e Economics 15 (2024)
Figu e 1. Es ima ed GARCH(1,1) ola ili ies o po olio bond excess e u ns o di e en ma-
u i ies.
This implies ha he e exis s a ec o β=0min Rmsuch ha E((βY +1)2|F )is cons an ,
ha is, E(u +1(θ0)|F )=0wi hθ=(β,c)and u +1(θ):=(βY +1)2−c.
In he empi ical analysis, we use he Fama bond po olio e u ns om he Cen e
o Resea ch in Secu i y P ices (CRSP) (2023) wi h he ollowing ma u i ies: 1 o 2 yea s,
2 o 3 yea s, 3 o 4 yea s, 4 o 5 yea s, and 5 o 10 yea s.14 The da a is a mon hly equency
co e ing he pe iod Janua y 1952–Decembe 2020. We cons uc excess bond e u ns by
sub ac ing he 1-mon h isk- ee a e ( e ie ed om Kenne h R. F ench—Da a Lib a y
(2023)).
We s a by i ing a GARCH(1,1) o each o hese excess bond e u ns. The il e ed
GARCH ola ili ies a e plo ed in Figu e 1. As he g aph e eals, he e appea s o be
a s ong como emen in hese GARCH ola ili ies. This is p obably no oo su p ising
since he i s p incipal componen in hese i e bond e u ns explains in excess o 95%
o hei ola ili y.
o no-a bi age es ic ions in modeling and o ecas ing bond yields. The o ecas ing p ope ies a e u -
he de e io a ed by inco po a ing s ochas ic ola ili y. As Joslin and Le (2021)demons a e, hisisla gely
a ibu ed o he ac ha hese models impose a igh link be ween isk compensa ion and in e es a e
ola ili y, and ecommend he use o un es ic ed ac o models. This is he app oach ha we ollow he e.
14The da a o he bond po olio e u ns is ob ained om he Wha on Resea ch Da a Se ices (WRDS),
using da abase CRSP T easu ies—Fama bond po olios ©2023 (CRSP).
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 879
condi ions imposed on he se ies unc ions gl(·)a e as in de Jong and Bie ens (1994).
The con inui y and dominance condi ions on u(θ)a e use ul o gua an ee he con inu-
i y o θ→E(gl(x)u(y,θ)) o each l. Con inui y o hese unc ions and compac ness o
a e essen ial o claim he s a ed esul in Lemma B.4.
B.2 P oo s o main esul s
P oo o P oposi ion A.1. (i) I su ices o show ha (A.1)holds o k0 o claim ha i
holds o all k≥k0. Since αl≡0 o alll≥k0,weha e
Eu(y,θ)|x:=ρ(x,θ)=
k0−1

l=1
αl(θ)gl(x)and
ρ(x,θ)≡0⇔[αl=0, ∀l=1, ,k0−1].
By he law o i e a ed expec a ions, αl(θ)=E(gl(x)u(y,θ)) =0, ∀l=1, ,k0−1and
his es ablishes he claim since [ρ(x,θ)≡0⇔θ=θ0]holds by assump ion.
To es ablish he second claim, ecall ha ρ(x,θ)=∞
l=1αl(θ)gl(x).Also,by helaw
o i e a ed expec a ions, E(g(k)(x)u(y,θ)) =E(g(k)(x)ρ(x,θ)) so ha
Eg(k)(x)u(y,θ)=α1(θ),,αk(θ).
Hence, by he de ini ion o θk,
Eρ(x,θk)2=E
l≥k+1
αl(θk)gl(x)2
=
l≥k+1
αl(θk)2→0, as k→∞,(B.1)
whe e he second equali y holds by he s a ed assump ion ha (gl)la e o hono mal
and he con e gence ollows om (d).
Conside an a bi a y small and open neighbo hood o No θ0and le =
min NE(ρ(x,θ))2. By he con inui y assump ion (c), he compac ness o  N,and
he iden i ica ion p ope y in (4), we can claim ha >0. Also, om (B.1), i is clea ha
he e exis s k0∈Nsuch ha E[ρ(x,θk)]2< o all k≥k0. I hen ollows ha o k≥k0,
we ha e θk∈N, which p o es he claim.
(ii) Fi s , we es ablish he necessa y condi ion. I he i s -o de local iden i ica ion
condi ion ails, hen
RankEE∇θu(y,θ0)|xE∇θu(y,θ0)|x<p,
implying ha he e exis s δ= 0∈Rpsuch ha E(∇θu(y,θ0)|x)·δ=0almos su ely.
The e o e, o any k∈N,
Eg(k)(x)·∇θu(y,θ0)·δ=Eg(k)(x)·E∇θu(y,θ0)|x·δ=0.
As a esul ,
RankEg(k)(x)·∇θu(y,θ0)≤p−1, ∀k.

880 Do onon and Gospodino Quan i a i e Economics 15 (2024)
Since k→ Rank(E(g(k)(x)·∇
θu(y,θ0))) akes in ege alues, i is nondec easing and
bounded om abo e, i eaches i s maximum, say ≤p−1, as kinc eases. This shows
he necessa y condi ion.
Nex , we es ablish he su icien condi ion. Unde he s a ed condi ion, he e exis s
δ=0such ha
Egl(x)·∇θu(y,θ0)·δ=Egl(x)·E∇θu(y,θ0)|x·δ=0, o all l≥1. (B.2)
Since E(∇θu(y,θ0)|x)∈(L2(P))p,i si h componen can be w i en as l≥1αl,igl(x),
wi h αl,i’s being scala s. Taking he ele an linea combina ions (o e l) o he equali ies
in (B.2), we ha e
EE∇θu(y,θ0)|xE∇θu(y,θ0)|x·δ=0
and his comple es he p oo .
P oo o Theo em 4.1.Le R=(R1|R2)and conside he ans o ma ion θ=Rη :=
R1η1+R2η2,wi hθ,η∈Rp,η1∈R and η2∈Rp− ,andse ˆ
θ=Rˆη,andθ0=Rη0.
Hence, ¯
k(ˆ
θ)=¯
k(Rˆη)=¯
k(R1ˆη1+R2ˆη2). By a i s -o de Taylo expansion o η1→
¯
k(R1η1+R2ˆη2)a ound η01 and a second-o de Taylo expansion o η2→ ¯
k(R1η01 +
R2η2)a ound η02,weha e
¯
k(ˆ
θ)=¯
k(θ0)+1
√n∇θ¯
k(R1¯η1+R2ˆη2)R1√n(ˆη1−η01)+∇θ¯
k(θ0)R2(ˆη2−η02)
+1
2¯
H(k)(¯
θ)·√n· ecR2(ˆη2−η02)( ˆη2−η02 )R
2,
whe e ¯η1∈(η01,ˆη1)and ¯
θ∈(θ0,ˆ
θ)and bo h may di e om ow o ow.
Le ˜
θ=R1¯η1+R2ˆη2,¯
D1=1
√n∇θ¯
k(˜
θ)·R1,¯
D2=∇θ¯
k(θ0)R2,z0n=√n· ec(R2(˜η2−
η02)( ˜η2−η02)R
2)and we w i e
¯
k(ˆ
θ)=¯
k(θ0)+¯
D1√n(ˆη1−η01)+¯
D2(ˆη2−η02)+1
2¯
H(k)(¯
θ)z0n.(B.3)
By p emul iplying his equa ion by ¯
D
1ˆ
Wkand sol ing o √n(ˆη1−η01 ),weob ain
√n(ˆη1−η01)
=−¯
D
1ˆ
Wk¯
D1−1¯
D
1ˆ
Wk¯
k(θ0)−¯
k(ˆ
θ)+¯
D2(ˆη2−η02)+1
2¯
H(k)(¯
θ)z0n.(B.4)
Plugging his back in o (B.3), we ha e
¯
M(k)ˆ
W1/2
k¯
k(ˆ
θ)
=¯
M(k)ˆ
W1/2
k¯
k(θ0)+¯
M(k)ˆ
W1/2
k¯
D2(ˆη2−η02)+1
2¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)z0n,(B.5)
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 881
wi h ¯
M(k)=Ik−¯
P(k)and ¯
P(k)=ˆ
W1/2
k¯
D1(¯
D
1ˆ
Wk¯
D1)−1¯
D
1ˆ
W1/2
k. Then mul iplying each
side o (B.5) by i s own anspose and ea anging yields
1
4z
0n¯
H(k)(¯
θ)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)z0n
=¯
k(ˆ
θ)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
k(ˆ
θ)−¯
k(θ0)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
k(θ0)
−(ˆη2−η02)¯
D
2ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
D2(ˆη2−η02)−2¯
k(θ0)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
D2(ˆη2−η02)
−¯
k(θ0)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)z0n−(ˆη2−η02)¯
D
2ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)z0n.
By de ini ion, ¯
k(ˆ
θ)ˆ
Wk¯
k(ˆ
θ)≤¯
k(θ0)ˆ
Wk¯
k(θ0).Hence,
1
4z
0n¯
H(k)(¯
θ)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)z0n
=¯
k(ˆ
θ)ˆ
W1/2
k¯
P(k)ˆ
W1/2
k¯
k(ˆ
θ)−¯
k(θ0)ˆ
W1/2
k¯
P(k)ˆ
W1/2
k¯
k(θ0)
−(ˆη2−η02)¯
D
2ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
D2(ˆη2−η02)−2¯
k(θ0)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
D2(ˆη2−η02)
−¯
k(θ0)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)z0n−(ˆη2−η02)¯
D
2ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)z0n.
We show in he Supplemen al Ma e ial ha
¯
k(ˆ
θ)ˆ
W1/2
k¯
P(k)ˆ
W1/2
k¯
k(ˆ
θ)−¯
k(θ0)ˆ
W1/2
k¯
P(k)ˆ
W1/2
k¯
k(θ0)
=OP(¯
λkk/√n)+OP¯
λkkˆ
θ−θ02(B.6)
and, since ¯
λkis bounded and k/√n→0, only he second e m ma e s.
Using his ac and le ing H:=H(k)(θ0)and
1n:=¯
H(k)(¯
θ)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)−HW1/2
kM(k)W1/2
kH,
2n:=¯
H(k)(¯
θ)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
k(θ0)−HW1/2
kM(k)W1/2
k¯
k(θ0),
we can w i e
1
4z
0nHW1/2
kM(k)W1/2
kHz0n
≤OP(¯
λkk)ˆη−η02−(ˆη2−η02 )¯
D
2ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
D2(ˆη2−η02)
−2¯
k(θ0)ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
D2(ˆη2−η02)−¯
k(θ0)W1/2
kM(k)W1/2
kHz0n−
2nz0n
−(ˆη2−η02)¯
D
2ˆ
W1/2
k¯
M(k)ˆ
W1/2
k¯
H(k)(¯
θ)z0n−1
4z
0n1nz0n.(B.7)
F om (B.4), we can show ha ˆη1−η012=OP(ˆη2−η022
2)so ha ˆη−η02=
OP(ˆη2−η022). Also, om he second-o de local iden i ica ion p ope y, we ha e
1
4z
0nHW1/2
kM(k)W1/2
kHz0n≥1
4γkz0,n2
2=1
4γknˆη2−η024
2.
882 Do onon and Gospodino Quan i a i e Economics 15 (2024)
Le z1n=γ1/4
kn1/4(ˆη2−η02). By he Cauchy–Schwa z inequali y, (B.7)yields
1
4z1n4
2≤1
γ1/4
kn1/4OP(¯
λkk)z1n2+1
√nγk
¯
D
2ˆ
Wk¯
D2
2·z1n2
2
+2
(nγk)1/4
ˆ
W1/2
k¯
k(θ0)
2
ˆ
W1/2
k¯
D2
2·z1n2
+
γ−1/2
kHW1/2
kM(k)W1/2
k¯
k(θ0)
2·z1n2
2+1
√γk2n2·z1n2
2
+1
γ3/4
kn1/4
ˆ
W1/2
k¯
D2
2
ˆ
W1/2
k¯
H(k)(¯
θ)
2·z1n3
2+1
4γk1n2·z1n4
2.
Since γk/k=O(1), by Lemma B.3,weha e
1
√nγk
¯
D
2ˆ
Wk¯
D2
2=OP(¯
λkk/n),
1
(nγk)1/4
ˆ
W1/2
k¯
k(θ0)
2
ˆ
W1/2
k¯
D2
2=OP¯
λkk3/4/n1/4,
1
γ3/4
kn1/4
ˆ
W1/2
k¯
D2
2
ˆ
W1/2
k¯
H(k)(¯
θ)
2=OP¯
λkk1/4/n1/4,
1
γk1n2=OPˆ
Wk−Wk2+OP(¯
λk/√n)+OP¯
λkˆ
θ−θ02,
and
1
√γk2n2=OP(¯
λkk/n)+OP¯
λk√kˆ
θ−θ02+OP√kˆ
Wk−Wk2.
Since ¯
λkis bounded, k3/n →0and√kˆ
Wk−Wk2=oP(1), i ollows ha
1
γk1n2=oP(1),and 1
√γk2n2=OPk1/4/n1/4z1n2=oP(1)z1n2.
Hence,
z1n4
2≤
γ−1/2
kHW1/2
kM(k)W1/2
k¯
k(θ0)
2·z1n2
2
+oP(1)·z1n2+oP(1)·z1n2
2
+oP(1)·z1n3
2+oP(1)·z1n4
2.(B.8)
Since γ−1/2
kHW1/2
kM(k)W1/2
k¯
k(θ0)=OP(1), we can eadily claim ha z1n=OP(1).
Indeed, (B.8) amoun s o
1+oP(1)z1n2≤OP(1)
z1n2+oP(1)
z1n2
2+oP(1)
z1n2+oP(1).
Hence, i z1n2>1, his inequali y implies (1+oP(1))z1n2≤OP(1)+oP(1).Thus,
we ei he ha e (z1n2<1)o (1+oP(1))z1n2≤OP(1), which ensu es ha z1n2=
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 883
OP(1), ha is,
γ1/4
kn1/4(ˆη2−η02)=OP(1).
Using (B.4), we ob ain ha √n(ˆη1−η01 )=OP(1). Recalling ha ˆ
θ−θ0=R1(ˆη1−η01)+
R2(ˆη2−η02),weha e
ˆ
θ−θ02=OPn−1/2+OPγ−1/4
kn−1/4=OPγ−1/4
kn−1/4.
Also, by he de ini ion o R1and R2as spanning he ange o he anspose a ma ix and
he null space o ha same ma ix, espec i ely, we ha e R
1R2=0. Hence,
R
1(ˆ
θ−θ0)=R
1R1(ˆη1−η01)=OPn−1/2
and
R
2(ˆ
θ−θ0)=R
2R2(ˆη2−η02)=OPγ−1/4
kn−1/4.
To comple e he p oo , i only emains o es ablish (B.6), which is done in he Supple-
men al Ma e ial.
The alidi y o he esul s in Theo em 4.1 hinges on e i ying Assump ions 3(ii) and
4(iii, i ). Fo his eason, some u he ema ks on he a es o con e gence in Theo em
4.1, specialized o he i s -s ep and wo-s ep GMM es ima o s, a e wa an ed.
Rema k 1. Fo he i s -s ep GMM es ima o , he weigh ing ma ix ˆ
Wk:=Wk,0 is non-
andom wi h bounded eigen alues om abo e and away om ze o, and Assump ions
3(ii) and 4(i ) a e i ially e i ied. Assump ion 4(iii) is also sa is ied i , o ins ance,
Vkhas bounded eigen alues and his es ima o , say ˜
θ, is cha ac e ized by ˜
θ−θ0=
OP(k−1/4n−1/4). No e ha he eigen alues o Vka e bounded unde Assump ion A.1(i )
and i Vkhas uni o mly bounded diagonal elemen s. This la e condi ion is implied by
Assump ion A.1(ii).
Rema k 2. Fo he wo-s ep es ima o wi h ˆ
Wk:=ˆ
V−1
k, we assume ha he smalles
eigen alue o Vkis bounded away om 0, ha is, λmin(Vk)≥λ>0 o allk.Thisisa
easonable assump ion since λmax(Vk)is an inc easing sequence and we equi e ha
λmin(Vk)and λmax(Vk)a eo hesameo de o magni ude op ese eVk om being
ill-condi ioned. In his case,
λmaxV−1
k=1/λmin(Vk)≤1/λ.
Fu he mo e, Lemma B.2(ii, ) ensu es ha
λmaxˆ
V−1
k/λmaxV−1
k=1+oP(1),andλminˆ
V−1
k/λminV−1
k=1+oP(1).
No e ha in his lemma, n=(k3/n)1/4=o(1). This shows ha Assump ion 3(ii) holds.
Rema k 3. Finally, om Lemma B.2(i ), we ha e ha √kˆ
V−1
k−V−1
k2=OP(k5/4/n1/4),
which ensu es ha Assump ion 4(i ) holds i k5/n →0asn→∞.
884 Do onon and Gospodino Quan i a i e Economics 15 (2024)
P oo o Theo em 4.2. Since θ0is in he in e io o and ˆ
θcon e ges in p obabili y
o θ0,ˆ
θis also an in e io op imum wi h p obabili y app oaching one. The e o e, his
es ima o sol es
∇θ¯
k(ˆ
θ)ˆ
Wk¯
k(ˆ
θ)=0. (B.9)
By a mean- alue expansion o ∇¯
k(ˆ
θ)and a second-o de Taylo expansion ¯
k(ˆ
θ)
a ound θ0,weha e
∇θ¯
k(ˆ
θ)=∇θ¯
k(θ0)+¯
H(k)(˙
θ)√n(ˆ
θ−θ0)
and
¯
k(ˆ
θ)=¯
k(θ0)+∇θ¯
k(θ0)( ˆ
θ−θ0)+1
2¯
H(k)(¨
θ)√n(ˆ
θ−θ0)2,
wi h ˙
θ,¨
θ∈(θ0,ˆ
θ).Le ˙
h:=¯
H(k)(˙
θ),¨
h:=¯
H(k)(¨
θ),and ¯
D2:=∇
θ¯
k(θ0).The i s -o de
condi ion (B.9)yields
n−1/4∇¯
k(ˆ
θ)ˆ
Wk¯
k(ˆ
θ)
=n−1/4¯
D
2ˆ
Wk¯
k(θ0)+n−1/4¯
D
2ˆ
Wk¯
D2(ˆ
θ−θ0)+1
2¯
D
2ˆ
Wk¨
hn1/4(ˆ
θ−θ0)2
+˙
hˆ
Wk¯
k(θ0)n1/4(ˆ
θ−θ0)+˙
hˆ
Wk¯
D2n1/4(ˆ
θ−θ0)2
+1
2˙
hˆ
Wk¨
hn3/4(ˆ
θ−θ0)3=0.
In his amewo k, γk=HWkH.Wi hH:=H(k)(θ0),wecanw i e
1
2(γkn)3/4(ˆ
θ−θ0)3+1
2γk˙
hˆ
Wk¨
h−HWkH(γkn)3/4(ˆ
θ−θ0)3
+3
2γ3/4
kn1/4Hˆ
Wk¯
D2√γkn(ˆ
θ−θ0)2
+1
γ3/4
kn1/4(˙
h−H)ˆ
Wk¯
D2√γkn(ˆ
θ−θ0)2+1
√γk
HWk¯
k(θ0)(γkn)1/4(ˆ
θ−θ0)
+1
√γk˙
hˆ
Wk−HWk¯
k(θ0)(γkn)1/4(ˆ
θ−θ0)+1
2γ3/4
kn1/4¯
D
2ˆ
Wk(¨
h−H)√γkn(ˆ
θ−θ0)2
+1
√γkn¯
D
2ˆ
Wk¯
D2(γkn)1/4(ˆ
θ−θ0)+n−1/4¯
D
2ˆ
Wk¯
k(θ0)=0.
Simila o he lines o he p oo o Lemma B.3, i is no ha d o see ha
1
γk˙
hˆ
Wk¨
h−HWkH=oP(1),Hˆ
Wk¯
D2=OP(k),
˙
h−H2=OP√kn−1/2∨ˆ
θ−θ02=OPk1/4/n1/4,
(˙
h−H)ˆ
Wk¯
D2=OPk3/4/n1/4,and 1
√γk˙
hˆ
Wk−HWk¯
k(θ0)=oP(1).

Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 885
Then, le ing z1n:=(γkn)1/4(ˆ
θ−θ0)and Zn:=1
√γkHWk¯
k(θ0),weha e
z1nz2
1n+2Zn=oP(1). (B.10)
Since (z1n,Zn)=OP(1), by he P okho o ’s heo em, each subsequence o has a u he
subsequence ha con e ges in dis ibu ion o, say, (V,Z). Thus, along his con e ging
subsequence, (B.10) implies ha
VV2+2Z=0.
The e o e, i is no di icul o see ha |V|=1Z≤0√−2Zand, since as a Gaussian andom
a iable Zhas a symme ic dis ibu ion, |V|=1Z≥0√2Z. The ac ha his limi dis i-
bu ion is no speci ic o he subsequence implies ha he whole sequence con e ges o
(V,Z). By he con inuous mapping heo em, i ollows ha z2
1n
d
→V2=1Z≥0(2Z)and
his comple es he p oo .
P oo o Theo em 5.2. We p oceed in wo s eps by showing i s ha ˆ
Zis bounded by
wo s a is ics ˆ
Z1and ˆ
Z2, ha is,
ˆ
Z1+oP(1)≤ˆ
Z≤ˆ
Z2+oP(1). (B.11)
We hen show in he second s ep ha ˆ
Z1and ˆ
Z2con e ge in dis ibu ion o N(0, 1),
which es ablishes he s a ed esul .
S ep 1: By de ini ion,
¯
k(ˆ
θ)ˆ
V−1
k¯
k(ˆ
θ)≤¯
k(θ0)ˆ
V−1
k¯
k(θ0)=¯
k(θ0)V−1
k¯
k(θ0)+¯
k(θ0)ˆ
V−1
k−V−1
k¯
k(θ0).
No e ha
¯
k(θ0)ˆ
V−1
k−V−1
k¯
k(θ0)≤
ˆ
V−1
k−V−1
k
2
¯
k(θ0)

2
2.
F om Theo em 4.1, he i s -s ep GMM es ima o ˜
θis such ha ˜
θ−θ0=OP(k−1/4n−1/4).
Hence, Lemma B.2(i ) implies ha ˆ
V−1
k−V−1
k2=OP(k3/4n−1/4).Thus,

ˆ
V−1
k−V−1
k
2
¯
k(θ0)

2
2=OPk7/4n−1/4=√kOPk5/4n−1/4=oP(√k).
As a esul , we ha e
ˆ
Z=¯
k(ˆ
θ)ˆ
V−1
k¯
k(ˆ
θ)−k
√2k≤¯
k(θ0)V−1
k¯
k(θ0)−k
√2k+oP(1):=ˆ
Z2+oP(1). (B.12)
On he o he hand, using (B.5), we can w i e
¯
k(ˆ
θ)ˆ
V−1
k¯
k(ˆ
θ)
=¯
k(ˆ
θ)ˆ
V−1/2
k¯
P(k)ˆ
V−1/2
k¯
k(ˆ
θ)−¯
k(θ0)ˆ
V−1/2
k¯
P(k)ˆ
V−1/2
k¯
k(θ0)+¯
k(θ0)ˆ
V−1
k¯
k(θ0)
+(ˆη2−η02)¯
D
2ˆ
V−1/2
k¯
M(k)ˆ
V−1/2
k¯
D2(ˆη2−η02)
886 Do onon and Gospodino Quan i a i e Economics 15 (2024)
+1
4z
0n¯
H(k)(¯
θ)ˆ
V−1/2
k¯
M(k)ˆ
V−1/2
k¯
H(k)(¯
θ)z0n
+2¯
k(θ0)ˆ
V−1/2
k¯
M(k)ˆ
V−1/2
k¯
D2(ˆη2−η02)+¯
k(θ0)ˆ
V−1/2
k¯
M(k)ˆ
V−1/2
k¯
H(k)(¯
θ)z0n
+(ˆη2−η02)¯
D
2ˆ
V−1/2
k¯
M(k)ˆ
V−1/2
k¯
H(k)(¯
θ)z0n
:=(1)+(2)+(3)+(4)+(5)+(6)+(7).
F om (B.6), and Lemma B.2,weha e
(1)=OPk3/4n−1/4=oP(1),(3)=OPk1/2n−1/2=oP(1),
(4)=1
4z
0nHV−1/2
kM(k)V−1/2
kHz0n+oP(1),(5)=OPk3/4n−1/4=oP(1),
(6)=¯
k(θ0)V−1/2
kM(k)V−1/2
kHz0n+oP(1),(7)=OPk1/4n−1/4=oP(1)
and om he lines abo e, (2)=¯
k(θ0)V−1
k¯
k(θ0)+oP(√k).Asa esul ,wew i e
¯
k(ˆ
θ)ˆ
V−1
k¯
k(ˆ
θ)=¯
k(θ0)V−1
k¯
k(θ0)+¯
k(θ0)V−1/2
kM(k)V−1/2
kHz0n
+1
4z
0nHV−1/2
kM(k)V−1/2
kHz0n+oP(√k). (B.13)
Le he ank ac o iza ion o M(k)V−1/2
kHbe M(k)V−1/2
kH=H1H2,whe eH1and
H2a e a (k, h)-ma ix and a ( h,p2)-ma ix, espec i ely, wi h he same ank h=
Rank(M(k)V−1/2
kH)≤p2. By second-o de local iden i ica ion, h= 0so ha
M(k)V−1/2
kH=0.
Thus, (B.13) can be w i en as
¯
k(ˆ
θ)ˆ
V−1
k¯
k(ˆ
θ)=¯
k(θ0)V−1
k¯
k(θ0)+¯
k(θ0)V−1/2
kH1H2z0n
+1
4z
0nH
2H
1H1H2z0n+oP(√k).
Le ing m(u):=¯
k(θ0)V−1
k¯
k(θ0)+¯
k(θ0)V−1/2
kH1u+1
4uH
1H1u,andM1:=Ik−
H1(H
1H1)−1H
1, we can claim ha
min
u∈R h
m(u)+oP(√k)=¯
k(θ0)V−1/2
kM1V−1/2
k¯
k(θ0)+oP(√k)≤¯
k(ˆ
θ)ˆ
V−1
k¯
k(ˆ
θ).
Le ing ˆ
Z1:=¯
k(θ0)V−1/2
kM1V−1/2
k¯
k(θ0)−k
√2k,weob ain(B.11).
S ep 2: We now show ha bo h ˆ
Z1and ˆ
Z2a e asymp o ically s anda d no mal. We
i s conside ˆ
Z2.Weha e
ˆ
Z2=1
n√2k
n

=s: ,s=1
k(xs,ys,θ0)V−1
k k(x ,y ,θ0)
+
1
n
n

=1
k(x ,y ,θ0)V−1
k k(x ,y ,θ0)−k
√2k:=U1n+U2n.
Quan i a i e Economics 15 (2024) Robus speci ica ion es ing 887
The asymp o ic no mali y o U1n ollows eadily om he cen al limi heo em s a ed by
Theo em 5.1. In addi ion, i is no ha d o see ha E(U2n)=0. Using simila a gumen s as
in he p oo o P oposi ion S.2 in he Supplemen al Ma e ial, we can show ha E(U2
2n)=
o(1). This es ablishes ha U2n=oP(1). We can hen conclude ha ˆ
Z2is asymp o ically
s anda d no mal.
We now conside ˆ
Z1. No e i s ha , since M1is an o hogonal p ojec ion ma ix on a
space o dimension k− h, he eexis sa(k,k− h)-ma ix S1such ha S
1S1=Ik− hand
M1=S1S
1. In ha espec , ¯
k(θ0)V−1/2
kM1V−1/2
k¯
k(θ0)=¯
k(θ0)V−1/2
kS1S
1V−1/2
k¯
k(θ0).
Also, Va (S
1V−1/2
k¯
k(θ0)) =Ik− h. Using Theo em 5.1 and simila o he lines abo e o
ˆ
Z2, we can claim ha
ˆ
Z3:=¯
k(θ0)V−1/2
kS1S
1V−1/2
k¯
k(θ0)−(k− h)
2(k− h)
d
→N(0, 1).
Since 0 ≤ h≤p2wi h p ixed, we can see ha ˆ
Z1=ˆ
Z3+oP(1). The e o e, ˆ
Z1
d
→N(0, 1).
P oo o Theo em 5.3. F om Lemma B.4, he eexis k0and δ0>0such ha k>k
0
and
E( k0(x ,y ,ˆ
θ))E( k0(x ,y ,ˆ
θ)) ≥δ0.Weha e
k3/2n−1|ˆ
Z|≥2−1/2kn−1¯
k(ˆ
θ)ˆ
V−1
k¯
k(ˆ
θ)−k
≥2−1/2kn−1¯
k(ˆ
θ)¯
k(ˆ
θ)/λmax(ˆ
Vk)+21/2k2n−1
≥2−1/2/¯
λn−1¯
k0(ˆ
θ)¯
k0(ˆ
θ)+o(1), wi h p obabili y app oaching one.
Also,
n−1¯
k0(ˆ
θ)¯
k0(ˆ
θ)≥E k0(x ,y ,ˆ
θ)E k0(x ,y ,ˆ
θ)
+21
n
n

=1
k0(x ,y ,ˆ
θ)−E k0(x ,y ,ˆ
θ)
E k0(x ,y ,ˆ
θ)
≥δ0−2




1
n
n

=1
k0(x ,y ,ˆ
θ)−E k0(x ,y ,ˆ
θ)



2
E k0(x ,y ,ˆ
θ)
2
=δ0+oP(1)OP(1).
I ollows ha , wi h p obabili y app oaching one, k3/2n−1|ˆ
Z|≥(2−1/2/¯
λ)δ0+oP(1)and
his concludes he p oo by se ing δ:=(2−1/2/¯
λ)δ0.
P oo o nondec easing γkin Equa ion (15). We need o show ha o k≥k0,
γk≤γk+1. Fo his, we no e ha R2is ixed o k≥k0. The e o e, i su ices o show
ha o any a∈Rkand b∈R, and le ing :=(a,b)∈Rk+1,
aM(k)a≤ M(k+1) .
888 Do onon and Gospodino Quan i a i e Economics 15 (2024)
Le μ=E(gk+1(x)·(∇θu(y,θ0))),G1=G(k)and G=G(k+1).Weha e
G=G1
μ,M(k)=Ik−G1G
1G1−1G
1,andM(k+1)=Ik+1−GGG−1G.
W i e A:=G
1G1. Using he Woodbu y o mula, we can claim ha
GG−1=A+μμ−1=A−1−A−1μμA−1
1+μA−1μ.
Thus,
M(k+1) =aa+b2−aG1A+μμ−1G
1a−2baG1A+μμ−1μ
−b2μA+μμ−1μ
=aM(k)a+b−aG1A−1μ2
1+μA−1μ.
I ollows ha , o any a∈Rkand b∈R,
M(k+1) ≥aM(k)a
and his comple es he p oo .
Re e ences
And ews, Donald (1984), “Non-s ong mixing au o eg essi e p ocesses.” Jou nal o Ap-
plied P obabili y, 21, 930–934. [852]
And ews, Donald (1991), “Asymp o ic no mali y o se ies es ima o s o nonpa ame ic
and semipa ame ic eg ession models.” Econome ica, 59, 307–345. [854]
An oine, Be ille, Ke in P oulx, and E ic Renaul (2020), “Pseudo- ue SDFs in condi-
ional asse p icing models.” Jou nal o Financial Econome ics, 18, 656–714. [853]
Be man, Simeon (1964), “Limi heo ems o he maximum e m in s a iona y se-
quences.” Annals o Ma hema ical S a is ics, 35, 502–516. [866]
Bie ens, He man (1982), “Consis en model speci ica ion es s.” Jou nal o Econome ics,
20, 105–134. [850]
Bie ens, He man (1990), “A consis en condi ional momen es o unc ional o m.”
Econome ica, 58, 1443–1458. [869]
Bie ens, He man and We ne Plobe ge (1987), “Asymp o ic heo y o in eg a ed condi-
ional momen es s.” Econome ica, 65, 1129–1151. [850]
Bouge ol, Philippe and Nico Pica d (1992), “S a iona i y o GARCH p ocesses and some
nonnega i e ime se ies.” Jou nal o Econome ics, 52, 115–127. [868]