Dynamic regression discontinuity under treatment effect heterogeneity

Hsu, Yu-Chin; Shen, Shu
A icle
Dynamic eg ession discon inui y unde ea men e ec
he e ogenei y
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: Hsu, Yu-Chin; Shen, Shu (2024) : Dynamic eg ession discon inui y unde
ea men e ec he e ogenei y, Quan i a i e Economics, ISSN 1759-7331, The Econome ic Socie y,
New Ha en, CT, Vol. 15, Iss. 4, pp. 1035-1064,
h ps://doi.o g/10.3982/QE2150
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Quan i a i e Economics 15 (2024), 1035–1064 1759-7331/20241035
Dynamic eg ession discon inui y unde ea men e ec
he e ogenei y
Yu-Chin Hsu
Ins i u e o Economics, Academia Sinica, Depa men o Finance, Na ional Cen al Uni e si y,
Depa men o Economics, Na ional Chengchi Uni e si y, and CRETA, Na ional Taiwan Uni e si y
Shu Shen
Depa men o Economics, Uni e si y o Cali o nia
Reg ession discon inui y is a popula ool o analyzing economic policies o ea -
men in e en ions. This esea ch ex ends he classic s a ic RD model o a dy-
namic amewo k, whe e obse a ions a e eligible o epea ed RD e en s and,
he e o e, ea men s. Such dynamics o en complica e he iden i ica ion and es-
ima ion o long- e m a e age ea men e ec s. Empi ical pape s wi h such de-
signs ha e so a igno ed he dynamics o adop ed es ic i e iden i ying assump-
ions. This pape p esen s iden i ica ion s a egies unde a ious se s o weake
iden i ying assump ions and p oposes associa ed es ima ion and in e ence me h-
ods. The p oposed me hods a e applied o e isi he seminal s udy o Cellini, Fe -
ei a, and Ro hs ein (2010) on long- e m e ec s o Cali o nia local school bonds.
Keywo ds. Long- e m ea men e ec s, dynamic eg ession discon inui y, semi-
pa ame ic, a ying coe icien logi .
JEL classi ica ion. C31.
1. In oduc ion
Reg ession discon inui y (RD) models, which can be aced back o This le hwai e and
Campbell (1960), a e popula in policy e alua ions o o he se ings o ea men e ec
analysis. The se -up exploi s discon inui y in he design o many policies o nonpa a-
me ically iden i y ea men e ec s o obse a ions nea he eligibili y cu o . Classic
Yu-Chin Hsu: [email p o ec ed]
Shu Shen: [email p o ec ed]
This pape was i s p esen ed a a Uni e si y o Cali o nia, Be keley, causal in e ence g oup semina in
Ma ch 2020 unde he i le o “Dynamic Reg ession Discon inui y.” The au ho s a e g a e ul o h ee anony-
mous e e ees o aluable commen s and sugges ions on p e ious e sions o he pape . The au ho s also
hank semina pa icipan s om Uni e si y o Cali o nia, Be keley, Uni e si y o Cali o nia, I ine, Uni-
e si y o Ne ada, Reno, Uni e si y o Chicago, and Chambe lain Semina o help ul commen s. Yu-Chin
Hsu g a e ully acknowledges he esea ch suppo om Minis y o Science and Technology o Taiwan
(MOST107-2410-H-001-034-MY3, MOST110-2634-F-002-045), Na ional Science and Technology Council o
Taiwan (NSTC 112-2628-H-001-001), Academia Sinica In es iga o Awa d o Academia Sinica (AS-IA-110-
H01), and Cen e o Resea ch in Econome ic Theo y and Applica ions (113L8601) om he Fea u ed A eas
Resea ch Cen e P og am wi hin he amewo k o he Highe Educa ion Sp ou P ojec by he Minis y o
Educa ion o Taiwan. Shu Shen g a e ully acknowledges he esea ch suppo om UC Da is Small G an s
in Aid o Resea ch, C ea i e Ac i i ies, and Schola ship.
©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/QE2150
1036 Hsu and Shen Quan i a i e Economics 15 (2024)
s udies o RD iden i ica ion, es ima ion, and in e ence include Hahn, Todd, and an de
Klaauw (2001), Po e (2003), Lee (2008), Imbens and Lemieux (2008), Calonico, Ca a-
neo, and Ti iunik (2014), and Calonico, Ca aneo, Fa ell, and Ti iunik (2019), among
many o he s. Ca aneo and Ti iunik (2022) p o ide a comp ehensi e li e a u e e iew.
While he classic RD se -up, ei he sha p o uzzy, is s a ic in empi ical applica ions
we o en see si ua ions whe e each indi idual aces mul iple ounds o RD and, he e-
o e, could po en ially ecei e epea ed ea men s. Fo example, o e -app o ed mea-
su es such as unioniza ion (e.g., DiNa do and Lee (2004), Lee and Mas (2012)) o local
school bonds (e.g., Cellini, Fe ei a, and Ro hs ein (2010)) could be pu in on o o -
e s ecu en ly. A la ge body o li e a u e in poli ical science and economics (e.g., Lee
(2008), Pe e sson-Lidbom (2008), Fe ei a and Gyou ko (2009), Colonnelli, P em, and
Teso (2020)) uses RD o s udy he e ec o poli ical aces ha happen on a egula basis.
Dube, Giuliano, and Leona d (2019)andJohnson (2020) examine he pee e ec o RD
ea men s. Thei se ing is epea ed as well because he same indi idual can be exposed
o di e en pee ea men s a di e en poin s o ime.
The epea ed ea men design b ings complica ions o iden i ying long- e m e ec s
in he RD se ing jus as in o he non-RD se ings. As Heckman, Humph ies, and Ve a-
mendi (2016) discuss, in epea ed/sequen ial ea men se ings educed- o m analy-
sis can only iden i y a mixed bag o e ec s, called he long- e m o al e ec . Fo policy
analysis, esea che s o en wan o dis inguish a long- e m di ec e ec , o he “clean”
long- e m e ec o an ea lie ea men wi h no subsequen ea men s (Heckman,
Humph ies, and Ve amendi (2016)), om he mixed bag.
In he Cali o nia educa ion bond example s udied by Cellini, Fe ei a, and Ro hs ein
(2010) (CFR), o example, policymake s a e in e es ed in iden i ying he long- un e -
ec o passing a local educa ion bond on educa ion expendi u e, es sco es, and local
house p ices. Fo he pu pose o iden i ica ion, one wishes ha he educa ion bond o -
ing would only ake place once. Then, compa ing long- un ou comes be ween school
dis ic s ha ba ely pass and ba ely miss he o e sha e cu o , would gi e a e age long-
e m di ec e ec s o in e es s ( o hose school dis ic s a he o e sha e cu o ). In
eali y, educa ion bond measu es can be pu o wa d epea edly. As is discussed in CFR,
he di e ence in obse ed long- un ou comes a he RD cu o only cap u es an a e -
age o al e ec ha is also in luenced by subsequen bond au ho iza ions. Fo example,
school dis ic s ailing o pass he ini ial bond measu e a e mo e likely o pu o wa d
ano he bond measu e in la e pe iods (Cellini, Fe ei a, and Ro hs ein (2010)).
Among he empi ical li e a u e on epea ed RD se ings, CFR and subsequen s ud-
ies adop ing hei me hods (e.g., Da olia (2013), Abo , Kogan, La e u, and Peskowi z
(2020)) a e he only ones, as a as he au ho s know, ha seek p ope iden i ica ion o
a e age long- e m di ec e ec s. O he empi ical s udies ei he igno e he epea ed de-
sign o op o only s udy immedia e e ec s and/o long- e m o al e ec s. CFR p oposes
o iden i y a e age long- e m di ec e ec s om o al e ec s wi h a ecu si e RD s a egy
and a mo e pa ame ic e en s udy s a egy. Recen ly, Gallen, Joensen, Johansen, and
Ve amendi (2023) ex end CFR’s ecu si e iden i ica ion s a egy o a non-RD IV se ing.
This pape o malizes he epea ed RD design unde he po en ial ou come ame-
wo k. We ind ha , unde ea men e ec he e ogenei y, p ese ing he ecu si e CFR
Quan i a i e Economics 15 (2024) Dynamic eg ession discon inui y 1037
s a egy o iden i y long- e m a e age di ec e ec s equi es s ong assump ions. In he
Cali o nia educa ion bond applica ion, o example, using he ecu si e CFR s a egy
equi es he decision o pu ing o wa d o he bond measu es a e he ocal ound o
be exogeneous, among o he iden i ying condi ions such as pa h-independency and
homogenei y in bond e ec s. As Dong (2019) poin s ou in a s udy o RD design wi h
sample selec ion, al hough RD models could be ega ded as local andom expe imen s,
such andomness a ound unning a iable cu o s could no be used o a gue o he
igno ance o endogenous selec ion in o RD e en s. In he dynamic RD design, andom
pa icipa ion in subsequen ounds o RD is a ely plausible. Homogenei y in a e age
e ec s ac oss di e en RD ounds and ega dless o pas ea men pa hs is also e y
s ong.
Gi en he a o emen ioned conside a ions, he main con ibu ion o his pape is o
p opose a new iden i ica ion s a egy o long- e m a e age di ec e ec s. In con as o
ecu si e CFR, ou p oposed iden i ica ion s a egy does no impose any es ic ion on
po en ially endogenous RD pa icipa ion decisions. Ou s a egy also allows ea men
e ec s o depend on las pe iod’s ea men ake-up. Ou key iden i ica ion es ic ion is
a condi ional mean independence assump ion (CIA) ha equi es mean independence
be ween he second- ound unning a iable and he second- ound po en ial ou comes
wi hou he second- ound ea men , condi ional on RD pa icipa ion. Fo ac abil-
i y in longe - e m e ec iden i ica ion, we also impose a Ma ko ian- ype assump ion
o simpli y he pa h-dependency s uc u e o a e age ea men e ec s. Besides poin
iden i ica ion, we p o ide pa ial iden i ica ion esul s in he Supplemen al Appendix
(Hsu and Shen (2024)) unde Manski- ype (e.g., Manski and Peppe (2000)) mono onic-
i y condi ions.
Ou pape ela es o he dynamic ea men e ec li e a u e ou side he RD se ing.
As Han (2021) discusses, he bios a is ics li e a u e (e.g., Robins (1986,1987), Mu phy,
an de Laan, Robins, and C. P. P. R. G oup (2001), Mu phy (2003), Chak abo y and Mu -
phy (2014)) has a long his o y o s udying dynamic causal e ec s unde he assump ion
o sequen ial andomiza ion. As we shall explain in he pape , al hough sha p RD de-
signs a e o en unde s ood as local andom expe imen s, he dynamic RD se -up does
no enjoy sequen ial andomiza ion by design. In addi ion, imposing sequen ial an-
domiza ion o igno abili y (e.g., Blackwell (2013), Imai and Ra ko ic (2015), Imbens and
Lemieux (2008), Bojino , Rambachan, and Shepha d (2021)) on ea men s as an iden i-
ica ion condi ion can be undesi able in empi ical RD s udies.
Ou pape ela es o Heckman, Humph ies, and Ve amendi (2016) who e alua e
ea men e ec s in o de ed and uno de ed mul is age decision p oblems wi h an in-
s umen al a iable app oach, o Sun and Ab aham (2021), Callaway and San ’Anna
(2021), and A hey and Imbens (2022) who examine ea men e ec s in panel e en s ud-
ies wi h one single i e e sible ea men , o De Chaisema in and d’Haul oeuille (2020)
who s udy linea wo-way ixed-e ec eg essions o panel da a models wi h ea men
e ec he e ogenei y ac oss g oups o o e ime, and o De Chaisema in and d’Haul -
oeuille (2024) who in es iga e pa h-speci ic long- e m a e age ea men e ec s using
he pa allel end condi ion. All a o emen ioned pape s ha e di e en model se -ups
and iden i ying assump ions han ou s. Han (2021) p oposes o iden i y pa h-speci ic
1038 Hsu and Shen Quan i a i e Economics 15 (2024)
ATEs using a sequence o dynamic ea men selec ion equa ions and excluded ins u-
men s. We do no ha e na u al ins umen s in he epea ed RD se ing.
Ou dynamic RD model also ela es o RD models wi h epea ed designs o mul-
iple sco es/cu o s, including G embi, Nannicini, and T oiano (2016) who p opose a
di e ence-in-discon inui ies s a egy o pa ial ou he e ec o a con ounding policy,
L ,Sun,Lu,andLi(2019) who conside RD su i al analysis whe e indi idual ea men s
a e allowed o be alloca ed a di e en p e- ea men du a ion, mul isco e models s ud-
ied by Papay, Wille , and Mu nane (2011), Rea don and Robinson (2012), and Wong,
S eine , and Cook (2013), and he mul icu o RD model o Ca aneo, Keele, Ti iunik, and
Vazquez-Ba e (2016). Gi en ou ocus on long- e m di ec e ec s, his pape is signi i-
can ly di e en om he o he s.
Aside om iden i ica ion, ou pape con ibu es o he li e a u e by designing a new
wo-s ep semipa ame ic bounda y es ima ion p ocedu e. Speci ically, ou iden i ied
long- e m e ec s has a o m o in e se p opensi y sco e weigh ing (IPW), and so he
es ima ion ollows wo s eps. In he i s s ep, we model he p opensi y sco e unc ion
semipa ame ically and es ima e i wi h he local MLE app oach in Cai, Fan, and Li
(2000). In he second s ep, we plug in es ima ed p opensi y sco es o local-linea eg es-
sions. Ou p oposed i s -s ep local MLE algo i hm is pa icula ly sui able o he RD
se ing, because i allows he p opensi y sco e es ima o o s ay local o he RD cu o
along he dimension o he unning a iable (c . Gelman and Imbens (2019)) while no
o e bu dening he inal wo-s ep es ima o wi h he “cu se o dimensionali y.” In e ms
o in e ence, ou pape is he i s o apply he weigh ed boo s ap designed in Ma and
Koso ok (2005) o ke nel-based bounda y es ima ion, which is he main wo kho se o
he RD li e a u e. The weigh ed boo s ap me hod has also been adop ed by Chen and
Pouzo (2009), Che nozhuko , Fe nández-Val, Hode lein, Holzmann, and Newey (2015),
Che nozhuko , Fe nández-Val, and Kowalski (2015), and Fe nández-Val, an Vuu en,
and Vella (2021), among o he s, in o he es ima ion se ings.
The es o he pape is o ganized as ollows. Sec ion 2s a s wi h a simple wo-
pe iod dynamic RD model, explaining why long- e m di ec e ec s a e impo an policy
pa ame e s and why hei iden i ica ion canno be ob ained di ec ly om he RD de-
sign. Sec ion 2.2 o malizes he ecu si e CFR iden i ica ion s a egy unde ea men
e ec he e ogenei y and discuss i s limi a ions. Sec ions 2.3 and 2.4 p esen a new iden-
i ica ion and es ima ion s a egy o he one-pe iod-a e a e age di ec e ec in he
benchma k wo-pe iod model. Sec ion 3p esen s he gene al mul ipe iod dynamic RD
model and he iden i ica ion o longe - e m a e age di ec e ec s. Sec ion 4s udies es-
ima ion and in e ence o he gene al model. Sec ion 5 e isi s he empi ical s udy o
Cali o nia educa ion bonds using CFR’s published da a se . Sec ion 6concludes. Mon e
Ca lo simula ions and p oo s a e p o ided in he Supplemen al Appendix, which also
includes pa ial iden i ica ion esul s and se e al empi ical- ele an special cases.
2. A benchma k wo-pe iod model
In his sec ion, we use a simple wo-pe iod model o de ine he dynamic RD design un-
de he po en ial ou come amewo k. We ake he simple model o poin ou complica-
ions b ough by ha ing a epe i ion in he RD design, as well as he di e ences be ween

Quan i a i e Economics 15 (2024) Dynamic eg ession discon inui y 1039
dynamic RD models and o he non-RD dynamic models p e iously s udied in he li e -
a u e. We p opose a no el iden i ica ion s a egy in his wo-pe iod model and discuss
i s ad an ages compa ed o p e ious me hods.
Be o e we begin, i is impo an o poin ou ha he simple wo-pe iod model de-
ined in his sec ion is only a s a ing poin . Some impo an iden i ica ion esul s p o-
ided in his pape a e only ele an o he gene al case p esen ed in Sec ion 3.
2.1 The classic po en ial ou come amewo k
Conside a epea ed RD se ing, whe e he RD e en (e.g., elec ion, es ing, e c.) akes
place a he beginning o each pe iod, and he ea men is adminis a ed immedia ely
ollowing he e en o pa icipan s who pass a unning a iable h eshold. An ou come
is obse ed a he end o each pe iod.
In pe iod one, we assume o now ha e e yone akes pa in he RD e en , so he
obse ed ea men Di1and ou come Yi1o indi idual isa is y
Di1=1(Zi1≥0),Yi1=Yi1(0)·(1−Di1)+Yi1(1)·Di1,
whe e Zi1is he i s - ound unning a iable and Yi1(d1)is he po en ial i s -pe iod
ou come wi h Di1=d1, o d1=0, 1. Wi hou loss o gene ali y, all RD cu o s in his
pape a e no malized o ze o.
In pe iod wo, he po en ial ou come amewo k gi es
Di2=Di2(0)·(1−Di1)+Di2(1)·Di1,and
Yi2=Yi2(0, 0)·(1−Di1)·1−Di2(0)+Yi2(0, 1)·(1−Di1)·Di2(0)
+Yi2(1, 0)·Di1·1−Di2(1)+Yi2(1, 1)·Di1·Di2(1)
≡
2∈L2
Yi22·Di2,L2=(0, 0),(0, 1),(1, 0),(1, 1),
whe e Di2and Yi2a e obse ed ea men decisions and ou comes, while Di2(d1)and
Yi2(d1,d2),d1,d2=0, 1, a e hei po en ial coun e pa s. Di(.)is he pa h indica o o
indi idual iand L2is he se o all possible ea men pa hs in wo pe iods. This dynamic
po en ial ou come amewo k is common in dynamic causal e ec se ings ou side RD.
See, o example, Sec ion III o he bios a is ics ex book He nán and Robins (2023)and
p e ious wo k in econome ics, including Hahn, Todd, and an de Klaauw (2001)and
Bojino , Rambachan, and Shepha d (2021).
The RD se ing also b ings special ea u es o he dynamic model. Speci ically, he
second-pe iod ea men decision Di2(d1)is de e mined by a po en ially endogenous
RD pa icipa ion indica o Si2(d1)and a po en ially endogenous second- ound unning
a iable Zi2(d1)such ha Di2(d1)=Si2(d1)·1(Zi2(d1)≥0). The second-pe iod obse ed
pa icipa ion decision and ea men decision sa is y
Si2=Si2(0)·(1−Di1)+Si2(1)·Di1,
Di2=Si2(0)·1Zi2(0)≥0·(1−Di1)+Si2(1)·1Zi2(1)≥0·Di1.
1040 Hsu and Shen Quan i a i e Economics 15 (2024)
The second- ound unning a iable Zi2=Zi2(0)·(1−Di1)+Zi2(1)·Di1is only obse ed
gi en RD pa icipa ion, ha is, Si2=1.
Example (Cali o nia Educa ion Bonds). In he Cali o nia educa ion bond example
s udied in CFR, ou come measu es include local educa ion expendi u es, house p ices,
s uden achie emen s, e c. A bond measu e is app o ed i i s o e sha e exceeds he leg-
isla i e cu o (no malized o (0). No ma e whe he a school dis ic ge s i s bond ap-
p o ed in he i s ound (Di1) o no , i could choose o ini ia e a new bond measu e in
he nex elec ion yea (Si2=1) and choose how much campaign e o s o pu in o he
new measu e o imp o e i s o e sha e esul (Zi2(d1), o d1=0, 1).
We o mally de ine he ollowing indi idual ea men e ec s o he wo-pe iod
model:
immedia e e ec o Di1:θi;0,1 =Yi1(1)−Yi1(0);
immedia e e ec o Di2:θd1
i;0,2 =Yi2(d1,1)−Yi2(d1,0), o d1=0, 1;
one-pe iod-a e di ec e ec o Di1:θi;1,1 =Yi2(1, 0)−Yi2(0, 0);
one-pe iod-a e o al e ec o Di1:˜
θi;1,1 =˜
Yi2(1)−˜
Yi2(0),
whe e ˜
Yi2(d1)≡Yi2d1,Di2(d1)=Yi2(d1,0)1−Di2(d1)+Yi2(d1,1)Di2(d1).
No e ha he immedia e e ec o he second- ound ea men is pa h-dependen . Fol-
lowing Heckman, Humph ies, and Ve amendi (2016) and CFR, we de ine long- e m di-
ec e ec s by p ohibi ing ea men ake-ups a e he ocal ound. The o al e ec , on
he o he hand, does no limi ea men decisions a e he ocal ound. Di ec and o-
al e ec s a e also commonly seen in he media ion li e a u e.1See Hube (2020) o a
li e a u e e iew.
Assump ion 2.1. The e exis s an >0, such ha :
1. Zi1is con inuous in z1∈(−,)≡Nwi h P[Zi1≥0]∈(0, 1);
2. E[Yi1(d1)|Zi1=z1],E[Di2(d1)|Zi1=z1],and E[˜
Yi2(d1)|Zi1=z1]a e all con inuous
in z1∈N, o bo h d1=0, 1.
Assump ion 2.1 imposes adi ional RD smoo hness condi ions ha p o ide iden-
i ica ion o he a e age immedia e e ec E[θi;0,1|Zi1=0]and he a e age i s -s age
e ec E[Di2(1)−Di2(0)|Zi1=0]. Since he andom a iable ˜
Yi2(d1), which akes place
in he second pe iod bu only speci ies he i s - ound ea men s a us, can be iewed
1In he media ion li e a u e, θi;1,1 =Yi2(1, 0)−Yi2(0, 0)is he con olled di ec e ec , in con as o he
pu e di ec e ec , which would be Yi2(1, Di2(0))−Yi2(0, Di2(0))in ou no a ion. Flo es and Flo es-Lagunes
(2009,2010) s udy he pu e di ec e ec in he econome ics li e a u e. As Hube (2020) discusses, he pu e
di ec e ec is no in e es ing in he dynamic se ing.
Quan i a i e Economics 15 (2024) Dynamic eg ession discon inui y 1041
as a po en ial ou come o he i s - ound ea men , smoo hness condi ions in Assump-
ion 2.1 also iden i ies he a e age one-pe iod-a e o al e ec :
E[˜
θi;1,1|Zi1=z1]=E˜
Yi2(1)−˜
Yi2(0)|Zi1=0
=lim
z10E[Yi2|Zi1=z1]−lim
z10E[Yi2|Zi1=z1]. (2.1)
In he es o Sec ion2, we explo e condi ions ha iden i y he a e age di ec e ec
E[θi;1,1|Zi1=0], which has impo an policy implica ions as a gued in Heckman and
Na a o (2007) and CFR. Fo conciseness, we call E[θi;1,1|Zi1=0] he one-pe iod-a e
ATE. I has he ollowing ela ionship wi h he one-pe iod-a e a e age o al e ec :
E[˜
θi;1,1|Zi1=0]=E[θi;1,1|Zi1=0]+Eθ1
i;0,2Di2(1)|Zi1=0
−Eθ0
i;0,2Di2(0)|Zi1=0. (2.2)
Example (con inued). In he Cali o nia educa ion bond example, he one-pe iod-a e
ATE (E[θi;1,1|Zi1=0]) is he a e age e ec o passing an educa ion bond in he i s pe-
iod on he second-pe iod ou come wi h no bond au ho iza ion a e he i s pe iod,
among all school dis ic s a he i s -pe iod o e sha e cu o . The one-pe iod-a e ATE
in luences he one-pe iod-a e a e age o al e ec (E[˜
θi;1,1|Zi1=0]), which is iden i ied
by he pe iod wo ou come discon inui y obse ed a he pe iod one o e sha e cu o .
Howe e , he one-pe iod-a e o al e ec is also in luenced by he ac ha school dis-
ic s can pass new bond measu es in he second pe iod, and hence ecei e immedia e
e ec s om hose addi ional ea men s.
2.2 Recu si e iden i ica ion s a egy in CFR
CFR p opose a ecu si e iden i ica ion s a egy o long- e m di ec e ec s based on an
implici ea men e ec homogenei y assump ion, which equi es indi idual ea men
e ec s o a y only by he numbe o pe iods be ween po en ial ou comes and he ocal
ound o ea men .2Fo he wo-pe iod model, ha is o equi e
θ0≡θi,0,1 =θ0
i,0,2 =θ1
i,0,2 and θ1≡θi,1,1 o all i, (2.3)
whe e θ0and θ1a e unknown ixed pa ame e s.
The s ong assump ion educes he ela ionship in (2.2) o
E[˜
θi;1,1|Zi1=0]=θ1+θ0EDi2(1)−Di2(0)|Zi1=0.
The simpli ica ion implies ecu si e iden i ica ion o he one-pe iod-a e ATE θ1, since
all o he componen s o he equa ion a e di ec ly iden i iable h ough smoo hness con-
di ions.
CFR’s ecu si e s a egy can be ex ended o allow o indi idual ea men e ec he -
e ogenei y unde he po en ial ou come amewo k ou lined in he p e ious sec ion.
2See, o example, Sec ion IV.B o CFR. In CFR, ou di ec e ec is called he “ ea men -on- he- ea ed”
e ec and ou o al e ec is called he “in en - o- ea ” e ec .
1042 Hsu and Shen Quan i a i e Economics 15 (2024)
Lemma 2.1. Unde Assump ion 2.1, he homogeneous ATE equi emen :
ATE0≡E[θi;0,1|Zi1=0]=Eθ1
i;0,2|Zi1=0=Eθ0
i;0,2|Zi1=0, (2.4)
and he andom second- ound ea men selec ion equi emen :
Eθd1
i;0,2|Di2(d1),Zi1=0=Eθd1
i;0,2|Zi1=0,d1=0, 1, (2.5)
he ecu si e iden i ica ion s a egy in CFR can be p ese ed:
ATE0=lim
z10E[Yi1|Zi1=z1]−lim
z10E[Yi1|Zi1=z1],
ATE1≡E[θi;1,1|Zi1=0]=lim
z10E[Yi2|Zi1=z1]−lim
z10E[Yi2|Zi1=z1]
−ATE0·lim
z10E[Di2|Zi1=z1]−lim
z10E[Di2|Zi1=z1].
To p ese e CFR’s ecu si e iden i ica ion s a egy unde indi idual ea men e -
ec he e ogenei y, condi ions (2.4)and(2.5) a e equi ed. The condi ions include he
s ong homogenei y assump ion in (2.3) as a special case.3I he homogenei y ATE con-
di ion in (2.4) is iola ed due o diminishing ma ginal e u ns o epea ed ea men s
(i.e., E[θ1
i;0,2|Zi1=0]<E
[θ0
i;0,2|Zi1=0]=E[θi;0,1|Zi1=0]), hen ATE1iden i ied in
Lemma 2.1 is smalle han he ue alue o E[θi;1,1|Zi1=0].
The andom ea men selec ion condi ion in (2.5) should no o be con used wi h
he local andomness in ui ion o he RD design. A ea men in e en ion sa is ying
he sha p RD design is only locally andom a i s own unning a iable cu o . In o he
wo ds, he RD design associa ed wi h Di2(d1)only indica es local andomness among
indi iduals a Zi2(d1)=0, o d1=0, 1. Equa ion (2.5) is di e en . I is no implied by
he RD design and is a s ong condi ion used o ob ain he ecu si e iden i ica ion e-
sul .
Example (con inued). In he Cali o nia educa ion bond example, da a e eal ha
school dis ic s ba ely passing hei educa ionbond in he i s ound do no pu o wa d
ano he bond in he second pe iod (i.e., E[Si2(1)|Zi1=0]=limz10E[Si2|Zi1=z1]=0).
The e o e, equa ion (2.5) essen ially es ic s ha , o a school dis ic i ha ma ginally
ailed he i s - ound o ing, he decision o pu ing o wa d ano he measu e in he
nex elec ion cycle (Si2(0)) is no ela ed o he immedia e ea men e ec o he new
3A mo e gene al mul ipe iod e sion o he lemma is discussed in he Supplemen al Appendix, whe e we
also ex end Lemma 2.1 o include condi ioning co a ia es Xi∈X. The ex ension eplaces he homogeneous
ATE assump ion in (2.4) and he andom ea men selec ion assump ion in (2.5)wi h
Eθd1
i;0,2|Di2(d1),Xi=x,Zi1=0=Eθd1
i;0,2|Xi=x,Zi1=0, and (2.6)
CATE0(x)=E[θi;0,1|Xi=x,Zi1=0]
=Eθd1
i;0,2|Xi=x,Zi1=0,∀d1=0, 1, x∈X. (2.7)
Quan i a i e Economics 15 (2024) Dynamic eg ession discon inui y 1049
The decomposi ion is new o he li e a u e, as a as he au ho s know. I is di e en
om he decomposi ion used behind he ecu si e CFR iden i ica ion s a egy, whe e
he long- e m o al e ec is decomposed o i s co esponding di ec e ec plus a sum
o a ious sho e - e m di ec e ec s adjus ed by i s -s age ea men decisions.6In he
es o he sec ion, we use equa ion (3.1) o de elop an iden i ica ion s a egy because
he new decomposi ion in ol es much ewe pa h-dependen ea men e ec s in i s
o mula ion.
Aside om u ilizing he new decomposi ion esul , we impose a Ma ko ian- ype
condi ion o u he educe he numbe o pa h-dependen ea men e ec pa ame-
e s in ol ed in iden i ica ion. Le ηi;0,1,ηk−1
i;0,k,ηi;τ,1,ηk−1
i;τ,k,˜ηi;τ,1,and ˜ηk−1
i;τ,kbe i s -
s age coun e pa s o immedia e e ec s θi;0,1 and θk−1
i;0,k, di ec e ec s θi;τ,1 and θk−1
i;τ,k,
and o al e ec s ˜
θi;τ,1 and ˜
θk−1
i;τ,k, espec i ely.7The ollowing assump ion summa izes
he iden i ying es ic ions used in he gene al mul ipe iod dynamic RD model.
Assump ion 3.1 (Longe - e m ATEs). The e exis s >0such ha o all z1∈N,we
ha e:
1. (Ma ko ian) o any k=3, 4, ,K,k−2∈Lk−2,and d=0, 1, immedia e e ec s
and τ-pe iod-a e o al e ec s sa is y ha E[θ(k−2,d)
i;0,k|Dik(k−2,d)=1, Zi1=z1]=
E[θd
i;0,2|Di2(d)=1, Zi1=z1]≡μd
0and E[˜
θ(k−2,d)
i;τ,k|Dik(k−2,d)=1, Zi1=z1]=
E[˜
θd
i;τ,2|Di2(d)=1, Zi1=z1]≡˜μd
τ, o all τ=1, ,K−k;simila condi ions also
hold o immedia e and long- e m i s -s age e ec s;
2. (CIA:mul ipe iod) o any d1=0, 1, z2∈R,and x∈X,E[Mi(d1,0)|Xi=x,Zi2(d1)=
z2,Si2(d1)=1, Zi1=z1]=E[Mi(d1,0)|Xi=x,Si2(d1)=1, Zi1=z1],whe e he an-
dom a iable Mi(d1,0)can be Di3(d1,0),˜
Yi(2+τ)(d1,0) o all τ=1, ,K−2, o
˜
Di(3+τ)(d1,0) o all τ=1, ,K−3;
3. (Smoo hness:mul ipe iod) o all d1,d2=0, 1, and x∈X,E[Mi(d1,d2)|Xi=
x,Di2(d1)=d2,Zi1=z1]is con inuous in z1,whe e he andom a iable Mi(d1,d2)
can be Di3(d1,d2),˜
Yi(2+τ)(d1,d2) o all τ=1, ,K−2, o ˜
Di(3+τ)(d1,d2) o all
τ=1, ,K−3;
The second and hi d pa s o Assump ion 3.1 a e anilla ex ensions o Assump-
ion 2.2 iewing Di3(d1,d2),˜
Yi(2+τ)(d1,d2),and ˜
Di(2+τ)(d1,d2)as po en ial ou comes
associa ed wi h he i s wo ea men s bu aking place in a la e pe iod. The key new
6When τ=2, o example, he decomposi ion unde lying he ecu si e CFR iden i ica ion s a egy is
˜
θi;2,1 =θi;2,1 +θ1
i;1,2Di2(1)+θ(1,0)
i;0,3(1−Di2(1))Di3(1, 0)+θ(1,1)
i;0,3Di2(1)Di3(1, 1)−θ0
i;1,2Di2(0)−θ(0,0)
i;0,3(1−
Di2(0))Di3(0, 0)+θ(0,1)
i;0,3Di2(0)Di3(0, 1).
7Le ˜
Di(k+1+τ)(k)be he (k+1+τ)- h pe iod quasi-po en ial ea men decision wi h only he i s
k ounds o p e ious ea men s a us speci ied, o any τ≥1 and k≥1. Then ηi;0,1 =Di2(1)−Di2(0),
ηi;τ,1 =Di(2+τ)(1, 0τ)−Di(2+τ)(0, 0τ), and ˜ηi;τ,1 =˜
Di(2+τ)(1)−˜
Di(2+τ)(0) o all τ≥1. Fo all k≥
2, ηk−1
i;0,k=Di(k+1)(k−1,1)−Di(k+1)(k−1,0),ηk−1
i;τ,k=Di(k+1+τ)(k−1,1,0τ)−Di(k+1+τ)(k−1,0,0τ), and
˜ηk−1
i;τ,k=˜
Di(k+1+τ)(k−1,1)−˜
Di(k+1+τ)(k−1,0) o all τ≥1.

1050 Hsu and Shen Quan i a i e Economics 15 (2024)
condi ion in Assump ion 3.1 is he Ma ko ian es ic ion. In he Cali o nia educa ion
bond applica ion, he Ma ko ian es ic ion allows an educa ion bond’s a ious imme-
dia e and long- e m a e age e ec s o depend a bi a ily on las elec ion cycle’s bond
au ho iza ion bu no any o he bond au ho iza ions u he in he pas . Al hough non-
i ial, he Ma ko ian condi ion is much less es ic i e han he homogeneous ATE con-
di ion used in Lemma 2.1 (o Lemma A.1) o he ecu si e CFR iden i ica ion s a egy.
Simila Ma ko ian- ype es ic ions a e also used in De Chaisema in and d’Haul -
oeuille (2024)andImai, Kim, and Wang (2023) o non-RD dynamic ea men e ec
se ings. When τ=2, he Ma ko ian assump ion oge he wi h he decomposi ion in
(3.2) imply ha
E[˜
θi;2,1|Zi1=0]=E[θi;2,1|Zi1=0]+˜μ1
1·EDi2(1)|Zi1=0−˜μ0
1·EDi2(0)|Zi1=0
+μ0
0·E[ηi;1,1|Zi1=0].
Lemma 3.2. Unde he assump ions used in Lemma 2.2,Assump ion A.1 o iden i ying
long- e m a e age o al e ec s,and Assump ion 3.1,we ha e ha o τ=2, ,K−1,
E[θi;τ,1|Zi1=0]=lim
z10E[Yi(τ+1)|Zi1=z1]−lim
z10E[Yi(τ+1)|Zi1=z1]
−˜μ1
τ−1·lim
z10E[Di2|Zi1=z1]+˜μ0
τ−1·lim
z10E[Di2|Zi1=z1]
−
τ−2

s=1˜μ0
s·E[ηi;τ−1−s,1|Zi1=0]−μ0
0·E[ηi;τ−1,1|Zi1=0], (3.3)
whe e
μ0
0=lim
z10EYi2Si2Di2−λ0(Xi)
1−λ0(Xi)E[Di2|Zi1=z1]Zi1=z1,
˜μ0
s=lim
z10EYi(2+s)Si2Di2−λ0(Xi)
1−λ0(Xi)E[Di2|Zi1=z1]Zi1=z1,and
˜μ1
s=lim
z10EYi(2+s)Si2Di2−λ1(Xi)
1−λ1(Xi)E[Di2|Zi1=z1]Zi1=z1,
o all s≥1. In addi ion, ea ing i s -s age decisions as ou comes,E[ηi;1,1|Z1=0]is
iden i ied by Lemma 2.2 and E[ηi;k,1|Z1=0] o all k=2, ,τ−1is iden i ied by (3.3)
ecu si ely.
The p oo is gi en in he Supplemen al Appendix. In Sec ion A.4 o he Supplemen al
Appendix, we also discuss se e al impo an special cases o he dynamic RD model,
including ha ing an abso bing s a e ea men and no obse ing some ini ial ounds o
RD da a.
Quan i a i e Economics 15 (2024) Dynamic eg ession discon inui y 1051
4. Es ima ion and in e ence
Es ima ion o he one-pe iod-a e ATE is discussed in Sec ion 2.4 o he benchma k
wo-pe iod model. In he nex , we i s s udy in e ence o he p oposed one-pe iod-a e
ATE es ima o . Then we ex end he wo-pe iod es ima ion and in e ence s a egy o he
gene al mul ipe iod se ing in oduced in Sec ion 3.
4.1 In e ence in he benchma k wo-pe iod se ing
The in e ence p ocedu e p oposed in his sec ion o he one-pe iod-a e ATE es ima-
o de ined in Sec ion 2.4 adap s he weigh ed boo s ap p ocedu e ollowing Ma and
Koso ok (2005). The p ocedu e is ac able in empi ical applica ions as i keeps es i-
ma ion and in e ence o di e en long- e m ATEs wi hin a uni o m o ma . The p o-
cedu e does no pu sue bandwid h choice op imali y in he sense o asymp o ic mean
squa ed e o s (AMSE), howe e . In he Supplemen al Appendix, we discuss an al e -
na i e AMSE-op imal es ima ion and in e ence p ocedu e o he one-pe iod-a e ATE
based on Calonico, Ca aneo, and Ti iunik (2014), Calonico, Ca aneo, and Fa ell (2018,
2020,2022).8The weigh ed boo s ap p ocedu e discussed in his sec ion could also be
adap ed o ecu si e CFR es ima o s.
4.1.1 Assump ions and asymp o ic p ope ies Le φγ1,ni(D2i,S2i,Z1i,Xi)and
φγ0,ni(D2i,S2i,Z1i,Xi)be in luence unc ions o es ima o s ˆγ1and ˆγ0de ined in Sec-
ion 2.4, espec i ely, such ha
√nhˆγ1−γ1=1
√nh
n

i=1
φγ1,ni(D2i,S2i,Z1i,Xi)+op(1),
√nhˆγ0−γ0=1
√nh
n

i=1
φγ0,ni(D2i,S2i,Z1i,Xi)+op(1).
Le ˜
φα1,ni(Y2i,D2i,S2i,Z1i,Xi)and ˜
φα0,ni(Y2i,D2i,S2i,Z1i,Xi)be in luence unc-
ions o in easible es ima o s ˜α1and ˜α0de ined by he ollowing:
˜α1,˜
β1=a gmin
α,β
n

i=1
1(Z1i≥0)KZ1i
hAiα,β;γ12,
˜α0,˜
β0=a gmin
α,β
n

i=1
1(Z1i<0)KZ1i
hAiα,β;γ02.
The in easible es ima o s a e gene a ed by he same local linea eg essions as used in
Sec ion 2.4 o es ima o s ˆα1and ˆα0, bu assuming known i s -s ep p opensi y sco es.
8Ex ending he obus in e ence p ocedu e o longe - e m ATE es ima o s could be ealized h ough cal-
cula ing i s -o de linea app oxima ions o each τ-pe iod-a e es ima o (e.g., Sec ions 4.1 and 4.2 o
Calonico, Ca aneo, and Ti iunik (2014)) o bias co ec ion and op imal bandwid h choice calcula ion. I
would be an in e es ing opic o u u e esea ch.
1052 Hsu and Shen Quan i a i e Economics 15 (2024)
De ine g adien e ms 1
γ=limz10E[∇γ[Y2S2(D2−p(X,γ))
1−p(X,γ)]|γ=γ1|Z1=z1]and 0
γ=
limz10E[∇γ[Y2S2(D2−p(X,γ))
1−p(X,γ)]|γ=γ0|Z1=z1]. By he del a me hod, we ob ain he ollow-
ing in luence unc ion ep esen a ion o ˆα0and ˆα1such ha o d1=0, 1, we ha e
√nhˆαd1−αd1
=1
√nh
n

i=1˜
φαd1,ni(Y2i,S2i,D2i,Z1i,Xi)−d1
γ·φγd1,ni(D2i,S2i,Z1i,Xi)+op(1)
≡1
√nh
n

i=1
φαd1,ni(Y2i,S2i,D2i,Z1i,Xi)+op(1). (4.1)
The ep esen a ion implies ha he asymp o ic a iance o ˆ
¯
θ1,1 could be es ima ed by
ˆ
V11 =1
nh
n

i=1ˆ
φα1,ni(Y2i,D2i,S2i,Z1i,Xi)2+ˆ
φα0,ni(Y2i,D2i,S2i,Z1i,Xi)2,
whe e ˆ
φαd1,ni(Y2i,D2i,S2i,Z1i,Xi)is he es ima ed e sion o φαd1,ni(Y2i,D2i,S2i,Z1i,
Xi)wi h all unknown pa ame e s eplaced wi h co esponding es ima o s; d1=0, 1.
We nex p o ide de ailed assump ions and asymp o ic p ope ies o he p oposed
wo-s ep es ima o wi h espec o a a ying coe icien logi i s s age. Fo no a ional
simplici y, we assume o he es o he pape ha he Xi ec o includes a cons an .
Then a a ying coe icien logi i s -s age model is o mula ed as p(x,γ)=L(xγ)wi h
L(a)=exp(a)/(1+exp(a)).
Assump ion 4.1. λ(x;z1)=L(xγ(z1)) is he co ec speci ica ion on z1∈N o some
>0.
Assump ion 4.2. Densi y z1(z1)is wice con inuously di e en iable in z1on N,and
z1(z1)is bounded away om ze o on N o some >0.
Assump ion 4.3. Assume ha :
1. The ke nel unc ion K(·)is a nonnega i e symme ic bounded ke nel wi h suppo
[−1, 1];K(u)du =1.
2. The bandwid h sa is ies ha h→0, nh3→∞,and nh5→0as n→∞.
Assump ion 4.1 equi es ha he a ying coe icien logi model is co ec ly spec-
i ied. Assump ion 4.2 imposes s anda d smoo hness condi ions on he densi y o he
unning a iable. Assump ion 4.3 imposes s anda d condi ions on he ke nel unc ion
and unde smoo hed bandwid h. Unde smoo hing is equi ed such ha he bias o ke -
nel es ima o s becomes asymp o ically negligible. In p ac ice, we ecommend using he
iangula ke nel (i.e., K(x)=|x|·1(|x|<1)) and unde smoo hing he obus RD band-
wid h in oduced in Calonico, Ca aneo, and Ti iunik (2014) (CCT), which is o o de
n1/5. As discussed ea lie , unde smoo hing is no AMSE op imal. In he Supplemen al
Quan i a i e Economics 15 (2024) Dynamic eg ession discon inui y 1053
Appendix, we p opose an al e na i e p ocedu e ha uses bias co ec ion and AMSE-
op imal bandwid h in he second s ep. The al e na i e p ocedu e also equi es a highe -
o de local polynomial and a la ge bandwid h in he i s s ep such ha es ima ion e -
o s om he i s s ep do no a ec AMSE o he inal es ima o .
The ollowing lemma p o ides asymp o ic p ope ies o he i s -s ep a ying coe -
icien logi es ima o s. Simila esul s could be de i ed i he p opensi y sco e unc ion
λ(.; .) ollows o he semipa ame ic models such as a ying coe icien P obi .
Lemma 4.1. Suppose ha Assump ions 4.1–4.3 and B.1–B.2 hold.Then o d=0, 1, we
ha e
√nhˆγd−γd
hˆ
βd
FS −hβd=1
√nh
n

i=1d−1S2i·1(Z1i≥0)d·1(Z1i<0)1−d
·K(Z1i/h)D2i−LX
iγd+βdZ1iXi
Z1iXi/h+op(1),
whe e disgi eninequa ion(D.2) in he Supplemen al Appendix.In addi ion, o d=
0, 1, we ha e
√nhˆγd−γd
hˆ
βd
FS −hβd⇒N0, d−1dd−1,
whe e disgi eninequa ion(D.3) in he Supplemen al Appendix.
Aside om assump ions s a ed ea lie , he lemma also equi es Assump ions B.1 and
B.2 in he Supplemen al Appendix. The o me imposes smoo hness condi ions on he
a ying coe icien in neighbo hoods igh abo e and below he RD cu o . The la e
imposes momen condi ions on he condi ioning co a ia es.
Fo asymp o ic p ope ies o ˆα0and ˆα1, we impose addi ional Assump ions B.1 and
B.4, s a ed in he Supplemen al Appendix. The o me includes smoo hness condi ions
o he in easible es ima o s ˜α0and ˜α1using ue alues o i s -s ep p opensi y sco e
unc ions. The la e imposes condi ions ha con ol he impac o i s -s ep es ima ion
e o s on he asymp o ic p ope ies o easible wo-s ep es ima o s ˆα0and ˆα1.Gi en he
assump ions, √nh(ˆαd1−αd1), o d1=0, 1, has linea ep esen a ion as in (4.1)wi h
˜
φαd1,ni(Y2i,D2i,S2i,Z1i,Xi)
=(10)·−1
z·1(Z1i≥0)d1·1(Z1i<0)1−d·K(Z1i/h)
·Y2i−E[Y2i|Z1i]−Y2iS2iD2i−LX
iγd1
1−LX
iγd1+EY2iS2iD2i−LX
iγd1
1−LX
iγd1Z1i
·1
Z1i/h,
whe e z= z1(0)·μz,0 μz,1
μz,1 μz,2 ,andμz,j=u≥0ujK(u)du o j=1, 2, . The in luence
unc ion ep esen a ion hen implies he ollowing asymp o ic esul s.
1054 Hsu and Shen Quan i a i e Economics 15 (2024)
Theo em 4.1. Suppose ha Assump ions 4.1–4.3 and B.1–B.4 hold.Fo d1=0, 1,we hen
ha e
√nhˆαd1−αd1d
→N(0, Vαd1),d1=0, 1;
√nh(ˆ
¯
θ1,1 −¯
θ1,1)d
→N(0, Vα1+Vα0),
whe e Vα0=limn→∞ limz10h−1E[φ2
α0,ni(Y2i,D2i,S2i,Z1i,Xi)|Z1i=z1]and Vα1=
limn→∞limz10h−1E[φ2
α1,ni(Y2i,D2i,S2i,Z1i,Xi)|Z1i=z1].
The exac exp essions o Vα0and Vα1a e edious o calcula e in gene al. In he nex
sec ion, we adap he weigh ed boo s ap p ocedu e i s in oduced in Ma and Koso ok
(2005) o simula e he limi ing dis ibu ion o he p oposed es ima o . S udies adop -
ing he p ocedu e in o he se ings include Chen and Pouzo (2009), Che nozhuko e
al. (2015), Che nozhuko , Fe nández-Val, and Kowalski (2015), and Fe nández-Val, an
Vuu en, and Vella (2021), among many o he s. Ou pape is he i s o apply weigh ed
boo s ap o ke nel-based bounda y es ima ion, which is he main wo kho se o he RD
li e a u e.
4.1.2 Weigh ed boo s ap in e ence Le {Wi}n
i=1be a sequence o pseudo- andom a i-
ables independen o he sample pa h wi h uni mean and a iance. De ine he weigh ed
boo s ap es ima o o ¯
θ1,1 as
ˆ
¯
θw
1,1 =ˆα1,w−ˆα0,w,
ˆα1,w,ˆ
β1,w=a g min
α,β
n

i=1
Wi·1(Z1i≥0)KZ1i
hAiα,β;ˆγ1,w2,
ˆα0,w,ˆ
β0,w=a g min
α,β
n

i=1
Wi·1(Z1i<0)KZ1i
hAiα,β;ˆγ0,w2,
whe e
ˆγ1,w,ˆ
β1,w
FS =a gmax
γ,β
n

i=1
Wi·S2i1(Z1i≥0)KZ1i
h
·D2ilogp(Xi,γ+βZ1i)+(1−D2i)·log1−p(Xi,γ+βZ1i),
ˆγ0,w,ˆ
β0,w
FS =a gmax
γ,β
n

i=1
Wi·S2i1(Z1i<0)KZ1i
h
·D2ilogp(Xi,γ+βZ1i)+(1−D2i)·log1−p(Xi,γ+βZ1i).
The weigh ed boo s ap p ocedu e is simple o ca y ou . Gi en a simula ed copy o
{Wi}n
i=1, weigh ed boo s ap epea s he o iginal wo-s ep ke nel-based es ima ion p o-
cedu e (i.e., i s -s ep local MLE p opensi y sco e es ima ion and second-s age local
linea eg essions), eplacing he o iginal ke nel weigh s K(Z1i/h)wi h new weigh s

Quan i a i e Economics 15 (2024) Dynamic eg ession discon inui y 1055
Wi·K(Z1i/h). The p ocedu e could also be easily ex ended o adjus o wi hin-clus e
co ela ions by assigning andom Wia he clus e le el.
Following Ma and Koso ok (2005), √nh(ˆ
¯
θw
1,1 −ˆ
¯
θ1,1)and √nh(ˆ
¯
θ1,1 −¯
θ1,1)ha e he
same limi ing dis ibu ion unde sui able condi ions. The nex heo em o malizes he
alidi y o he weigh ed boo s ap es ima o gi en a a ying coe icien logi i s s age.
Theo em 4.2. Suppose ha Assump ions 4.1–4.3 and B.1–B.4 hold and ha {Wi}n
i=1is a
sequence o i.i.d.pseudo- andom a iables independen o he sample pa h wi h E[Wi]=
Va [Wi]=1 o all i.We hen ha e
√nhˆ
¯
θw
1,1 −ˆ
¯
θ1,1d
→N(0, Vα1+Vα0)
condi ional on he sample pa h wi h p obabili y app oaching one.
Al hough Wican ollow any dis ibu ion wi h uni mean and a iance, in he simu-
la ion and empi ical sessions whe e he i s -s ep p opensi y sco e unc ion is modeled
wi h a ying-coe icien logi , we use a disc e e dis ibu ion whe e Wi=0.5 o 3 wi h
p obabili ies 0.8 and 0.2, espec i ely. The bina y andom a iable wi h posi i e suppo
ensu es ha he weigh ed logi objec i e unc ions emain globally conca e.
4.2 Es ima ion and in e ence in he gene al mul ipe iod se ing
Long- e m ATEs in he mul ipe iod se ing o E[θi;τ,1|Zi1=0] o τ≥2 a e iden i ied in
Lemma 3.2. This sec ion elabo a es hei es ima ion and in e ence.
Le us s a wi h case o τ=2. To es ima e he wo-pe iod-a e ATE, o ¯
θ2,1 ≡
E[θi;2,1|Zi1=0], one can i s es ima e he one-pe iod-a e a e age i s -s age e ec
¯η1,1 ≡E[ηi;1,1|Zi1=0] ollowing he es ima ion p ocedu e o one-pe iod-a e ATE,
o ¯
θ1,1, desc ibed in Sec ion 2.4, ea ing i s -s age ea men decisions as ou comes.
Then one can es ima e o he componen s o equa ion (3.3). Simila ly, he es ima ion
o any τ-pe iod-a e ATE o τ≥3 in ol es es ima ing he a e age i s -s age e ec
¯ηk,1 ≡E[ηi;k,1|Zi1=0] o all k=1, 2, ,τ−1, which can be done ecu si ely ollowing
Lemma 3.2, and sepa a e es ima ion o o he componen s in equa ion (3.3).
We o mally de ine he es ima o o ¯
θ2,1. Rew i ing Lemma 3.2 wi h τ=2gi es
¯
θ2,1 =α1
1−α0
1−˜μ0
nu/˜μde¯η1,1,whe e˜μ0
nu =lim
z10EYi2Si2Di2−λ0(Xi)
1−λ0(Xi)Zi1=z1,
˜μde =E[Di2|Zi1=z1],
α1
1=lim
z10EYi3−Yi3Si2Di2−λ1(Xi)
1−λ1(Xi)Zi1=z1,
α0
1=lim
z10EYi3−Yi3Si2Di2−λ0(Xi)
1−λ0(Xi)Zi1=z1.
Le ˆα1
1,ˆα0
1,ˆ
˜μ0
nu,ˆ
˜μde,ˆα1
s,andˆα0
s be es ima o s o α1
1,α0
1,˜μ0
nu,˜μde,α1
s,andα0
s,
espec i ely. All o hem can be de ined using he same wo-s ep semipa ame ic p o-
cedu e desc ibed in Sec ion 2.4 o he es ima ion o ˆα0and ˆα1, whe e he i s -s ep
1056 Hsu and Shen Quan i a i e Economics 15 (2024)
p opensi y es ima ion uses ke nel-based local MLE. Le φα1
1,ni,φα0
1,ni,φ˜μ0
nu,ni,φ˜μde,ni,
φα1
s,ni,andφα0
s,ni deno e he in luence unc ions o he es ima o s, espec i ely. De -
ini ions o φα1
1,ni,φα0
1,ni,φ˜μ0
nu,ni,φα1
s,ni,andφα0
s,ni a e simila o in luence unc ions
gi eninequa ion(4.1) o ˆα0and ˆα1. The in luence unc ion o ˆ
˜μde is de ined as
φ˜μde,ni =1
z1(0)K(Z1i
h)(D2i−E[D2i|Z1i]).
Le ˆ
¯
θ2,1 =ˆα1
1−ˆα0
1−ˆ
˜μ0
nu/ˆ
˜μde(ˆα1
s −ˆα0
s)be he es ima o o ¯
θ2,1. By he del a
me hod, we ha e √nh(ˆ
¯
θ2,1 −¯
θ2,1)=1
√nh n
i=1φ¯
θ2,1,ni(Y3i,Y2i,D3i,D2i,S2i,Z1i,Xi)+
op(1),whe eφ¯
θ2,1,ni(Y3i,Y2i,D3i,D2i,S2i,Z1i,Xi)=φα1
1,ni −φα0
1,ni −α1
s−α0
s
˜μde φ˜μ0
nu,ni +
˜μ0
nu(α1
s−α0
s)
(˜μde)2φ˜μde,ni −˜μ0
nu
˜μde (φα1
s,ni −φα0
s,ni ). The asymp o ic no mali y o ˆ
¯
θ2,1 ollows om
he in luence unc ion ep esen a ion. Gi en he in luence unc ion ep esen a ion, i
is easy o see ha he weigh ed boo s ap p ocedu e could be applied he e, oo, and in
gene al, o he in e ence p oblem o any τ-pe iod-a e ATE wi h τ≥2.
Mon e Ca lo simula ions o he p oposed es ima ion and in e ence p ocedu e a e
summa ized in he Supplemen al Appendix.
5. Empi ical example:The e ec o CA school bonds
This sec ion e isi s he s udy o local educa ion bonds using he da a se published by
CFR. As desc ibed in CFR, school dis ic s in Cali o nia became eligible o issuing gen-
e al obliga ion bonds h ough P oposi ion 46 in 1984. CFR s udy he e ec s o bond au-
ho iza ion on local house p ices, s uden achie emen s, and o he ou comes using Cal-
i o nia da a om 1987 o 2006. Due o da a limi a ions, we s udy wo ou come a iables:
o al expendi u e pe pupil and capi al loading pe pupil in a school dis ic .
The e a e 1551 bond measu es om 655 school dis ic s wi h nonmissing o e sha e
in o ma ion. Al hough he da a se s a s om he i s yea s o educa ion bonds, he ex-
pendi u e ou comes we use a e no obse ed un il 1995. Among all he bond measu es,
282 a e p oposed a e 5 o mo e yea s o inac ion (no measu e) and ha e nonmissing
expendi u e ou come da a up o 4 pe iods a e he measu e yea . We ocus on long- e m
e ec s o hese bond measu es.
Figu e 1gi es a isual illus a ion o he da a. The bo om ow o he igu e shows
immedia e and long- e m o al i s -s age e ec s o passing an educa ion bond mea-
su e. No school dis ic s ha ba ely passed he o e sha e cu o in he i s ial au ho-
ized ano he bond he ollowing yea . The p obabili ies inc ease sligh ly in yea s 3 and
4. A ound 27% o school dis ic s ha ba ely missed he o e sha e cu o in he i s
ial success ully au ho ized hei i s educa ion bond in he ollowing yea . The nega-
i e i s -s age e ec s imply ha long- e m a e age o al e ec s obse ed di ec ly om
ou come discon inui y a e smalle han he long- e m ATEs o policy in e es s.
Table 1 epo s nonpa ame ic es ima ion esul s o immedia e and long- e m a -
e age di ec e ec s ollowing he ex ended ecu si e CFR s a egy o malized in Lem-
mas 2.1 and A.1. The es ima ed a e age e ec s o he capi al ou lays ou come a e highly
s a is ically signi ican . E ec s o he o al expendi u es ou come a e only ma ginally
Quan i a i e Economics 15 (2024) Dynamic eg ession discon inui y 1057
Figu e 1. A e age o al e ec s and his og ams. No e: The da a se is om Cellini, Fe ei a, and
Ro hs ein (2010). The ke nel bandwid h o each ow is se o he same alue, which is he a e age
o CCT bandwid hs among all ou RD eg essions o he ow. The da a sample o he i s wo
ows is a subse o ha o he las ow, because o missing alues in he expense ou comes.
signi ican . Es ima es in Table 1a e la ge han hose epo ed in Table 4 o CFR. The di -
e ence comes om wo sou ces. Fi s , Table 4 o CFR is es ima ed by pooling di e en
ounds o RDs. Unde ea men e ec he e ogenei y, such a pooling s a egy is no ap-
1058 Hsu and Shen Quan i a i e Economics 15 (2024)
Table 1. A e age di ec e ec s: nonpa ame ic CFR based on Lemmas 2.1 and A.1.
Immedia e One-pe iod-a e Two-pe iod-a e Th ee-pe iod-a e
k=4.25 4.5 4.25 4.5 4.25 4.5 4.25 4.5
To al expendi u es pe pupil
897 767 1502 1385 2101 2025 3581 3626
(664)(
601)(
885)(
826)(
1384)(
1294)(
1888)(
1775)
Capi al ou lays pe pupil
361 275 497 455 1002 977 2379 2399
(248)(
216)(
203)(
186)(
569)(
497)(
1026)(
898)
No e:Theda ase is omCellini, Fe ei a, and Ro hs ein (2010). S anda d e o s a e calcula ed using weigh ed boo s ap
wi h 1000 boo s ap epe i ions. Unde smoo hed CCT bandwid h is calcula ed ollowing sugges ions in Sec ion E o he Sup-
plemen al Appendix, wi h he CCT bandwid h epo ed in Figu e 1.
p op ia e, because ma ginal indi iduals a di e en ounds o RD cu o s ha e di e en
ea men e ec s. Second, Table 4 o CFR is es ima ed pa ame ically wi h global poly-
nomials while Table 1is es ima ed nonpa ame ically using he local linea me hod.
Table 2 epo s es ima ion esul s ollowing he p oposed me hod. Va ying-
coe icien logi is used in i s -s ep p opensi y sco e es ima ion. Panel A o he able
epo s es ima ion and in e ence esul s when he CIA condi ion in Sec ion 2holds wi h-
ou any condi ioning co a ia e. Panel B o he able epo s esul s when he i s -pe iod
ou come is used as he i s -s age condi ioning co a ia e. The es ima es a e semipa a-
me ic and ely on he i s -s ep a ying-coe icien logi unc ional o m o p opensi y
Table 2. A e age di ec e ec s: he p oposed p ocedu e.
Immedia e One-pe iod-a e Two-pe iod-a e Th ee-pe iod-a e
k=4.25 4.5 k =4.25 4.5 4.25 4.5 4.25 4.5
Panel A:
To al expendi u es pe pupil
897 767 1070 959 1440 1266 2532 2331
(664)(
601)(
776)(
716)(
1453)(
1244)(
2874)(
1992)
Capi al ou lays pe pupil
361 275 405 398 1245 1240 2850 2884
(248)(
216)(
158)(
149)(
448)(
401)(
882)(
760)
Panel B:
To al expendi u es pe pupil
897 767 903 808 1016 863 1682 1511
(664)(
601)(
964)(
882)(
2073)(
1863)(
4684)(
3629)
Capi al ou lays pe pupil
361 275 433 416 1387 1364 3176 3195
(248)(
216)(
176)(
164)(
495)(
472)(
1156)(
1146)
No e:Theda ase is omCellini, Fe ei a, and Ro hs ein (2010). S anda d e o s a e calcula ed using weigh ed boo s ap
wi h 1000 boo s ap epe i ions. Unde smoo hed CCT bandwid h is calcula ed ollowing sugges ions in Sec ion E o he Sup-
plemen al Appendix, wi h he CCT bandwid h epo ed in Figu e 1. Panel A epo s es ima ion esul s wi h nonpa ame ic
i s -s ep p opensi y sco e es ima o s wi h no condi ioning co a ia es. Panel B uses he i s -pe iod ou come as he condi ion-
ing co a ia e in he i s -s ep p opensi y sco e es ima ion.