Covariate adjustment in stratified experiments

Cy ynbaum, Max
A icle
Co a ia e adjus men in s a i ied expe imen s
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: Cy ynbaum, Max (2024) : Co a ia e adjus men in s a i ied expe imen s,
Quan i a i e Economics, ISSN 1759-7331, The Econome ic Socie y, New Ha en, CT, Vol. 15, Iss. 4,
pp. 971-998,
h ps://doi.o g/10.3982/QE2475
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/320322
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Quan i a i e Economics 15 (2024), 971–998 1759-7331/20240971
Co a ia e adjus men in s a i ied expe imen s
Max Cy ynbaum
Depa men o Economics, Yale Uni e si y
This pape s udies co a ia e adjus ed es ima ion o he a e age ea men e ec in
s a i ied expe imen s. We wo k in a gene al amewo k ha includes ma ched u-
ples designs, coa se s a i ica ion, and comple e andomiza ion as special cases.
Reg ession adjus men wi h ea men -co a ia e in e ac ions is known o weakly
imp o e e iciency o comple ely andomized designs. By con as , we show ha
o s a i ied designs such eg ession es ima o s a e gene ically ine icien , po-
en ially e en inc easing es ima o a iance ela i e o he unadjus ed bench-
ma k. Mo i a ed by his esul , we de i e he asymp o ically op imal linea co a i-
a e adjus men o a gi en s a i ica ion. We cons uc se e al easible es ima o s
ha implemen his e icien adjus men in la ge samples. In he special case o
ma ched pai s, o example, he eg ession including ea men , co a ia es, and
pai ixed e ec s is asymp o ically op imal. We also p o ide no el asymp o ically
exac in e ence me hods ha allow esea che s o epo smalle con idence in e -
als, ully e lec ing he e iciency gains om bo h s a i ica ion and adjus men .
Simula ions and an empi ical applica ion demons a e he alue o ou p oposed
me hods.
Keywo ds. Ma ched pai s, analysis o co a iance, blocking, obus s anda d e -
o , ea men e ec s.
JEL classi ica ion. C10, C14, C90.
1. In oduc ion
This pape s udies co a ia e adjus ed es ima ion o he a e age ea men e ec (ATE) in
s a i ied expe imen s. Resea che s o en make use o bo h s a i ied ea men assign-
men and ex pos co a ia e adjus men o imp o e he p ecision o expe imen al es i-
ma es. Indeed, ou o a su ey o o e 50 expe imen al pape s published in he AER and
AEJ be ween 2018–2023, we ound ha 57% use s a i ied andomiza ion, and 80% used
some o m o ex pos co a ia e adjus men . An in luen ial pape by Lin (2013)showed
in a design-based se ing ha he eg ession es ima o wi h ull ea men -co a ia e in-
e ac ions is always asymp o ically weakly mo e e icien han di e ence o means es i-
ma ion o comple ely andomized designs. Negi and Woold idge (2021)ex ended hese
esul s o es ima ion o he ATE using da a sampled om a supe popula ion. Howe e ,
ques ions emain abou he in e ac ion be ween s a i ica ion and eg ession adjus -
men and he implica ions o combining hese me hods o bo h es ima o e iciency
and he powe and alidi y o in e ence me hods. To s udy hese ques ions, we wo k in
Max Cy ynbaum: [email p o ec ed]
I hank he anonymous e e ees o help ul sugges ions du ing he e ision p ocess.
©2024 The Au ho . 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/QE2475
972 Max Cy ynbaum Quan i a i e Economics 15 (2024)
he s a i ied andomiza ion amewo k o Cy ynbaum (2023), which includes ma ched
uples designs (e.g., ma ched pai s), coa se s a i ica ion, and comple e andomiza ion
as special cases.
We show ha he Lin (2013) in e ac ed eg ession adjus men is gene ically ine i-
cien in he amily o linea ly adjus ed es ima o s, wi h asymp o ic e iciency only in he
limi ing case o comple e andomiza ion. Mo i a ed by his inding, we cha ac e ize he
e icien linea co a ia e adjus men o a gi en s a i ied design, p o iding se e al new
es ima o s ha achie e he op imal a iance.
Ou i s esul de i es he op imal linea adjus men coe icien o a gi en s a i i-
ca ion. We show ha asymp o ically he in e ac ed eg ession es ima o uses he w ong
objec i e unc ion, minimizing a ma ginal a iance objec i e ha is o ally insensi i e
o he s a i ica ion. By con as , he op imal adjus men coe icien minimizes a mean-
condi ional a iance objec i e, condi ional on he co a ia es used o s a i y. In ui i ely,
he e icien co a ia e adjus men is ailo ed o he s a i ica ion, igno ing luc ua ions
o he es ima o ha a e p edic able by he s a i ica ion co a ia es. Sec ion 3.2 d aws
an in e es ing connec ion wi h pa ially linea eg ession (Robinson (1988)), showing
ha e icien linea adjus men o a s a i ied design is asymp o ically equi alen o
doubly- obus semipa ame ic adjus men o an i.i.d. design. In ui i ely, s a i ica ion
con ibu es he nonpa ame ic componen o he semipa ame ic adjus men unc ion.
Ou second se o esul s de elops easible e sions o he op imal linea adjus men
de i ed in Sec ion 3.1. Fi s , we show ha i he condi ional expec a ion o he adjus -
men co a ia es is linea in a known se o ans o ma ions o he s a i ica ion a iables,
hen adding he la e o he in e ac ed eg ession es o es op imali y. Nex , we elax his
assump ion, p o iding ou di e en eg ession es ima o s ha a e asymp o ically e -
icien unde weak condi ions. Fo ma ched pai s expe imen s o in se ings wi h lim-
i ed ea men e ec he e ogenei y, he nonin e ac ed eg ession wi h a ull se o pai
ixed e ec s is asymp o ically e icien . Mo e gene ally, we show asymp o ic op imal-
i y o wi hin-s a um (inconsis en ly) pa ialled e sions o he Lin and y anny-o - he-
mino i y es ima o s (Lin (2013)). We also de ine a “g oup OLS” es ima o , ex ending a
p oposal o Imbens and Rubin (2015) o ma ched pai s expe imen s o a la ge class o
designs. We show ha his g oup OLS es ima o is also asymp o ically op imal.
Ou inal con ibu ion is o de elop no el asymp o ically exac in e ence me hods
o co a ia e adjus ed es ima ion unde s a i ied designs. Con idence in e als based
on he usual he e oskedas ici y obus a iance es ima o a e known o be conse a i e
in s a i ied expe imen s (Bai, Romano, and Shaikh (2021)). By con as , he co e age
p obabili ies o ou p oposed con idence in e als con e ge o he speci ied nominal
le el, wi h no o e co e age. Ou app oach applies o a gene ic amily o linea co a ia e
adjus men s and andomiza ion schemes, including as special cases nonin e ac ed e-
g ession adjus men , he Lin (2013) in e ac ed eg ession, and all o he o he es ima o s
conside ed in his pape . Simula ions and an empi ical applica ion o he expe imen
in Baysan (2022a) sugges ha he usual obus con idence in e als can subs an ially
o e co e in s a i ied expe imen s, while ou con idence in e als ha e close o nomi-
nal co e age.
Quan i a i e Economics 15 (2024) Co a ia e adjus men in s a i ied expe imen s 973
We p esen se e al ex ensions o ou main esul s in he Supplemen al Appendix
(Cy ynbaum (2024)). In he i s , we conside es ima ion and in e ence in s a i ied ex-
pe imen s wi h noncompliance. As a simple co olla y o ou esul s on ATE es ima ion,
we cha ac e ize he op imal linea ly adjus ed Wald es ima o o he LATE (Imbens and
Ang is (1994)), cons uc a easible implemen a ion o he e icien adjus men , and
p o ide asymp o ically exac in e ence me hods. We also s udy e icien linea adjus -
men o inely s a i ied designs wi h noncons an ea men p opo ions, as in Cy yn-
baum (2023), and b ie ly conside he p oblem o e icien nonlinea adjus men .
The e has been signi ican in e es in ea men e ec es ima ion unde di e en
expe imen al designs in he ecen li e a u e. Some pape s s udying co a ia e adjus -
men unde s a i ied andomiza ion include Bugni, Canay, and Shaikh (2018), Foga-
y (2018), Liu and Yang (2020), Lu and Liu (2024), Wei, Tu, and Liu (2022), Reluga, Ye,
and Zhao (2024), Wang, Wang, and Liu (2021), Ting, Shao, Yi, and Zhao (2022), Ke, Liu,
and Yang (2024), and Chang (2023). These wo ks di e om ou pape in a leas one
o he ollowing ways: (1) s udying in e ence on he sample a e age ea men e ec
(SATE) a he han he ATE in a supe popula ion, (2) es ic ing o coa se s a i ica ion
(s a um size going o in ini y), o (3) p o ing weak e iciency gains bu no op imali y.
In a ini e popula ion se ing, Ke, Liu, and Yang (2024) shows asymp o ic e iciency o
a p ojec ion-based es ima o nume ically equi alen o he “pa ialled Lin” app oach
conside ed in Sec ion 3.4.2. In he same se ing, Lu and Liu (2024) p o e e iciency o a
y anny-o - he-mino i y s yle eg ession simila bu no equi alen o one he consid-
e ed in Sec ion 3.4.4. Bo h pape s gi e conse a i e in e ence on he SATE, while we
p o ide asymp o ically exac in e ence on he ATE using a gene alized pai s-o -pai s
(Abadie and Imbens (2008)) s yle app oach. Rema ks 3.19 and 3.22 in Sec ion 3.4 below
p o ide a de ailed compa ison.
Rela i e o he abo e pape s, he supe popula ion amewo k conside ed he e c e-
a es some new echnical challenges. Fo example, as poin ed ou in Bai, Romano, and
Shaikh (2021), ma ching uni s in o da a-dependen s a a pos -sampling p oduces a
complica ed dependence s uc u e be ween he ea men assignmen s and andom co-
a ia es. We deal wi h his using a igh -ma ching condi ion (Equa ion (2.1)) and ma -
ingale CLT analysis simila o Cy ynbaum (2022). This se ing also has analy ical ad-
an ages, which allow us o es ablish new concep ual esul s. Fo example, he popu-
la ion le el cha ac e iza ion o he op imal adjus men coe icen in Sec ion 3.1 allows
us o gi e explici necessa y and su icien condi ions o he e iciency o se e al com-
monly used eg ession es ima o s. The e iciency o in e ac ed eg ession unde a “ ich
co a ia es” condi ion, as well as he equi alence be ween op imal linea adjus men o
s a i ied designs and doubly- obus semipa ame ic adjus men appea o be new ob-
se a ions in his li e a u e. To he bes o ou knowledge, we gi e he i s asymp o ically
exac in e ence on he ATE o gene al co a ia e adjus ed es ima o s unde inely s a i-
ied andomiza ion.
Independen ly, Bai, Jiang, Romano, Shaikh, and Zhang (2024b) s udy co a ia e ad-
jus men unde ma ched pai s andomiza ion in a supe popula ion amewo k. They
also ind ha eg ession adjus men wi hou pai ixed e ec s may be ine icien , while
adding pai ixed e ec s es o es e iciency. Rela i e o ou wo k, hey addi ionally s udy
974 Max Cy ynbaum Quan i a i e Economics 15 (2024)
egula ized eg ession adjus men unde high-dimensional asymp o ics, which we do
no conside . By con as , we s udy mo e gene al o ms o s a i ica ion, allowing coa se
and ine s a i ica ion wi h a bi a y ea men p opo ions p= 1/2. Fo such designs,
he s a a ixed-e ec s es ima o may s ill be ine icien . To ix his, we in oduce no el
o ms o linea adjus men ha a e e icien unde gene al s a i ied designs.
The es o he pape is o ganized as ollows. In Sec ion 2, we de ine no a ion and
in oduce he amily o s a i ied designs ha we will conside h oughou he pape .
Sec ion 3gi es ou main esul s, cha ac e izing op imal co a ia e adjus men , and con-
s uc ing e icien es ima o s. Sec ion 4p o ides asymp o ically exac in e ence on he
ATE o gene ic linea ly adjus ed es ima o s. In Sec ions 5and 6, we s udy he ini e-
sample p ope ies o ou me hod, including bo h simula ions and an empi ical applica-
ion o he expe imen in Baysan (2022a). Sec ion 7concludes wi h some ecommenda-
ions o p ac i ione s.
2. F amewo k and s a i ied designs
Fo a bina y ea men d∈{0, 1},le Yi(1),Yi(0)deno e he ea ed and con ol po en-
ial ou comes, espec i ely. Fo ea men assignmen Di,le Yi=Yi(Di)=DiYi(1)+
(1−Di)Yi(0)be he obse ed ou come. Le Xideno e co a ia es. Conside da a
(Xi,Yi(1),Yi(0))n
i=1sampled i.i.d. om a supe popula ion o in e es . We a e in e es ed
in es ima ing he a e age ea men e ec in his popula ion, ATE =E[Y(1)−Y(0)].A -
e sampling uni s i=1, ,n, ea men s D1:na e assigned by s a i ied andomiza ion.
In pa icula , we use he “local andomiza ion” amewo k in oduced in Cy ynbaum
(2022).
De ini ion 2.1 (Local andomiza ion). Le ea men p opo ions p=a/k wi h gcd(a,
k)=1.1Suppose ha nis di isible by k o no a ional simplici y. Pa i ion he ex-
pe imen al uni s in o n/k g oups gwi h {1, ,n}=ggdisjoin ly and |g|=k.Le
ψ(X)∈Rdψdeno e a ec o o s a i ica ion a iables. Suppose ha he g oups ha sa -
is y a homogenei y condi ion wi h espec o ψ(X)such ha
1
n
g
i,j∈gψ(Xi)−ψ(Xj)2
2=op(1). (2.1)
Requi e ha he g oups only depend on he s a i ica ion a iables ψ1:nand da a-
independen andomness πn,so ha g=g(ψ1:n,πn) o each g. Independen ly, o
each |g|=k, d aw ea men a iables (Di)i∈gby se ing Di=1 o exac ly aou o k
uni s, comple ely a andom. Fo a s a i ica ion sa is ying hese condi ions, we deno e
D1:n∼Loc(ψ,p).
Example 2.2 (Ma ched uples). Equa ion (2.1) equi es uni s in a g oup o ha e simila
ψ(Xi) alues and can be hough o as a igh -ma ching condi ion. Cy ynbaum (2023)
p o ides an i e a i e pai ing algo i hm o ma ch uni s in o g oups ha p o ably sa is y
1gcd(a,k)s ands o g ea es common di iso .

Quan i a i e Economics 15 (2024) Co a ia e adjus men in s a i ied expe imen s 975
his condi ion o any k. D awing ea men s D1:n∼Loc(ψ,p)p oduces a “ma ched k-
uples” design o p=a/k. Ma ched pai s co esponds o he case p=1/2.
Example 2.3 (Comple e andomiza ion). We say a iables D1:na e comple ely andom-
ized wi h ea men p obabili y pi D1:nis d awn uni o mly om all ec o s d1:nwi h
di=1 o exac ly p opo ion po he uni s. Fo mally, P(D1:n=d1:n)=1/n
pn o all such
ec o s. We deno e comple e andomiza ion by D1:n∼CR(p). Comple e andomiza ion
may be ob ained in ou amewo k by se ing ψ=1 and o ming g oups |g|=ka an-
dom, which au oma ically sa is ies Equa ion (2.1). Fo example, assigning 2 ou o 3 uni s
in each g oup o ea men gi es a “ andom ma ched iples” ep esen a ion o comple e
andomiza ion wi h p=2/3.
Rema k 2.4 (Coa se s a i ica ion). Simila ly, coa se s a i ica ion wi h la ge ixed s a a
S(X)∈{1, ,m}canalsobeob ainedinou amewo kbyse ingψ(X)=S(X)and
ma ching uni s wi h iden ical S(X) alues in o g oups a andom. Because o his, ou
amewo k enables a uni ied asymp o ic analysis o a wide ange o s a i ica ions.
Expe imen iming Suppose ha he expe imen e does he ollowing:
(1) Samples uni s and obse es hei baseline co a ia es.
(2) Pa i ions he uni s in o da a-dependen g oups g=g(ψ1:n,πn) ha sa is y Equa-
ion (2.1) o some s a i ica ion a iables ψ(X).
(3) D aws ea men assignmen s D1:n∼Loc(ψ,p),obse esou comesYi(Di),and
o ms an es ima e o he ATE, po en ially adjus ing o co a ia es h(X).
We a e agnos ic abou he exac ime a which he co a ia es a e obse ed, subjec o
he cons ain s abo e. Fo example, i could be ha only ψ(X)is obse ed a he design
s age, while he ull ec o Xis collec ed la e wi h he ou comes, and he expe imen e
chooses o adjus o h(X)⊆X. Al e na i ely, he ull ec o Xcould be obse ed a
he design s age, bu he expe imen e chooses o only s a i y on ψ(X), and adjus s o
h(X)⊆Xa s ep (3). We may o may no ha e ψ(X)⊆h(X).2
Conside he unadjus ed es ima o gi en by he coe icien 
θon Din he eg ession
Y∼1+D.Be o ediscussingco a ia eadjus men ,we i s s a eahelp ul a iancede-
composi ion o 
θ ha will be used ex ensi ely below. Le c(X)=E[Y(1)−Y(0)|X]de-
no e he condi ional a e age ea men e ec (CATE) and σ2
d(X)=Va (Y(d)|X) he he -
e oskedas ici y unc ion. De ine he balance unc ion:
b(X;p)=EY(1)|X1−p
p1/2
+EY(0)|Xp
1−p1/2
. (2.2)
We o en deno e b=b(X;p)in wha ollows. Cy ynbaum (2022)shows ha i D1:n∼
Loc(ψ,p) hen √n(
θ−ATE )⇒N(0, V)wi h
V=Va c(X)+EVa (b|ψ)+Eσ2
1(X)
p+σ2
0(X)
1−p. (2.3)
2Ou asymp o ic amewo k le s h(X),ψ(X)be ixed as n→∞.
976 Max Cy ynbaum Quan i a i e Economics 15 (2024)
The a iance Vis in ac he Hahn (1998) semipa ame ic a iance bound3 o
he ATE (wi h co a ia es ψ(X)), p o iding a o mal sense in which s a i ica ion does
nonpa ame ic eg ession adjus men “by design.” The middle e m is he mos im-
po an o ou analysis below. Fo example, in his no a ion he di e ence in asymp-
o ic e iciency be ween s a i ica ions ψ1and ψ2( o ixed p)issimplyE[Va (b|ψ1)] −
E[Va (b|ψ2)]. No e also ha E[Va (b|ψ)] ≤Va (b) o any ψ, showing how s a i ica ion
emo es he a iance due o luc ua ions ha a e p edic able by ψ(X).
Mo ing beyond he di e ence o means es ima o 
θ, suppose ha a he analysis
s age, he expe imen e has access o co a ia es h(X), which may s ic ly con ain ψ(X).
One may y o u he imp o e he e iciency o ATE es ima ion by eg ession adjus -
men using hese co a ia es, ei he using he s anda d eg ession Y∼1+D+ho he
eg ession Y∼1+D+h+Dh (wi h demeaned co a ia es) s udied in Lin (2013). We
s udy he in e ac ion be ween co a ia e adjus men and s a i ica ion in Sec ion 3.1 be-
low, cha ac e izing he op imal linea adjus men .
3. Main esul s
3.1 E icien linea adjus men in s a i ied expe imen s
In his sec ion, we begin by s udying he e iciency o commonly used co a ia e-adjus ed
es ima o s o he ATE unde s a i ied andomiza ion. Lin (2013) showed ha in a com-
ple ely andomized expe imen , equi alen o D1:n∼Loc(ψ,p)wi h ψ=1, eg ession
adjus men wi h ull ea men -co a ia e in e ac ions is asymp o ically weakly mo e e -
icien han di e ence o means es ima ion. Negi and Woold idge (2021)ex ended his
esul o ATE es ima ion in he supe popula ion amewo k ha we use in his pape .
In e es ingly, we show ha his esul is a ypical. Fo a gene al s a i ied expe imen
wi h ψ=1, Lin (2013) s yle eg ession adjus men may be s ic ly ine icien ela i e o
di e ence o means. The issue is ha he in e ac ed eg ession sol es he w ong op-
imiza ion p oblem, minimizing a ma ginal a iance objec i e when, due o he s a -
i ica ion, i should ins ead minimize a mean-condi ional a iance objec i e, condi-
ional on he s a i ica ion a iables ψ. In ac , he Lin es ima o is o ally insensi i e
o he s a i ica ion, es ima ing he same adjus men coe icien o any s a i ied design
D1:n∼Loc(ψ,p). Because o his, in e ac ed eg ession is gene ically subop imal and in
some cases can e en be s ic ly less e icien han di e ence o means. Be o e p oceed-
ing, we s a e ou main assump ion.
Assump ion 3.1 (Smoo hness and momen condi ions). Assume he ollowing:
(i) The condi ional expec a ions E[h(X)|ψ]and E[Y(d)|ψ] o d∈{0, 1}a e Lipschi z
con inuous in he s a i ica ion a iables ψ.
(ii) The momen s E[Y(d)4]<∞ o d∈{0, 1}and E[|h (X)|4]<∞ o all 1≤ ≤
dim(h),|ψ(X)|2<K<∞a.s.and Va (h)0.
3A ms ong (2022) shows ha his a iance bound also holds o s a i ied designs.
Quan i a i e Economics 15 (2024) Co a ia e adjus men in s a i ied expe imen s 977
Now we a e eady o de ine he Lin es ima o and s a e ou i s esul . Deno e
hi=h(Xi)and demeaned co a ia es ˜
hi=hi−En[hi],wi hEn[hi]≡n−1n
i=1hi.TheLin
es ima o 
θLis he coe icien on Diin he in e ac ed eg ession
Yi∼1+Di+˜
hi+Di˜
hi. (3.1)
De ine he wi hin ea men a m co a ia e means ¯
h1=En[hiDi]/En[Di]and ¯
h0=
En[hi(1−Di)]/En[1−Di]. The Lin es ima o 
θLcan be ela ed o he di e ence o means
es ima o 
θas

θL=
θ−γ
L(¯
h1−¯
h0). (3.2)
He e, he adjus men coe icien γLis γL=(1−p)(
a1+
a0)+p
a0,whe e
a0and 
a1a e
he coe icien s on ˜
hiand Di˜
hiin Equa ion (3.1). The ollowing heo em cha ac e izes
he asymp o ic p ope ies o his es ima o unde s a i ied designs.
Theo em 3.2. Le Assump ion 3.1 hold.I D1:n∼Loc(ψ,p), hen he Lin es ima o
√n(
θL−ATE )⇒N(0, V)wi h
V=Va c(X)+EVa b−γ
Lh|ψ+Eσ2
1(X)
p+σ2
0(X)
1−p.
The adjus men coe icien sa is ies γL
p
→γLwi h γL=a gminγ∈RdhVa (b−γh).
The a iance Vdi e s om he a iance o he unadjus ed es ima o only in he mid-
dle e m, which changes om E[Va (b|ψ)] in he unadjus ed case o E[Va (b−γ
Lh|ψ)]
o he in e ac ed eg ession. C ucially, he second s a emen o Theo em 3.2 shows ha
he adjus men coe icien γLa emp s o minimize a ma ginal a iance, ins ead o he
mean-condi ional a iance ha shows up in Vabo e. Because o his, he es ima o may
be ine icien o s a i ica ions ψ=1, since in gene al
γL=a gmin
γ∈Rdh
Va b−γh=a gmin
γ∈Rdh
EVa b−γh|ψ≡γ∗.
Obse e ha he Lin es ima o is comple ely insensi i e o he expe imen al design,
es ima ing he same adjus men coe icien γL=a gminγVa (b−γh) o any s a i ica-
ion a iables ψ(X). The ollowing example shows ha his can lead o s ic ine iciency
ela i e o di e ence o means es ima ion.
Example 3.3 (Random assignmen o class size). Suppose Y(d)a e s uden es sco es
unde andom assignmen o one o wo class sizes d∈{0, 1}.Le h(X)be pa en ’s weal h
and ψ(X)p e ious yea (baseline) es sco es. Suppose pa en ’s weal h is p edic i e o
u u e es sco es ma ginally so ha Co (h,Y(d)) >0. Then Co (h,b)>0 and he Lin
coe icien is γL=Va (h)−1Co (h,b)>0. Howe e , i on a e age pa en ’s weal h has
no p edic i e powe o es sco es condi ional on a s uden ’s baseline sco es (a p oxy
o abili y), hen E[Co (h,Y(d)|ψ)] =0. In his case, eg ession adjus men o pa en ’s
978 Max Cy ynbaum Quan i a i e Economics 15 (2024)
weal h h(X)in an expe imen s a i ied on he ea lie sco es ψ(X)will be s ic ly less
e icien han unadjus ed es ima ion since
Vlin −Vunadj =EVa b−γ
Lh|ψ−EVa (b|ψ)
=−2γLECo (h,b|ψ)+γ2
LEVa (h|ψ)=γ2
LEVa (h|ψ)>0.
An impo an special case occu s when he design is comple ely andomized (ψ=1)
o i he co a ia es and s a i ica ion a iables a e independen h(X)⊥⊥ ψ(X).In his
case, he Lin es ima o is weakly mo e e icien han di e ence o means since we ha e
EVa b−γ
Lh|ψ=Va b−γ
Lh=min
γVa b−γh≤Va (b).
An analogue o Theo em 3.2 also holds o he nonin e ac ed eg ession es ima o
Yi∼1+Di+hiunde s a i ied designs D1:n∼Loc(ψ,p). The nonin e ac ed es ima o is
known o be ine icien ela i e o di e ence o means e en o comple ely andomized
expe imen s unless p=1/2 o ea men e ec s a e homogeneous. Fo comple eness,
we gi e asymp o ic heo y and op imali y condi ions o his es ima o unde s a i ied
andomiza ion in Sec ion A.3 in he Supplemen al Appendix.
We no ed abo e ha he Lin es ima o 
θLcan be w i en in he canonical o m

θL=
θ−γ
L(¯
h1−¯
h0). In ac , mos commonly used adjus ed es ima o s can be w i en
in he s anda d o m 
θadj =
θ−γ(¯
h1−¯
h0) o some γ,up oo de Op(n−1) ac o s. The
ollowing heo em desc ibes he asymp o ic p ope ies o gene al co a ia e-adjus ed
es ima o s 
θadj o his o m. To a oid ca ying a ound ac o s o pin ou a iance ex-
p essions, in wha ollows, we scale adjus ed es ima o s by he no maliza ion cons an
cp=p(1−p).
Theo em 3.4. Le Assump ion 3.1 hold.Suppose γp
→γand conside he adjus ed es i-
ma o

θadj =
θ−γ(¯
h1−¯
h0)cp.
I D1:n∼Loc(ψ,p) hen √n(
θadj −ATE )⇒N(0, V(γ)) wi h
V(γ)=Va c(X)+EVa b−γh|ψ+Eσ2
1(X)
p+σ2
0(X)
1−p. (3.3)
We de ine a linea ly-adjus ed es ima o o be asymp o ically e icien i i globally
minimizes he asymp o ic a iance V(γ)in he p e ious heo em.
De ini ion 3.5 (Op imal linea adjus men ). The es ima o 
θadj =
θ−γ(¯
h1−¯
h0)cpis
e icien o he design D1:n∼Loc(ψ,p)and co a ia es h(X)i γp
→γ∗ o an op imal
adjus men coe icien
γ∗∈a gmin
γ∈Rdh
EVa b−γh|ψ.
In pa icula , V(γ∗)=minγ∈RdhV(γ).
Quan i a i e Economics 15 (2024) Co a ia e adjus men in s a i ied expe imen s 985
In he es o his sec ion, we de elop es ima o s ha a e ully e icien o any inely
s a i ied design, wi hou imposing any assump ions on ea men e ec he e ogenei y
o ea men p opo ions.
3.4.2 Pa ialled Lin es ima o Fi s , we de ine a pa ialled e sion o he Lin es ima o .
Le g(i)deno e he g oup ha uni ibelongs o and de ine he wi hin-g oup pa ialled
co a ia es
ˇ
hi=hi−1
k
j∈g(i)
hj.
Fo example, i k=2 his is jus he wi hin-pai co a ia e di e ence ˇ
hi=(1/2)(hi−
hm(i)),whe eiis ma ched o m(i). We can hink o ˇ
hias an inconsis en bu app oxi-
ma ely unbiased signal o he nonpa ame ically esidualized co a ia e hi−E[hi|ψi].
Nex , we use hese pa ialled co a ia es in he Lin eg ession
Y∼1+Di+ˇ
hi+Diˇ
hi. (3.8)
De ine he pa ialled Lin es ima o 
θPL o be he coe icien on Diin his eg ession. Fo
e e ence, simila o he Lin eg ession, we may w i e his in he s anda d o m 
θPL =

θ−γ
PL(¯
h1−¯
h0)cpwi h adjus men coe icien γPL =(
a1+
a0)1−p
p+
a0p
1−p,whe e
a0
and 
a1a e coe icien s on ˇ
hiand Diˇ
hi.
Ou main esul in Theo em 3.23 below shows ha he pa ialled Lin es ima o 
θPL
is asymp o ically e icien in he sense o De ini ion 3.5,wi hγPL
p
→γ∗ o he op imal
adjus men coe icien γ∗.
Rema k 3.17 (In ui ion o op imali y). Theo em 3.4 showed ha an es ima o 
θ−
γ(¯
h1−¯
h0)cpis e icien i γp
→γ∗and γ∗sol es he condi ional-mean a iance p oblem
γ∗∈a gminγE[Va (b−γh|ψ)]. By using wi hin-s a um pa ialled eg esso s ˇ
hi,we o ce
he Lin es ima o o only use co a ia e signal hi−E[hi|ψi] ha is mean-independen o
he s a i ica ion a iables.
Rema k 3.18 (T ea men -s a a in e ac ions). As an al e na i e o 
θPL, one may a -
emp o use he Lin eg ession Yi∼(1, hi,gn(i)) +Di(1, hi,gn(i)) wi h lea e-one-ou
s a a ixed e ec s gn(i)=(1(i∈gj))n/k−1
j=1. Un o una ely, his p oduces collinea e-
g esso s o p=a/k i ei he a=1o a=k−1, which includes he case o ma ched
pai s. To see he issue, one can show by F isch–Waugh ha in con as o Equa ion
(3.8) abo e, his es ima o pa ials co a ia es hisepa a ely in each ea men a m, using
ˇ
hi1=hi−a−1j∈g(i)hjDji Di=1and ˇ
hi0=hi−(k−a)−1j∈g(i)hj(1−Dj)i Di=0.
Fo ins ance, i a=1 hen his is ˇ
hi=hi−hi=0 o alli, showing collinea i y. In he
case 1 <a<k−1 whe e his es ima o is easible, i is asymp o ically equi alen o he
pa ialled Lin es ima o . Howe e , ini e-sample p ope ies will be wo se due o noisie
wi hin-a m pa ialling.

986 Max Cy ynbaum Quan i a i e Economics 15 (2024)
Rema k 3.19. A calcula ion shows ha ou es ima o 
θPL is nume ically equi alen o
a eg ession es ima o p oposed in Ke, Liu, and Yang (2024), which he au ho s de i e
al e na ely h ough an op imal p ojec ion a gumen . They s udy es ima ion o he SATE
unde s a i ied andomiza ion in a ini e popula ion amewo k, p o iding conse a i e
in e ence. They do no de i e he exac o m o he asymp o ic a iance, ins ead lea ing
i as an in ini e sum, which hey assume con e ges o some limi . By con as , we de-
i e he exac o m o he asymp o ic a iance unde he da a-adap i e s a i ica ions in
De ini ion 2.1, enabling asymp o ically exac in e ence on he ATE using 
θPL.
3.4.3 G oup OLS es ima o Nex , we gene alize an es ima o p oposed by Imbens and
Rubin (2015) o co a ia e adjus men in ma ched pai s expe imen s o mo e gene al
s a i ied designs. Fo each g oup o uni s g=1, ,n/k in he design D1:n∼Loc(ψ,p),
de ine he wi hin-g oup di e ence o means o ou comes and co a ia es
yg=1
k
i∈g
YiDi
p−1
k
i∈g
Yi(1−Di)
1−pand hg=1
k
i∈g
hiDi
p−1
k
i∈g
hi(1−Di)
1−p.
Fo any g oup-indexed a ay (xg)g,deno eEg[xg]=k
ngxg. De ine he g oup OLS es i-
ma o 
θGby he eg ession
yg=
θG+γ
Ghg+eg(3.9)
wi h Eg[(1, hg)eg]=0. Fo mo i a ion, no e ha i h=0 hen his becomes yg=
θG+eg
and 
θGis jus he unadjus ed es ima o 
θG=¯
Y1−¯
Y0. Mo e gene ally, he adjus ed e -
sion can be w i en 
θG=Eg[yg]−γ
GEg[hg]=
θ−γ
G(¯
h1−¯
h0)wi h adjus men coe i-
cien γG=Va g(hg)−1Co g(hg,yg). The es ima o s 
θGand 
θPL a e nume ically iden ical
o he case o ma ched pai s, bu no o p=1/2. The main esul o his sec ion shows
ha 
θGis asymp o ically equi alen o he pa ialled Lin es ima o 
θPL,andbo ha e
asymp o ically op imal.
Rema k 3.20 (In ui ion o efficiency). The es ima o 
θGuses wi hin-g oup di e ences
o co a ia es ¯
hg1−¯
hg0 o p edic wi hin-g oup ou come di e ences ¯
Y1g−¯
Y0g. Simila o
he pa ialled Lin s a egy, by doing his we only measu e he a ia ion in co a ia es and
po en ial ou comes o hogonal o he s a i ica ion a iables. This o ces leas squa es
o compu e a condi ional a iance-co a iance ade-o , sol ing he op imal adjus men
p oblem in De ini ion 3.5. In pa icula , he p oo o Theo em 3.23 shows ha i D1:n∼
Loc(ψ,p), hen he adjus men coe icien
γG=Va g(hg)−1Co g(hg,yg)p
→cpa gmin
γ
EVa b−γh|ψ.
Rema k 3.21. Imbens and Rubin (2015) p opose 
θGin he case o ma ched pai s p=
1/2. Thei analysis uses a oy sampling model whe e he pai s hemsel es a e d awn
“p e-ma ched” om a supe popula ion. By con as , we model he expe imen al uni s as
being sampled om a supe popula ion, wi h uni s ma ched in o da a-dependen s a a
pos -sampling. This mo e ealis ic model complica es he analysis, p oducing di e en
Quan i a i e Economics 15 (2024) Co a ia e adjus men in s a i ied expe imen s 987
limi ing a iances and equi ing di e en in e ence p ocedu es. In a design-based se -
ing, Foga y (2018) shows ha he Imbens and Rubin (2015) es ima o is weakly mo e
e icien han di e ence o means o ma ched pai s designs. By con as , we ex end his
es ima o o a la ge amily o ine s a i ica ions s ic ly con aining ma ched pai s, and
show ha i is asymp o ically op imal among linea ly adjus ed es ima o s.
3.4.4 Ty anny-o - he-mino i y es ima o Finally, we de ine y anny-o - he-mino i y
(ToM) adjus men , ex ending Lin (2013). To do so, de ine he adjus men coe icien :
γTM =Va n(ˇ
hi)−1Co n(ˇ
hi,Yi|Di=1)1−p
p+Co n(ˇ
hi,Yi|Di=0)p
1−p. (3.10)
De ine he ToM es ima o 
θTM =
θ−γ
TM(¯
h1−¯
h0)cp. The main di e ence be ween he
ToM and pa ialled Lin adjus men coe icien s is ha γTM es ima es he condi ional
a iance E[Va (h|ψ)] only once, using he sample a iance Va n(ˇ
hi) o he ull expe i-
men al sample. By con as , pa ialled Lin es ima es his e m sepa a ely in each ea -
men a m, using Va n(ˇ
hi|Di=1)and Va n(ˇ
hi|Di=0). Because o his, we expec 
θTM o
be mo e s able han 
θPL in small expe imen s.
Rema k 3.22. Lu and Liu (2024) p opose an al e na e ToM eg ession adjus men o
s a i ied expe imen s. To compa e he app oaches, o p opensi y p=a/k de ine he
wi hin-a m pa ialling ˇ
hi1=hi−a−1i∈gDihiand ˇ
hi0=hi−(k−a)−1i∈g(1−Di)hi.
Thei es ima o akes he o m 
θLL =
θ−γ
LL(¯
h1−¯
h0). In ou no a ion, hei adjus men
coe icien γLL =
S−1
hh 
ShY has

Shh =EnDiˇ
hi1ˇ
h
i1
p2
a
a−1+(1−Di)ˇ
hi0ˇ
h
i0
(1−p)2
k−a
k−a−1
and simila ly o 
ShY . Thei app oach is in easible i a=1o a=k−1. Fo example, his
p ohibi s i s use in ma ched pai s and ma ched iples expe imen s.
3.4.5 Main esul The main esul o his sec ion shows ha all h ee es ima o s abo e
a e asymp o ically equi alen and e icien in he sense o De ini ion 3.5.
Theo em 3.23. Suppose Assump ions 3.1 and 3.14 hold.I D1:n∼Loc(ψ,p), hen 
θPL −

θG=op(n−1/2)and 
θPL −
θTM =op(n−1/2).We ha e √n(
θPL −ATE )⇒N(0, V∗)wi h he
op imal a iance
V∗=Va c(X)+min
γEVa b−γh|ψ+Eσ2
1(X)
p+σ2
0(X)
1−p.
Me hods o asymp o ically exac in e ence on he ATE using hese es ima o s a e
discussed in Sec ion 4below. Ou simula ions and empi ical esul s show ha he pa -
ialled Lin, G oup OLS, and ToM es ima o s beha e e y simila ly in ini e samples.
988 Max Cy ynbaum Quan i a i e Economics 15 (2024)
3.5 Fu he adjus men o s a i ica ion a iables
In his sec ion, we p o ide modi ied e sions o he p e ious es ima o s ha allow u -
he adjus men o co a ia es z(ψ) ha a e unc ions o he s a i ica ion a iables. As
discussed abo e, his canno imp o e i s -o de e iciency bu may imp o e ini e sam-
ple pe o mance by co ec ing o any emaining imbalances in ψno con olled by he
s a i ica ion.
Deno e zi=z(ψi). Fo each es ima o 
θkabo e wi h k∈{FE, PL, G,TM
},wede-
ine a modi ied es ima o o he o m τk=
θk−α
k(¯
z1−¯
z0)cp. Fo he ixed e ec s
es ima o , de ine τFE o be he coe icien on Diin he eg ession Yi∼(1, Di,ˇ
hi,zi).
Fo he pa ialled Lin es ima o , de ine τPL o be he coe icien on Diin he eg es-
sion Yi∼(1, ˇ
hi,zi)+Di(1, ˇ
hi,zi). De ine he modi ied ToM es ima o o be as in Equa-
ion (3.10), wi h (ˇ
hi,zi)in place o ˇ
hi. Finally, de ine he modi ied g oup OLS es ima o
τG=
θG−α
G(¯
z1−¯
z0)cp,wi hαG=αPL. Ou nex heo em shows ha hese es ima o s
a e asymp o ically equi alen o he o iginal e sions o each es ima o ha do no ad-
jus o z(ψ). Howe e , he simula ions in Sec ions 5and 6show ha hey may pe o m
be e in small expe imen s.
Theo em 3.24. Suppose Assump ions 3.1 and 3.14 hold,as well as Va (z)0and
E[|z|2
2]<∞.Then i D1:n∼Loc(ψ,p),we ha e τk=
θk+op(n−1/2) o k∈{FE, PL, G,
TM}.Each es ima o has he o mτk=
θk−α
k(¯
z1−¯
z0)cpwi hαFE
p
→a gminαVa ( −αz)
o as in Theo em 3.15 and αPL,αG,αTM
p
→a gminαVa (b−αz).
F om he second s a emen o he heo em, we can in e p e he modi ied es ima-
o s as aking a conse a i e app oach ha igno es s a i ica ion on ψand adjus s o
imbalances in z(ψ)as i he expe imen we e comple ely andomized.
4. In e ence
In his sec ion, we p o ide asymp o ically exac con idence in e als o he ATE in s a -
i ied expe imen s using gene ic linea ly adjus ed es ima o s. O e co e age is known o
be a p oblem o in e ence based on he usual Eicke –Hube –Whi e (EHW) a iance es-
ima o in s a i ied expe imen s. Fo example, Bai, Romano, and Shaikh (2021)shows
ha he EHW a iance es ima o s o Y∼1+D+hand he ixed-e ec s eg ession
Y∼D+h+zna e asymp o ically conse a i e o ma ched pai s designs i h=0. To
he bes o ou knowledge, we gi e he i s asymp o ically exac in e ence me hods o
co a ia e-adjus ed (h= 0) ATE es ima ion unde gene al s a i ied designs. Ou main
in e ence esul applies o any es ima o o he o m 
θ−γ(¯
h1−¯
h0)cp+op(n−1/2).In
pa icula , his enables asymp o ically exac in e ence on he ATE using any o he es i-
ma o s in his pape . Ou con idence in e als a e sho e han hose p oduced by EHW
in he simula ions and empi ical applica ion below, aking ull ad an age o he e i-
ciency gains om bo h s a i ica ion and co a ia e adjus men .
To de ine ou in e ence me hods, conside such an es ima o 
θ(γ)=
θ−γ(¯
h1−¯
h0)cp
wi h γp
→γ. De ine he augmen ed po en ial ou comes Ya
i(d)=Yi(d)−cpγhi o d∈
Quan i a i e Economics 15 (2024) Co a ia e adjus men in s a i ied expe imen s 989
{0, 1}and he augmen ed ou come Ya
i=Yi−cpγhi. Then appa en ly

θ(γ)=¯
Y1−¯
Y0−cpγ(¯
h1−¯
h0)=¯
Ya
1−¯
Ya
0. (4.1)
Ou s a egy is o apply he in e ence esul s o Cy ynbaum (2023) o di e ence o
means es ima ion 
θ=¯
Y1−¯
Y0 o he di e ence o augmen ed po en ial ou comes
¯
Ya
1−¯
Ya
0.Todoso,le Gndeno e he se o g oups in De ini ion 2.1. Fo each g∈Gn, de ine
he g oup cen oid ¯
ψg=|g|−1i∈gψi.Le ν:Gn→Gnbe a bijec i e ma ching be ween
g oups sa is ying ν(g)=g,ν2=Id, and he homogenei y condi ion
1
n
g∈Gn|¯
ψg−¯
ψν(g)|2
2=op(1). (4.2)
In p ac ice, νis ob ained by simply ma ching he g oup cen oids ¯
ψgin o pai s using he
De igs (1988) ma ching algo i hm. Le Gν
n={g∪ν(g):g∈Gn}be he unions o pai ed
g oups o med by his ma ching. De ine a(g)=i∈gDiand k(g)=|g|. Finally, de ine
he a iance es ima o componen s:

1=n−1
g∈Gν
n
1
a(g)−1
i=j∈g
Ya
iYa
jDiDj(1−p)
p2,

0=n−1
g∈Gν
n
1
(k−a)(g)−1
i=j∈g
Ya
iYa
j(1−Di)(1−Dj)p
(1−p)2,

10 =n−1
g∈Gn
k
a(k−a)(g)
i,j∈g
Ya
iYa
jDi(1−Dj).
Nex , de ine he a iance es ima o :

V=Va n(Di−p)Ya
i
p−p2−
1−
0−2
10. (4.3)
Ou in e ence s a egy begins wi h he sample a iance o he adjus ed es ima o ,
which is consis en o he asymp o ic a iance o 
θadj unde an i.i.d. design, bu oo
la ge unde s a i ied designs. We co ec his sample a iance using he es ima o s
abo e, which measu e how well he s a i ica ion a iables p edic augmen ed ou comes
in local egions o he co a ia e space. This sec ion’s main esul shows ha 
Vis consis-
en o he limi ing a iance o Theo em 3.4, enabling asymp o ically exac in e ence on
he ATE using adjus ed es ima o s.
Theo em 4.1 (In e ence). Unde he condi ions o Theo em 3.4,i D1:n∼Loc(ψ,p), hen

V=V+op(1).
By Theo em 4.1 and ou p e ious asymp o ic esul s in Theo em 3.4, he con idence
in e al 
C=[
θ(γ)±
V1/2c1−α/2/√n]wi h cα=−1(α)is asymp o ically exac in he sense
ha P(ATE ∈
C)=1−α+o(1).
990 Max Cy ynbaum Quan i a i e Economics 15 (2024)
5. Simula ions
In his sec ion, we use simula ions o es he ini e-sample pe o mance o he es ima-
o s s udied abo e. We conside quad a ic ou come models o he o m
Yi(d)=ψ
iQdψi+ψ
iLd+cd·u(Xi)+d
iEd
i|Xi=0
o d∈{0, 1}.Thecomponen ui=u(Xi) ep esen s co a ia e signal ha is independen
o he s a i ica ion a iables ψ(Xi). A e implemen ing he design D1:n∼Loc(ψ,p),
we ecei e access o scala co a ia es hi ha a e co ela ed wi h bo h ψiand Yi(d).In
pa icula , suppose ha hi=ψ
iQhψi+ψ
iLh+uiwi h E[ui|ψi]=0. In he ollowing sim-
ula ions, we le ψi∼N(0, Im),ui∼N(0, 1),andd
i∼N(0, 1/10)wi h (ψi,ui,d
i)join ly
independen . We use ea men p opo ions p=2/3 unless o he wise speci ied. Wi h
m≡dim(ψ),le A∈Rm×mha e Aij =1 o i=jand Aii =0. We simula e he ollowing
DGPs:
Model 1: Quad a ic coe icien s Qh=(1/m2)Aand Q0=Q1=(1/m)A. Linea co-
e icien s L0=1m,L1=21m,Lh=1m. Reg esso signal c1=c0=−3.
Model 2: As in Model 1 bu c0=−4andc1=−1.
Model 3: As in Model 2 bu p=1/2.
Model 4: As in Model 1 bu c0=2andc1=4.
Model 5: As in Model 1 bu c0=2andc1=4andp=1/2.
Model 6: As in Model 1 bu Qh=(1/100)A.
We begin by compa ing he e iciency p ope ies o di e en linea ly adjus ed es-
ima o s. Unadj e e s o simple di e ence o means (unadjus ed). The Lin es ima o is
s udied in Theo em 3.2.Nai e e e s o he nonin e ac ed eg ession Y∼(1, D,h),(The-
o em A.4). FE e e s o he ixed-e ec s es ima o (Theo em 3.15)andPlin he pa ialled
Lin es ima o (Theo em 3.23). GO e e s o G oup OLS and ToM e e s o Ty anny-o -
he-Mino i y es ima ion (Theo em 3.23). S a a con ols e e o modi ied e sions o
each o he p e ious es ima o s ha u he adjus o pa ame ic s a a con ols z(ψ),
as discussed in Sec ion 3.5. In ou simula ions, we se z(ψ)=ψ.Ad e e s o an adap-
i e5es ima o ha se s 
θadj =
θLi 
V(γL)≤
V(γPL)and 
θadj =
θPL o he wise,6including
pa ame ic con ols z(ψ)=ψin bo h cases.
Table 1s udies ini e sample e iciency. We p esen he mean squa ed e o (MSE)
a io, ela i e o unadjus ed es ima ion, o each o he adjus ed es ima o s abo e. The
bo om line o he able epo s he excess isk Rko each es ima o k ela i e o he op i-
mal es ima o . To de ine his, le MSEk,sbe he ela i e MSE o es ima o kin simula ion
5This es ima o is poin wise asymp o ically equi alen o
θPL. Issues wi h pos model-selec ion in e ence
(e.g., Leeb and Po sche (2005)) appea o be less wo ying he e, since e en unde he ixed al e na i e
γ∗=γL, he Lin es ima o is s ill √n-consis en and asymp o ically unbiased.
6We could also use a c oss- i e sion o 
V(γ) o educe bias. Howe e , he in-sample c i e ion pe o med
qui e well in ou simula ions.

Quan i a i e Economics 15 (2024) Co a ia e adjus men in s a i ied expe imen s 991
Table 1. Ra io o MSEs (%), adjus ed s. unadjus ed es ima ion.
No s a a con ols S a a con ols z(ψ)
(n,dim(ψ)) Model Unadj Nai e Lin FE Plin GO ToM Nai e Lin FE Plin GO ToM Ad
(600, 2) 1 100 113 102 49 48 49 48 36 35 35 37 37 36 34
2 100 126 102 64 57 58 57 60 46 52 47 47 47 45
3 100 116 116 38 38 38 38 48 48 36 36 37 36 37
4 100 27 31 31 27 27 27 26 26 38 32 33 32 26
5 100 28 28 18 18 18 18 21 21 19 19 19 19 19
6 100 100 100 11 11 11 11 7 7 9 9 9 9 7
(1200, 2) 1 100 114 103 44 44 44 44 35 34 31 33 33 33 32
2 100 126 102 60 56 56 56 61 47 50 47 46 47 45
3 100 116 116 38 38 38 38 48 48 37 37 37 37 37
4 100 26 30 29 25 25 25 23 24 36 30 30 30 24
5 100 28 28 17 17 17 17 20 20 17 18 17 18 18
6 100 101 101 9 9 9 9 7 7 8 8 8 8 7
(1200, 5) 1 100 142 127 85 84 84 84 25 24 41 46 55 46 24
2 100 145 123 94 86 87 86 45 34 57 54 62 54 34
3 100 137 137 81 81 81 81 40 40 54 54 57 54 40
4 100 27 31 31 27 27 27 25 20 54 45 49 45 20
5 100 32 32 24 24 24 24 18 18 38 38 39 38 18
6 100 138 138 67 67 67 67 15 15 36 36 39 37 15
Rk73 65 60 17 15 15 15 5 2 10 8 10 8 0.2
s.Thenwese Rk=(1/S)s(MSEk,s−minjMSEj,s), a e aging o e all simula ions in
he able. All esul s a e calcula ed using 2000 Mon e Ca lo epe i ions.
In models 1, 2, and 3, bo h Nai e and Lin s yle linea adjus men a e s ic ly in-
e icien ela i e o unadjus ed es ima ion. These models ha e ma ginal co a iance
Co (Y(d),h)>0 bu condi ional co a iance E[Co (Y(d),h|ψ)] <0, condi ional on he
s a i ica ion a iables. Because o his, he op imal adjus men coe icien γ∗<0, while
he Nai e and Lin eg essions es ima e posi i e adjus men coe icien s γN,γL>0, lead-
ing o e en wo se pe o mance han unadjus ed es ima ion in some cases. Fo Models
4and5, heNai e and Lin me hods a e compe i i e wi h he gene ic e icien me h-
ods om Sec ion 3.4. This is because in hese cases we made i so ha Co (Y(d),h)≈
E[Co (Y(d),h|ψ)],so ha “bychance”γ∗is close o γNand γL. Howe e , he pa ame -
ic coe icien s γNand γLa e es ima ed mo e p ecisely han he semipa ame ic objec
γ∗=E[Va (h|ψ)]−1E[Co (h,b|ψ)].Fo Model6,Lin wi h z(ψ)=ψcon ols is (app oxi-
ma ely) op imal by Theo em 3.9, since E[w|ψ]is (app oxima ely) linea in ψ.
Summa izing ou indings, he Lin,Plin,andNai e es ima o s wi h pa ame ic
S a a con ols z(ψ)=ψhad low excess isk ac oss speci ica ions, while he Ad es i-
ma o was he mos e icien o e all. The Nai e and Lin es ima o s wi hou s a a con-
ols o wi hin-s a um pa ialling had la ge MSE. The Plin,GO,andToM es ima o s
had simila MSE ac oss model speci ica ions. These gene ic me hods pe o med he
bes in egimes wi h la ge n,smalldim(ψ), and nonlinea E[h|ψ]. In hese cases, he
gap γL−γ∗be ween he subop imal Lin coe icien and op imal coe icien γ∗domi-
992 Max Cy ynbaum Quan i a i e Economics 15 (2024)
Table 2. P ope ies o in e ence.
No s a a con ols S a a con ols z(ψ)
Model Unadj Nai e Lin FE Plin GO ToM Nai e Lin FE Plin GO ToM Ad
%CI Leng h
s. Unadj
1 0 17 11 −5−5−5−5−49 −50 −34 −29 −26 −29 −50
2 0 18 10 −3−4−4−4−33 −41 −25 −25 −22 −25 −41
3 0 16 16 −6−6−6−6−36 −36 −24 −24 −24 −24 −36
40−46 −43 −42 −46 −46 −46 −50 −55 −22 −31 −26 −30 −55
50−44 −44 −49 −49 −49 −49 −56 −56 −34 −34 −31 −34 −56
60 16 16 −12 −12 −12 −12 −59 −59 −35 −35 −35 −35 −59
Co e age
(Exac )
1 0.95 0.95 0.95 0.96 0.96 0.96 0.96 0.95 0.95 0.96 0.96 0.95 0.96 0.95
2 0.95 0.95 0.95 0.95 0.96 0.96 0.96 0.95 0.96 0.95 0.96 0.95 0.96 0.96
3 0.95 0.95 0.95 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
4 0.95 0.95 0.94 0.96 0.95 0.95 0.95 0.95 0.95 0.96 0.96 0.96 0.96 0.95
5 0.94 0.95 0.95 0.96 0.96 0.96 0.96 0.95 0.95 0.96 0.96 0.97 0.96 0.95
6 0.95 0.95 0.95 0.97 0.97 0.97 0.97 0.96 0.96 0.97 0.97 0.96 0.97 0.96
Co e age
(EHW)
1 0.99 0.95 0.96 0.97 0.99 0.99 0.98 0.99 0.97
2 0.99 0.95 0.95 0.94 0.99 0.98 0.93 0.98 0.96
3 1.00 0.96 0.95 0.96 1.00 0.99 0.93 0.98 0.97
4 0.99 0.99 0.90 0.98 1.00 0.97 0.68 0.98 0.97
5 0.99 0.97 0.90 0.98 1.00 0.96 0.65 0.99 0.99
6 1.00 0.97 0.96 0.96 1.00 0.99 0.97 1.00 0.99
na es he addi ional a iabili y Va (γ∗)>Va (γL) equi ed o es ima e γ∗( his a iabil-
i y inc eases wi h dim(ψ)). Fo example, Plin wi h z(ψ)con ols pe o ms he bes when
(n,dim(ψ)) =(1200, 2),bu Lin wi h z(ψ)con ols is much be e when dim(ψ)=5. The
Ad es ima o used a a iance p e es o choose be ween Plin and Lin (including z(ψ)
con ols), allowing i o pe o m well in bo h egimes.
Table 2 epo s ini e-sample e iciency and co e age p ope ies o he asymp o i-
cally exac in e ence me hods de eloped in Sec ion 4.Wele n=1200 and dim(ψ)=5.
The i s panel shows % change in con idence in e al leng h ela i e o unadjus ed es i-
ma ion. All con idence in e als a e compu ed using he me hod in Theo em 4.1.Wesee
ha he ela i e e iciency o di e en es ima o s a e e lec ed by ou in e ence me hods.
In pa icula , asymp o ically exac in e ence allows esea che s o epo sho e con i-
dence in e als when a mo e e icien adjus men me hod is used. In he second panel,
we show co e age p obabili ies o ou asymp o ically exac con idence in e al ac oss
a ange o linea ly adjus ed es ima o s. The inal panel shows co e age p obabili ies o
con idence in e als based on he usual HC2 a iance es ima o , whe e applicable. The
HC2-based con idence in e als signi ican ly o e co e .
6. Empi ical applica ion
In his sec ion, we apply ou me hods o he expe imen in Baysan (2022a),7who es i-
ma es he e ec o a poli ical in o ma ion campaign on suppo o a 2017 Tu kish e e -
7The da a is a ailable om Baysan (2022b).
Quan i a i e Economics 15 (2024) Co a ia e adjus men in s a i ied expe imen s 993
Table 3. Empi ical esul s.
No s a a con ols S a a con ols z(ψ)
Model Unadj Nai e Lin FE Plin GO ToM Nai e Lin FE Plin GO ToM

θadj −0.0054 0.0040 0.0047 0.0041 0.0021 0.0034 0.0021 0.0041 0.0040 0.0038 0.0037 0.0031 0.0019
SE 0.0088 0.0074 0.0074 0.0074 0.0078 0.0077 0.0081 0.0074 0.0073 0.0077 0.0076 0.0078 0.0083
HC2 0.0155 0.0075 0.0073 0.0075 0.0149 0.0075 0.0070 0.0736 0.0071
endum emo ing checks and balances on execu i e powe . The campaign was adminis-
e ed by he opposi ion Republican People’s Pa y (CHP), who opposed he e e endum.
Randomiza ion was pe o med a he neighbo hood le el, s a i ied on qua iles o CHP
o e sha e in he p e ious 2015 elec ions. The main ou come is he “No” o e sha e in
he 2017 e e endum.8Due o he cos o adminis e ing he campaign, p=2/11 ou o
n=550 o al neighbo hoods we e ea ed. In he o iginal analysis, Baysan (2022a)pe -
o med nonin e ac ed co a ia e adjus men (Theo em A.4) o h(X)=numbe o egis-
e ed o e s, numbe o alid o es, numbe o o es o he CHP in 2015, CHP o e sha e
in 2015, o e u nou , and CHP o e sha e qua ile ixed e ec s.
In he i s block o Table 3, we eplica e he neighbo hood-le el analysis o Baysan
(2022a). 
θadj is he poin es ima e om each adjus men s a egy, SE is he asymp o -
ically exac s anda d e o om Sec ion 4, and EHW is he usual obus s anda d e o
(HC2). Es ima es in he “s a a con ols z(ψ)” sec ion include qua ile ixed e ec s, while
he le mos sec ion does no . The esul s in Sec ion 3.3 show ha Lin adjus men wi h
qua ile ixed e ec s is e icien in his case, and indeed his has he smalles es ima ed
s anda d e o . The gene ic e icien es ima o s ha e sligh ly la ge SE. The asymp o i-
cally exac s anda d e o s om Sec ion 4a e gene ally simila o o smalle han EHW,
excep o he Lin, FE, and Plin es ima o s wi h z(ψ)con ols. Howe e , ou simula ion
also showed ha EHW s anda d e o s may se e ely unde co e in hese cases.9
O e all, changing he adjus men me hod did no ha e an economically meaning ul
e ec on he conclusions o he s udy, and we eco e he null e ec o Baysan (2022a)in
all cases. The co a ia e hk=“CHP o e sha e in 2015” is highly p edic i e o Y=“CHP
o e sha e in 2017,” so adjus ing o his a iable ex pos p o ides a modes a iance
educ ion e en a e s a i ying on 2015 o e sha e qua iles. Howe e , he es ima ed op-
imal coe icien γ∗
k≈0.27 and Lin coe icien γL,k≈0.31 a e qui e simila , so (ine i-
cien ) Lin adjus men s ill pe o ms qui e well. The o he co a ia es such as hj=“ o e
u nou ” a e e y weak p edic o s o ou comes, so changing he adjus men coe icien
on hese a iables does no ma e much.
Nex , we ask how each es ima o would ha e pe o med in he expe imen in Baysan
(2022a) unde coun e ac ual andomiza ion p ocedu es, such as ine s a i ica ion.10 To
8Baysan (2022a) es ima es e ec s o he campaign on o e sha e a bo h he ballo box and neighbo hood
le el. We ocus on he neighbo hood le el e ec s.
9We also no e ha Bai, Tabo d-Meehan, and Liu (2024) ha e ound he EHW s anda d e o om a linea
eg ession wi h block ixed e ec s o be po en ially in alid in a ela ed p oblem.
10Algo i hms and in e ence me hods o ine s a i ica ion wi h p= 1/2 ha e only been de eloped e-
cen ly, o example, Bai (2022) and Cy ynbaum (2023).
994 Max Cy ynbaum Quan i a i e Economics 15 (2024)
Table 4. Simula ed designs.
No s a a con ols S a a con ols z(ψ)
Model Unadj Nai e Lin FE Plin GO ToM Nai e Lin FE Plin GO ToM
Coa se Es 0.0000 0.0001 0.0003 0.0001 0.0002 0.0003 0.0000 0.0001 0.0004 0.0001 0.0003 0.0002 0.0000
SE 0.0085 0.0076 0.0075 0.0075 0.0077 0.0077 0.0078 0.0076 0.0074 0.0078 0.0076 0.0079 0.0079
HC2 0.0144 0.0078 0.0078 0.0077 0.0141 0.0078 0.0077 0.0736 0.0080
Fine Es −0.0001 0.0000 0.0000 0.0000 0.0000 0.0004 −0.0001 0.0000 0.0002 0.0000 0.0002 0.0004 −0.0001
SE 0.0077 0.0081 0.0080 0.0076 0.0077 0.0077 0.0077 0.0075 0.0075 0.0075 0.0075 0.0077 0.0077
HC2 0.0144 0.0141 0.0142 0.0078 0.0145 0.0078 0.0078 0.0733 0.0078
Fine
p=
1/2
Es −0.0001 −0.0001 −0.0001 0.0001 0.0001 0.0001 0.0001 0.0000 −0.0001 0.0001 0.0001 0.0001 0.0001
SE 0.0072 0.0073 0.0073 0.0070 0.0070 0.0070 0.0070 0.0066 0.0066 0.0066 0.0066 0.0066 0.0066
HC2 0.0113 0.0111 0.0111 0.0059 0.0114 0.0060 0.0060 0.0565 0.0061
Fine
dim(ψ)=3
Es 0.0000 0.0001 0.0002 0.0001 0.0002 0.0000 0.0003 0.0001 0.0002 0.0000 0.0002 0.0001 0.0001
SE 0.0155 0.0155 0.0155 0.0155 0.0155 0.0155 0.0155 0.0146 0.0146 0.0146 0.0146 0.0147 0.0147
HC2 0.0145 0.0145 0.0145 0.0089 0.0145 0.0079 0.0078 0.0740 0.0078