Optimal HAR inference

Dou, Liyu
A icle
Op imal HAR in e ence
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: Dou, Liyu (2024) : Op imal HAR in e ence, Quan i a i e Economics, ISSN
1759-7331, The Econome ic Socie y, New Ha en, CT, Vol. 15, Iss. 4, pp. 1107-1149,
h ps://doi.o g/10.3982/QE1762
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Quan i a i e Economics 15 (2024), 1107–1149 1759-7331/20241107
Op imal HAR in e ence
Liyu Dou
Singapo e Managemen Uni e si y and The Chinese Uni e si y o Hong Kong, Shenzhen
This pape conside s he p oblem o de i ing he e oskedas ici y and au oco e-
la ion obus (HAR) in e ence abou a scala pa ame e o in e es . The main as-
sump ion is ha he e is a known uppe bound on he deg ee o pe sis ence in
da a. I de i e ini e-sample op imal es s in he Gaussian loca ion model and show
ha he obus ness-e iciency adeo s embedded in he op imal es s a e es-
sen ially de e mined by he maximal pe sis ence. I ind ha wi h an app op ia e
adjus men o he c i ical alue, i is nea ly op imal o use he so-called equal-
weigh ed cosine (EWC) es , whe e he long- un a iance is es ima ed by p ojec-
ions on o q ype II cosines. The p ac ical implica ions a e an explici link be ween
he choice o qand assump ions on he unde lying pe sis ence, as well as a co e-
sponding adjus men o he usual S uden - c i ical alue. I illus a e he esul s in
wo empi ical examples.
Keywo ds. He e oskedas ici y and au oco ela ion obus in e ence, long- un
a iance.
JEL classi ica ion. C12, C18, C22.
1. In oduc ion
This pape conside s he p oblem o de i ing app op ia e co ec ions o s anda d e -
o s when conduc ing in e ence wi h au oco ela ed da a. The esul ing he e oskedas-
ici y and au oco ela ion obus (HAR) in e ence has applica ions in OLS and GMM
se ings.1Compu ing HAR s anda d e o s in ol es es ima ing he “long- un a iance”
(LRV) in econome ic ja gon. Classical e e ences on HAR in e ence in econome ics in-
clude Newey and Wes (1987)andAnd ews (1991), among many o he s. The Newey–
Wes /And ews app oach is o use -andF- es s based on consis en LRV es ima o s and
o employ he c i ical alues de i ed om he no mal and chi-squa ed dis ibu ions.
Liyu Dou: [email p o ec ed]
I am deeply indeb ed o Ul ich Mülle o posing he ques ion and o his con inuous help, suppo , and
encou agemen . I would like o hank wo anonymous e e ees o cons uc i e commen s and sugges ions,
which ha e subs an ially imp o ed he pape . I also hank Xu Cheng, Paul Ho, Bo Hono é, Michal Kolesá ,
Jia Li, Mikkel Plagbo g-Mølle , Mikkel Søl s en, James S ock, Ma k Wa son, Ke-Li Xu, Jun Yu, and nume ous
pa icipan s in semina s o help ul commen s and sugges ions. I g a e ully acknowledge inancial suppo
om he Na ional Na u al Science Founda ion o China h ough G an 72103176.
1Fo ins ance, OLS/GMM wi h HAR in e ence has been used in many econome ic applica ions, such as
es ing long-ho izon e u n p edic abili y in inance (see, e.g., Koijen and Van Nieuwe bu gh (2011) and Ra-
pach and Zhou (2013)) and es ima ing impulse esponse unc ions by local p ojec ions in mac oeconomics
(see, e.g., Jo dà (2005)).
©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/QE1762
1108 Liyu Dou Quan i a i e Economics 15 (2024)
The esul ing HAR s anda d e o s a e asymp o ically jus i ied in a la ge a ie y o ci -
cums ances.
Small sample simula ions,2howe e , show ha he Newey–Wes /And ews app oach
can lead o alse ejec ions o he null a oo o en. A la ge subsequen li e a u e (su -
eyed in Mülle (2014)) employs al e na i e asymp o ics ha is o en mo e accu a e in
ini e samples and hus demons a es be e pe o mance o con olling he null ejec-
ion a e. To implemen hese p ocedu es in p ac ice, howe e , he use mus choose a
uning pa ame e . One example is he choice o bin he ixed-bscheme,3in which a
ixed-b ac ion o he sample size is used as he bandwid h in ke nel LRV es ima o s.
Ano he example is he choice o qin o hono mal se ies HAR es s,4in which he LRV is
es ima ed by p ojec ions on o qmean-ze o low- equency o hono mal unc ions. The
choice o he uning pa ame e embeds a adeo be ween bias and a iabili y o he
LRV es ima o . I subsequen ly leads o a size-powe adeo in he esul ing HAR in e -
ence. P e ious s udies add ess his adeo by es ic ing a en ion o HAR es s ha a e
based on ke nel and o hono mal se ies LRV es ima o s. They de i e he op imal uning
pa ame e based on second-o de asymp o ics and unde c i e ia ha a e age unc ions
o ype I and ype II e o s wi h di e en weigh s.5I is no clea , howe e , whe he he
esul ing HAR es s would emain op imal in ini e samples i hose es ic ions we e no
imposed. Mo eo e , as demons a ed la e , he choice o LRV es ima o o HAR es is
empi ically ele an . The e o e, i would be use ul o ha e guidelines o p ac i ione s o
implemen HAR in e ence wi h ce ain senses o op imali y.
The pu pose o his pape is o p o ide o mal ini e-sample e iciency esul s o HAR
in e ence abou a scala pa ame e o in e es , wi hou es ic ing he class o es s and
wi h commonly used no ions o op imali y in hypo hesis es ing. Speci ically, I de i e
op imal (weigh ed a e age powe maximizing scale in a ian ) HAR es s in he Gaussian
loca ion model, unde nonpa ame ic assump ions on he unde lying spec al densi y.
In addi ion, I ind ha wi h an app op ia e adjus men o he c i ical alue, i is nea ly
op imal o use a ype o - es wi h he LRV es ima ed by equal-weigh ed p ojec ions
on o q ype II cosines, which is known as he equal-weigh ed cosine (EWC) es in he
li e a u e (c . Mülle (2004,2007), Laza us, Lewis, S ock, and Wa son (2018)).
The main assump ion in his pape is ha he e exis s an uppe bound on he deg ee
o pe sis ence in da a. In ime-se ies e minology and om a spec al pe spec i e, his
amoun s o speci ying a wo s -case s eepes , o “uni o mly maximal” spec al densi y
unc ion in class F, which is he collec ion o all plausible spec a and is o a nonpa a-
me ic na u e, as opposed o possibly s ong pa ame ic classes.6In heo y, such p im-
2See, e.g., den Haan and Le in (1997,1998) o ea ly Mon e Ca lo e idence o he la ge size dis o ions o
HAR es s compu ed using he Newey–Wes /And ews app oach.
3See pionee ing pape s by Kie e , Vogelsang, and Bunzel (2000) and Kie e and Vogelsang (2002,2005).
Also, see Jansson (2004), Mülle (2004,2007), Phillips (2005), Phillips, Sun, and Jin (2006,2007), Sun, Phillips,
and Jin (2008), A chadé and Ca aneo (2011), Gonçal es and Vogelsang (2011), Sun and Kaplan (2012), Sun
(2014a); and Sun (2014b), among many o he s.
4See, e.g., Mülle (2004,2007), Phillips (2005), Ib agimo and Mülle (2010); and Sun (2013), among many
o he s.
5See, o example, Sun, Phillips, and Jin (2008) and Laza us, Lewis, and S ock (2021).
6Fo pa ame ic examples, Robinson (2005) assumes ha he unde lying pe sis ence is o he “ ac ional”
ype and de i es consis en LRV es ima o s unde ha class; Mülle (2014) assumes ha he unde lying
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1109
i i es may con ain smoo hness es ic ions (e.g., bounds on de i a i es) and/o shape
es ic ions (e.g., mono onici y). Bu , o con enience in ac ual implemen a ions, I sug-
ges he p ac i ione s ollow he con en ion o measu ing pe sis ence by i ing a simple
AR(1) model o da a o de e mine a easonable .
As i u ns ou , he ini e-sample e iciency bounds, he bes choice o q, and he c i i-
cal alue adjus men in he nea ly op imal EWC es a e essen ially go e ned by he max-
imal pe sis ence. The p ac ical implica ion is ha once he maximal pe sis ence in da a
is app op ia ely de e mined, he EWC es wi h adjus ed c i ical alue can be used wi h-
ou much loss o e iciency. The esul ing implemen a ion is s aigh o wa d and only
in ol es es ima ing an AR(1) model and making a simple adjus men o he S uden -
c i ical alue o he EWC es . Fu he mo e, his p ocedu e can be easily adap ed o
eg ession models. I discuss hese p ac ical ma e s in de ail below in Sec ion 5. In addi-
ion, I illus a e he implemen a ion in wo empi ical examples in Sec ion 6conce ning
con idence in e al cons uc ion and hypo hesis es ing wi h au oco ela ed da a.
This pape makes h ee main con ibu ions. Fi s , I es ablish a ini e-sample he-
o y o op imal HAR in e ence in he Gaussian loca ion model unde a simpli ying ap-
p oxima ion. To do so, I ollow Mülle (2014) and ecas HAR in e ence as a p oblem
o in e ence abou he co a iance ma ix o a Gaussian ec o . The spec um, as an
in ini e-dimensional nuisance pa ame e , complica es he solu ion o he p oblem. To
make p og ess, I use insigh s om he so-called leas a o able app oach and iden i y
he “leas a o able dis ibu ion” o e he class F. The esul ing op imal es embeds
obus ness-e iciency adeo s in hypo hesis es ing. This op imal adeo is a unc ion
o he unde lying p imi i e F, namely he se ial co ela ions one is willing o co ec o
unde he null, and he al e na i e dependency ha one desi es he es o o ien powe
owa d.
Second, I ind ha nea ly op imal in e ence can be ob ained by using he EWC
es , bu only a e an adjus men o he S uden - c i ical alue. The p ac ical impli-
ca ions a e an explici link be ween he choice o qand assump ions on he unde lying
spec um, as well as a co esponding adjus men o he S uden - c i ical alue. In de-
ail, conside a second-o de s a iona y scala ime se ies y . The spec al densi y o y
scaled by 2πis gi en by he unc ion :[−π,π]→[0, ∞).To es H0:E[y ]=0 agains
H1:E[y ]=0, he EWC es uses a -s a is ic
q
EWC =Y0




q

j=1
Y2
j/q
,(1)
whe e Y0is he sample mean o y and Yj,j=1, 2, ,qa e qweigh ed a e ages o y as
Yj=T−1√2T
=1cos(πj( −1/2)/T)y . These weigh ed a e ages can be app oxima ely
hough o as independen ly no mally dis ibu ed, each wi h a iance T−1 (πj/T ).As
men ioned ea lie , he choice o qembeds a bias and a iance adeo o he LRV es i-
ma o q
j=1Y2
j/q. The con en ional wisdom is o choose qsu icien ly small such ha
long- un p ope y can be app oxima ed by a s a iona y Gaussian AR(1) model, wi h coe icien a bi a ily
close o one and de i es uni o mly alid in e ence me hods ha maximize weigh ed a e age powe .
1110 Liyu Dou Quan i a i e Economics 15 (2024)
Figu e 1. Powe unc ion plo o a weigh ed a e age powe (WAP) bound induced es , op imal
EWC es , and size-adjus ed EWC es using q=3. No es: Unde he al e na i e, he mean o y is
δT−1/2and y ollows a Gaussian whi e noise. Unde he null, he “uni o mly maximal” unc ion
o Fco esponds o an AR(1) wi h coe icien 0.8. Sample size T=100.
{Yj}q
j=1can be ea ed as i.i.d. no mals. By doing so, one a oid possibly la ge bias in es i-
ma ing he LRV, and he esul ing EWC es has less size dis o ions when he S uden -
c i ical alue is employed. In con as , he new EWC es sugges s using a la ge qand an
app op ia ely enla ged c i ical alue o mo e powe ul in e ence. Bo h he choice o q
and he c i ical alue adjus men depend on he class F.
Figu e 1illus a es his second con ibu ion in es ing E[y ]=0, ∈Fagains he
local al e na i e E[y ]=δT−1/2 o T=100, whe e y ollows a Gaussian whi e noise and
he “uni o mly maximal” unc ion o Fco esponds o an AR(1) model wi h coe icien
0.8. In his con ex , o a oid size dis o ions la ge han 0.01, one needs o choose q=3
when he S uden - c i ical alue is employed. The new EWC es , howe e , has q=6and
in la es he S uden - c i ical alue by a ac o o 1.13. Mo eo e , i is nea ly as powe ul
as a weigh ed a e age powe bound induced es . I has a 28.9% e iciency gain o e he
size-adjus ed EWC es using q=3, in o de o achie e he same powe o 0.5.7
Thi d, I p opose a simple adjus men o he c i ical alue o he EWC es . The ad-
jus ed c i ical alue is compu ed easily, by in e ing a one-dimensional nume ical in-
eg al. Fo p ac ical con enience, I o e a ule o humb o adjus he S uden - c i ical
alue o he EWC es in Table 2, as ollows. Unde a se ies o classes Fwhe e he la ges
pe sis ence is pa ame e ized as an AR(1) wi h coe icien ρ=1−c/T,Table1lis s he
7By e iciency gain, I mean he inc ease o δ2in pe cen o he size-adjus ed EWC es using q=3in
o de o achie e he same powe o he new EWC es . I no e ha one canno di ec ly appeal o Pi man
e iciency measu e ( he inc ease o he numbe o obse a ions equi ed o achie e he same powe ) in he
con ex o Figu e 1, since he sample size Tis ixed a 100. A di e en calcula ion, howe e , shows ha o
T=77 he size-adjus ed EWC es using q=6 has powe o a ound 0.5, unde he same δsuch ha he EWC
es using q=3 yields powe o 0.5 o T=100.

Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1111
Table 1. Op imal qand adjus men ac o o he S uden - c i ical alue o le el αEWC.
c5 10203040 50 77
ρ0.95 0.9 0.8 0.7 0.6 0.5 0.23
α=0.05 (3, 1.55)(
4, 1.26)(
6, 1.13)(
7, 1.07)(
9, 1.05)(
12, 1.04)(
19, 1.02)
No e: Based on a se ies o classes F, in which he “uni o mly maximal” unc ion co esponds an AR(1) wi h coe icien
ρ=1−c/T. Sample size Tis 100, bu he esul ing op imal choice o q(almos ) emains unchanged o ixed c≤40 and o
T=200, 500, 1000.
op imal choice o qand he adjus men ac o o he S uden - c i ical alue o selec ed
c(and ρ o ixed T). I u ns ou ha he esul ing op imal choice o q(almos ) emains
unchanged as T a ies, o ixed c≤40. Mo e in e es ingly, o ixed qand T, head-
jus men ac o does no change subs an ially unde o he ypes o F.Table2collec s
he adjus men ac o s in (augmen ed) Table 1 o selec ed q. In he e en ha he same
qis op imally chosen unde di e en c, he la ges adjus men ac o (co esponding
o he la ges c) is sugges ed in he ule o humb. In case esea che s pick a alue o q
by some o he means, I sugges adjus ing he co esponding S uden - c i ical alue di-
ec ly acco ding o Table 2.O he wise,Sec ion5.1 p o ides guidance on de e mining a
easonable , he subsequen c i ical alue adjus men , and he selec ion o q.
This pape ela es o a la ge li e a u e. Fi s , unlike he majo i y o he HAR li e -
a u e, I conside op imal HAR in e ences wi hou es ic ing he class o es s. Second,
he majo i y o he li e a u e add esses he sampling a iabili y o LRV es ima o s ia he
so-called ixed-basymp o ics, and u he accoun s o bias by highe -o de adjus men
o he ixed-bc i ical alue.8In con as , I concu en ly ackle bias and a iance in es-
ima ing he LRV by a i s -o de adjus men in he spi i o employing ixed smoo hing
asymp o ics unde s ong pe sis ence in Sun (2014a). E en so, he esul ing adjus ed c i -
ical alue is easily compu ed wi hou simula ions unde a simpli ying s uc u e. Thi d,
his pape con ibu es o he uni o m size con ol li e a u e de eloped by Mülle (2014),
P eine s o e and Pö sche (2016), Pö sche and P eine s o e (2018,2019), and Mülle
and Wa son (2022). I analy ically de i e powe ul es s ha uni o mly con ol size o e
a guably la ge classes o models, while Mülle (2014) nume ically de e mines powe ul
es s unde a possibly es ic ed pa ame ic class o models. P eine s o e and Pö sche
(2016)andPö sche and P eine s o e (2018,2019) ocus on size dis o ions and powe
Table 2. Rule o humb o adjus men ac o o he S uden - c i ical alue o le el αEWC.
q346891011121620
α=0.05 1.55 1.37 1.17 1.15 1.09 1.09 1.07 1.05 1.03 1.03
No e:Eachqis jus i ied as he op imal choice o le el αEWC es , unde some class Fand o sample size T. An example
o he co esponding class Fis he one in which he “uni o mly maximal” unc ion co esponds o an AR(1) model wi h coe i-
cien ρ=1−c/T as in Table 1. Only he la ges adjus men ac o is displayed should he same qeme ge as he op imal choice
unde di e en c.
8See, o example, Velasco and Robinson (2001), Sun, Phillips, and Jin (2008), Sun (2011,2013,2014c);
and Laza us, Lewis, and S ock (2021).
1112 Liyu Dou Quan i a i e Economics 15 (2024)
de iciencies o gi en HAR es s, allowing o gene al classes o models. Mülle and Wa -
son (2022) de i e ini e-sample size con ol esul s in gene al spa ial se ings wi hou he
simpli ying s uc u e conside ed in his pape bu o e limi ed analy ical esul s on he
e iciency side.
The sugges ion o using a la ge qand enla ged c i ical alues o he EWC es mi -
o s ecen ecommenda ions o nonpa ame ic in e ence, such as hose o A ms ong
and Kolesá (2018,2020). In di e en con ex s, A ms ong and Kolesá and I bo h s ess
he ad an age o accep ing bias in es ima ing a nonpa ame ic unc ion and hen us-
ing a sui ably adjus ed c i ical alue o accoun o he maximum bias. Ou amewo ks
a e, howe e , di e en . I conside a Gaussian expe imen in which he he e oskedas ic-
i y is go e ned by an unknown nonpa ame ic unc ion and is hus mo e in he spi i o
Lehmann and S ein (1948), while he main ocus in A ms ong and Kolesá (2018)isan
unknown eg ession unc ion in he mean o a homoskedas ic Gaussian expe imen .
The emainde o he pape is o ganized as ollows. Sec ion 2se s up he model and
discusses p elimina ies. Sec ion 3de i es e iciency esul s unde an essen ial simpli i-
ca ion, which a e heo e ically in es iga ed in mo e gene al se ings in Sec ion 4.Sec-
ion 5con e s he heo e ical insigh s in o p ac ical guidance and discusses he imple-
men a ion in eg ession models. Sec ion 6p o ides empi ical illus a ions wi h a sel -
con ained guide o implemen a ion. In e es ed p ac i ione s can skip he heo e ical
discussions and ead Sec ion 6di ec ly. P oo s and compu a ional de ails a e p o ided
in he Appendices.
2. Model and p elimina ies
The pape conce ns in e ence abou μin he loca ion model,
y =μ+u , =1, 2, ,T,(2)
whe e μis he popula ion mean o y and u is a mean-ze o s a iona y Gaussian p o-
cess wi h absolu ely summable au oco a iances γ(j)=E[u uy−j]. The spec um o y
scaled by 2πis gi en by he e en unc ion :[−π,π]→ [0, ∞)de ined ia (λ)=
∞
j=−∞cos(jλ)γ(j).Wi hy=(y1,y2,,yT)and e=(1, 1, ,1),
y∼Nμe,( ),(3)
whe e ( )has elemen s ( )j,k=(2π)−1π
−π (λ)e−i(j−k)λdλwi h i =√−1. The loca-
ion model (2) is o en conside ed a s ylized se ing o p o ide heo e ical insigh s in o
HAR in e ence.9As simple as i is, his model is empi ically ele an in a numbe o si ua-
ions. Fo example, he s a is ical s udy o uncondi ional equal p edic i e abili y (UEPA)
conce ning compe ing o ecas s in inancial and mac oeconomic con ex s amoun s o
es ing an uncondi ional mean-ze o condi ion in (2)wi hy being he p oduced loss di -
e en ial se ies.
9Fo HAR s udies based on he Gaussian loca ion model see, o example, Velasco and Robinson (2001),
Jansson (2004), Sun, Phillips, and Jin (2008), Sun (2011,2013,2014c), Mülle (2014), Laza us, Lewis, and
S ock (2021), among many o he s.
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1113
Th oughou he pape , I mainly ocus on p esen ing and analyzing new e iciency
esul s o HAR in e ence in he uni a ia e Gaussian loca ion model (3), and discuss he
implica ions o conduc ing in e ence abou a noncons an eg esso in eg ession se -
ings. The ideas and me hods explo ed in he simples model (3) can o en be used as
a ounda ion o s udying HAR in e ence in mul i a ia e loca ion models and gene al
GMM se ings, po en ially in ol ing addi ional complica ions.10 Fo mal gene aliza ions
o his pape ’s e iciency esul s along hose lines a e, howe e , no s aigh o wa d and
beyond he scope o he pape .
The HAR in e ence p oblem in (3) conce ns es ing H0:μ=0(o he wise,sub ac
he hypo hesized mean om y ) agains H1:μ= 0 based on he obse a ion y.The
de i a ion o powe ul es s in his p oblem is complica ed by he ac ha he al e na i e
is composi e (μis no speci ied unde H1) and he p esence o he in ini e-dimensional
nuisance pa ame e . I ollow s anda d app oaches o deal wi h μand mainly ocus on
ackling he nuisance pa ame e in his pape .
I is use ul o ake a spec al ans o ma ion o he model (3). In pa icula , as in o-
duced in he In oduc ion, conside he one- o-one ans o ma ion om {y }T
=1in o he
sample mean Y0=T−1T
=1y and he T−1 weigh ed a e ages:
Yj=T−1√2
T

=1
cosπj( −1/2)/Ty ,j=1, 2, ,T−1. (4)
De ine as he T×Tma ix wi h i s column equal o T−1e,and(j+1) h column wi h
elemen s T−1√2cos(πj( −1/2)/T), =1, ,T,andι1as he i s column o IT.Then
Y=(Y0,Y1,,YT−1)=y∼Nμι1,0( ),(5)
whe e 0( )=( ). The HAR es ing p oblem becomes H0:μ=0 agains H1:μ=0
based on he obse a ion Y.
A common de ice o dealing wi h he composi e al e na i e in he na u e o μis o
sea ch o es s ha maximize weigh ed a e age powe o e μ. Fo analy ical ac abili y,
I ollow Mülle (2014) o conside a Gaussian weigh ing unc ion o μwi h mean ze o
and a iance η2. The scala η2go e ns whe he close o dis an al e na i es a e em-
phasized by he weigh ing unc ion. Fo a gi en , and hus known 0( )1,1, he choice
η2=(κ−1)0( )1,1 e ec i ely changes he es ing p oblem o H
0:Y∼N(0, 0( ))
agains H
1:Y∼N(0, 1( )),whe e1( )=0( )+(κ−1)ι1ι
10( )1,1.This ans-
o ms he p oblem in o one o in e ence abou co a iance ma ices. The hype pa am-
e e κspeci ies a weigh ed a e age powe c i e ion. As a gued by King (1987), i makes
sense o choose κin a way such ha good es s ha e app oxima ely 50% weigh ed a -
e age powe . The choice o κ=11 would induce he esul ing bes 5% le el (in easible)
es ( ejec i Y2
0>3.840( )1,1) o ha e powe o app oxima ely P(χ2
1>3.84/11)≈56%.
I hususeκ=11 h oughou he implemen a ions.
10Fo o mal discussions o HAR in e ence in gene al GMM se ings see, o example, Sun (2014b), Hwang
and Sun (2017,2018).
1114 Liyu Dou Quan i a i e Economics 15 (2024)
In mos applica ions, i is easonable o impose ha i he null hypo hesis is ejec ed
o some obse a ion Y, hen i should also be ejec ed o he obse a ion aY , o any
a>0. By s anda d es ing heo y,11 any es sa is ying his scale in a iance p ope y can
be w i en as a unc ion o Ys=Y/√YY. The densi y o Ysunde H
i,i=0, 1 is equal
o (see Ka iya (1980)andKing (1980))
hi, ys=Ci( )−1/2ysi( )−1ys−T/2(6)
o some cons an C.
By es ic ing o scale in a ian es s, he HAR es ing p oblem has been u he
ans o med in o H
0:“Yshas densi y h0, ” agains H
1:“Yshas densi y h1, .” The
p oblem emains nons anda d due o he p esence o nuisance pa ame e .Tomake
p og ess, I i s conside di ec ing powe a a la spec um 1=1(whi enoise)when
demons a ing how o de i e e iciency bounds in Sec ion 3. In his case, he al e na-
i e H
1 hen becomes a single hypo hesis H
1, 1:“Yshas densi y h1, 1,” whe e 1( 1)=
κT−1diag(1, κ−1,,κ−1). Mo eo e , unde he null, I assume belongs o an explici
unc ion class Fand seek scale in a ian es s ha uni o mly con ol size o e F.InSec-
ion 4.1, I obus i y insigh s om he whi e noise case o ones whe e powe is di ec ed
a a non la spec um ˜
1, o when minimax bounds a e conce ned i belongs o a class
G⊂Funde H1.
The es ing p oblem is now educed o dis inguishing he composi e null H
0
om a single al e na i e. A well-known gene al solu ion o his ype o p oblem p o-
ceedsas ollows(c .Lehmann and Romano (2005)). Suppose is some p obabili y
dis ibu ion o e F, and he composi e null H
0is eplaced by he single hypo he-
sis H
0,:“Yshas densi y h0, d( ).” Any ad hoc es ϕah ha is known o be o
le el αunde H
0also con ols size unde H
0,, because ϕah(ys)h0, (ys)d( )dys=
ϕah(ys)h0, (ys)dysd( )≤α. By Neyman–Pea son lemma, he likelihood a io es
o H
0,agains H
1, 1, deno ed by ϕ, 1, yields a bound on he powe o ϕah.Fu he -
mo e, i ϕ, 1also con ols size unde H
0, hen i mus be he bes es o H
0agains
H
1, 1and he esul ing powe bound is he lowes possible one. In he ja gon o s a is-
ical es ing, he dis ibu ion ha yields he bes es (should i exis ) is called he “leas
a o able dis ibu ion,” and I deno e i by ∗ h oughou he pape . Un o una ely, he e
is no sys ema ic way o de i ing such a dis ibu ion. I make p og ess along his line in
he ollowing sec ions.
3. Fini e-sample e iciency esul s
In his sec ion, I impose a simpli ying Whi le- ype diagonal s uc u e on he implied co-
a iance ma ices 0o he e ec i e obse a ion Y, speci y a p io i ha Fpossesses a
mos pe sis en spec um, and analy ically de i e he esul ing leas a o able dis ibu-
ion, and hus ob ain he op imal es . Mo e speci ically, I make he ollowing assump-
ions.
11See, o example, Chap e 6 in Lehmann and Romano (2005).
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1121
I one desi es o di ec powe a say ˜
1ins ead o 1, hen he c i ical alue adjus men s
will no change o any gi en q, and he op imal selec ion o qnow makes (12) each i s
la ges alue a {˜
ζj=(c a
q)2κ−1˜
1(πj/T )/˜
1(0)wj}q
j=1. In his sense, obus ness and e i-
ciency cons ain s a e sequen ially add essed. In con as , he op imal es ϕ∗,˜
1(see
de ails in Co olla y 4.4) and he esul ing op imal weigh s {w∗
j}in he -s a is ic o m (9)
a y simul aneously wi h ˜
1, ende ing i less ope a ionally s aigh o wa d as compa ed
o he weigh ed cosine es s.
The abo e discussion, o cou se, applies o he class o EWC es s, o which wj=q−1
o any gi en q.In ha case,i is u he pa ame e ized using he limi ing local- o-ze o
spec a unde local- o-uni y asymp o ics, ha is, 1/(π2j2+c2) o some c>0, (c a
q)2
mi o s he c i ical alue o an Fs a is ic (wi h p=1) om ixed-smoo hing asymp o ics
unde s ong (local- o-uni y) pe sis ence in Sun (2014a). Mo eo e , (c a
q)2is o a hump
shape as a unc ion o q o any gi en and ini e c(see Figu e 3 in Sun (2014a)). The
op imal choice o q hen amoun s o exploi ing ha hump shape and picking he qsuch
ha he ec o ha s acks q eplica es o (c a
q)2κ−1q−1and T−1−qze os majo izes all
o he candida e ec o s o he same o m. In ace o c=∞,c a
qmono onically dec eases
in q. As a esul , one shall u ilize all in o ma ion in he da a and op imally choose q=
T−1.
In ui i ely, his qshall also balance he wo s absolu e bias (| (0)−q−1q
j=1 (πj/
T)|) and a iabili y o he LRV es ima o q−1q
j=1Y2
jin a es ing-op imal sense. Acco d-
ingly, he adjus ed c i ical alue c a
qaccoun s o his maximum bias in he spi i o A m-
s ong and Kolesá . Howe e , since c a
qis de e mined in a a he mo e complica ed way
han in hei se ings, I choose no o discuss u he hese bias- a iance adeo s. Fu -
he mo e, I no e ha because he implied weigh s {w∗
j}q∗
j=1in he op imal es do no de-
pend on he associa ed c i ical alue c q∗in a s aigh o wa d way, and hus may b ing
ano he laye o complica ions, nei he will I o mally compa e q∗in he op imal es
wi h he MSE o es ing op imal qin he EWC class in his pape .
Commen 5. One may wonde whe he Theo em 3.3 is limi ed by only o ien ing
powe owa d 1. As i u ns ou , he insigh s can be gene alized o accommoda e pos-
sibly mo e complex al e na i es. I elega e he o mal analysis along his line o Sec-
ion 4.1.
3.2 The op imal EWC es
By using highe -o de expansions, Laza us, Lewis, and S ock (2021) de i e a size-powe
on ie o ke nel and o hono mal se ies HAR es s unde an asymp o ic amewo k.
The EWC es is shown o achie e ha on ie in hei con ex . I is, howe e , no clea
how he EWC es pe o ms in he cu en con ex . The e iciency bounds de i ed in he
las sec ion p o ide a na u al benchma k o gauge he pe o mance o an ad hoc es . In
his sec ion, I ake up he EWC es as he ad hoc es and discuss i s p ope ies.
I ha e wo ela ed goals. The i s is o s udy he (weigh ed a e age) powe p ope ies
o he EWC es ela i e o he op imal es in Theo em 3.3. As i u ns ou , he EWC es
is close o op imal, unde an app op ia e choice o qand wi h he adjus ed c i ical alue,
which is jus discussed in Commen 4 abo e. I e e o his new EWC es as he op imal

1122 Liyu Dou Quan i a i e Economics 15 (2024)
Table 4. Weigh ed a e age powe (WAP) o he op imal es and he op imal EWC es .
ρ0.50 0.60 0.70 0.80 0.90 0.95 0.98
qin op imal EWC 12 9 7 6431
c i ical alue adj. ac o 1.04 1.05 1.07 1.13 1.26 1.55 1.87
WAP o op imal EWC 0.507 0.492 0.472 0.438 0.351 0.233 0.088
WAP o op imal es 0.507 0.493 0.475 0.442 0.358 0.239 0.091
No e: The “uni o mly maximal” unc ion co esponds o an AR(1) wi h coe icien ρ.Nominalle elis5%. Sample size Tis
100.
EWC he ea e . My second goal is o d aw he ollowing p ac ical implica ions o he
op imal EWC es ia i s compa ison wi h he con en ional EWC es : One should use
he EWC es wi h a la ge qand app op ia ely enla ged c i ical alues o mo e powe ul
HAR in e ence.
3.2.1 Powe o heop imalEWC es Conside he ype o Fin Table 3, ha is, he “uni-
o mly maximal” unc ion o he class Fco esponds o an AR(1) wi h coe icien ρ.
As ρ a ies, Tables 4displays he c ucial ing edien s (qand adjus men ac o ela i e o
S uden - c i ical alue) in he op imal EWC es , i s esul ing weigh ed a e age powe
and he e iciency bound induced by he op imal es . Two obse a ions a e immedia e.
Fi s , he op imal EWC es is nea ly as powe ul as he op imal es . This obse a ion
emains when a di e en ype o class Fis conside ed in Table 5. Second, he selec ions
o qin he op imal EWC es and q∗in he op imal es a e no necessa ily equal. A e
all, as explained in Commen s 3 and 4 abo e, hey a e de e mined in a guably di e en
obus ness-e iciency adeo mechanisms. I is wo h no ing ha qin he op imal EWC
es can be somewha sensi i e o nume ical e o s in e alua ing (12)when is ela i ely
la . This, howe e , does no ha e any subs an ial consequence in heo y.
3.2.2 P ac ical implica ions Recall ha he con en ional wisdom in implemen ing he
EWC es is o use a su icien ly small qand o employ he S uden - c i ical alue. I
ind, howe e , ha i is be e o use a la ge qand o employ an enla ged c i ical alue.
Take he example om Figu e 2as an illus a ion: The “uni o mly maximal” unc ion
co esponds o an AR(1) wi h coe icien 0.8 and he sample size is ixed o be 100.
Acco ding o con en ional wisdom, one needs o use q=3 in he usual EWC es o
ob ain size dis o ions less han 0.01. The op imal EWC es , howe e , selec s a la ge
q=6 (highligh ed in Table 4), and he co esponding S uden - c i ical alue mus be
in la ed by a ac o o 1.13 o exac size con ol. An apple- o-apple compa ison hen
Table 5. Weigh ed a e age powe (WAP) o he op imal es and he op imal EWC es .
C10.0 5.6 3.2 1.8 1.0 0.6 0.2 0.1
WAP o op imal EWC 0.286 0.361 0.418 0.453 0.482 0.500 0.526 0.533
WAP o op imal es 0.288 0.364 0.419 0.457 0.484 0.501 0.527 0.536
No e: The “uni o mly maximal” unc ion o Fis (φ)=exp(−Cφ).Nominalle elis5%.Tis 100.
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1123
Figu e 5. Powe unc ion plo o he es ϕ∗, he op imal and con en ional EWC es s. No es:
Unde he al e na i e, he mean o y is δT−1/2(1−ρ1)−1and y ollows a Gaussian AR(1) wi h
coe icien ρ1. Unde he null, he o Fco esponds o an AR(1) wi h coe icien 0.8. Sample
size Tis 100.
e eals ha he size-adjus ed weigh ed a e age powe o he usual EWC es (0.39) has
abou 12% loss as compa ed o ha o he op imal EWC es (0.438, as highligh ed in
Table 4).
The supe io powe p ope y o he op imal EWC es is u he e iden when local
al e na i es a e conside ed. In pa icula , in he con ex o he abo e example, I con-
side μ=δT−1/2(1−ρ1)−1unde he al e na i e. Panels (a) and (b) o Figu e 5plo he
powe o he op imal es ϕ∗, he op imal EWC es , and he size-adjus ed EWC es us-
ing q=3 o a ious δunde ρ1=0andρ1=0.8, espec i ely. As can be seen in panel
(a), e en hough he op imal EWC es unde ejec s unde he null, i is mo e powe ul
han he EWC es using q=3 in de ec ing local de ia ions om he null. Speci ically,
by using he op imal EWC es , a 28.9% e iciency imp o emen is ob ained in o de o
achie e he same powe o 0.5. In he case in which ρ1=0.8, he e iciency gain is la ge
(47.6%), since he op imal EWC es hen exac ly con ols size by cons uc ion. Fu he -
mo e, gi en ha he op imal EWC es is nea ly as powe ul as he o e all op imal es ϕ∗
in e ms o weigh ed a e age powe unde he whi e noise al e na i e, i is no su p ising
o see ha he powe unc ions o hese wo es s a e almos iden ical.
4. Theo e ical gene aliza ions
The ini e-sample e iciency esul s in Sec ion 3a e de i ed unde seemingly es ic i e
assump ions. In pa icula , when powe is di ec ed a he whi e noise al e na i e a p io ,
he op imal es ϕ∗possesses a p ecise sense o op imali y, and he op imal EWC is nu-
me ically ound o be nea ly as powe ul as ϕ∗. Mo e impo an ly, he exis ing e iciency
esul s a e en i ely based on he Whi le- ype diagonal s uc u e. I is na u al o ask how
limi ed hese simpli ying assump ions a e in elici ing heo e ical insigh s in HAR in e -
ence. Said di e en ly, can he insigh s on e iciency in Sec ion 3be gene alized o mo e
1124 Liyu Dou Quan i a i e Economics 15 (2024)
gene al se ings? In his sec ion, I ake up hese ques ions and discuss he heo e ical
gene aliza ions.
4.1 Powe di ec ions and minimax e iciency esul s
Fi s o all, I de ise op imal es s ha di ec powe a a non la ˜
1, and, mo e gene ally,
a nonsingle on class G⊂Funde H1. Fo analy ical ac abili y, I main ain Assump ion
3.1 and also impose a Whi le- ype s uc u e on 1( ), which au oma ically holds unde
1.
Assump ion 4.1. Fo all ∈G,1( )=T−1diag(κ (0), (π/T ),, (π(T−1)/T).
Adop ing he con en ional scale in a iance and weigh ed a e age powe maximiz-
ing c i e ia, I now seek powe ul es s as unc ions o Ys=Y/√YYin he p oblem o
Hd
0:Y∼N0, T−1diag (0), (π/T ),, π(T−1)/T, ∈F(13)
agains Hd
1,G:Y∼N0, T−1diagκ (0), (π/T ),, π(T−1)/T, ∈G,
unde he ollowing assump ion ha nes s Assump ion 3.2 as a special case (G={ 1}).
Assump ion 4.2. (a) The e exis s a ∈Fsuch ha (0)
(φ)≤ (0)
(φ), o all φ∈[−π,π]and
∈F.
(b) The e exis s a ∈Gsuch ha (0)
(φ)≥ (0)
(φ), o all φ∈[−π,π]and ∈G.
(c) (πj/T )/ (πj/T )≥ (π(j+1)/T)/ (π(j+1)/T),j=0, 1, ,T−2.
(d) Fcon ains all kinked unc ions de ined by θ(φ)= θ(φ) (φ),in which, o θ∈
[0, π], θ(φ)=1i |φ|≤θand θ(φ)∈[1, ∞)i |φ|>θ.
(e) Gcon ains all kinked unc ions de ined by gθ(φ)= 
θ(φ) (φ),in which, o θ∈
[0, π], 
θ(φ)=1i |φ|≤θand 
θ(φ)∈[0, 1]i |φ|>θ.
In ui i ely, ela i e pe sis ence be ween he null and al e na i e hypo heses ma e s
in dis inguishing hem. In Sec ion 3, because he al e na i e is ixed a a cons an sin-
gle on, he “uni o mly maximal” spec al densi y plays a i al ole in de i ing “leas a-
o able” esul s. Now, unde Assump ion 4.2, / posi s he la ges possible ela i e pe -
sis ence and is hus expec ed o be an impo an p imi i e in elici ing simila e iciency
esul s. I u ns ou ha his in ui ion is co ec and I o malize i below as a minimax
esul .
I ollow Lehmann and Romano (2005) o de ine he necessa y no a ion in he HAR
con ex . Le 0and 1deno e dis ibu ions o o e Fand G, espec i ely. Le ϕ0,1
be he mos powe ul le el αweigh ed a e age powe maximizing scale in a ian es
o es ing H0,0agains H1,1,inwhichH0,0and H1,1a e simple hypo heses de ined
analogously o H
0,in Sec ion 2,andle β0,1be i s weigh ed a e age powe o a gi en
κ. Suppose 0and 1a e such ha sup ∈FE[ϕ0,1(y)] ≤αand in ∈GE[ϕ0,1(y)] =
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1125
β0,1, henϕ0,1maximizes in ∈GE[ϕ(y)] among all alid le el α es s ϕo H0.The
ollowing heo em makes u he op imal s a emen s abou he minimax weigh ed a -
e age powe bound β0,1among all possible candida es o 0and 1.
Theo em 4.3. Unde Assump ions 3.1,4.1,and 4.2,
(i) I κ (0)/ (0)≤ (π/T )/ (π/T ), hen he smalles β0,1among all possible pai s
o 0and 1is αand is a ained by he i ial andomized es .
(ii) I κ (0)/ (0)> ( )(π/T )/ (π/T ), hen he ollowing es ϕ∗iden i ies he leas
a o able pai o dis ibu ions (∗
0,∗
1)in he sense ha :
β∗
0,∗
1≤β0,1
o all possible pai s o 0and 1,in which ϕ∗ ejec s o la ge alues o
Y2
0+ (0)
q∗

j=1
Y2
j/ (πj/T )
Y2
0+κ (0)
q∗

j=1
Y2
j/ (πj/T )
(14)
o a unique 1≤q∗≤T−1, and wi h he c i ical alue c q∗such ha he es is o
le el αunde = and a ains i s minimax powe β∗
0,∗
1a = .
The p oo s a egy o Theo em 4.3 is simila o ha o Theo em 3.3.Ip o ebo ho
hem and he wo immedia e co olla ies wi hin a cohe en amewo k in Appendix A.
Co olla y 4.4. Le G={˜
1} o some ˜
1 ha is no necessa ily equal o he la 1so ha
=˜
1in he sense o Assump ion 4.2(b). Unde Assump ions 3.1,4.1,4.2:
(i) I κ˜
1(0)/ (0)≤˜
1(π/T )/ (π/T ), hen he bes weigh ed a e age powe maximiz-
ing scale in a ian es o H0:μ=0agains H1:μ=0is he i ial andomized
es .
(ii) I κ˜
1(0)/ (0)>˜
1(π/T )/ (π/T ), he bes le el αweigh ed a e age powe max-
imizing scale in a ian es ϕ∗o H0:μ=0agains H1:μ= 0 ejec s o la ge
alues o
Y2
0+ (0)
q∗

j=1
Y2
j/ (πj/T )
Y2
0+κ˜
1(0)
q∗

j=1
Y2
j/˜
1(πj/T )
(15)
o a unique 1≤q∗≤T−1, and wi h he c i ical alue c q∗such ha he es is o
le el αunde = .
1126 Liyu Dou Quan i a i e Economics 15 (2024)
Co olla y 4.5. Le Fand Gbe se s o sa is ying Assump ion 4.2 and wi h = 1.Unde
Assump ions 3.1 and 4.1, he weigh ed a e age powe o he op imal es in Theo em 3.3
gi es he minimax powe bound in he sense o Theo em 4.3.
4.2 Nea op imali y o EWC es s
I is ound in Sec ion 3.2 ha he so-called op imal EWC es nea ly a chi es he e -
iciency bound when powe is di ec ed a he whi e noise ( 1). I is emp ing o ask
whe he such indings emain in mo e ealis ic si ua ions, especially gi en ha e i-
ciency bounds a e de i ed in Sec ion 4.1 when powe is o ien ed owa d non la al e -
na i es.
I i s main ain he Whi le diagonal s uc u e wi h AR(1) as be o e (coe icien
ρ0) bu conside powe di ec ions a possibly nonmono onic ˜
1o AR(2) p ocesses wi h
oo s ρ1and ρ2. In panel (a) o Figu e 6, I do no endogenize he es s a is ics o each
(ρ0,ρ1,ρ2)combina ion. Mo e p ecisely, he boxplo s a e o he di e ence in weigh ed
a e age powe s o he ϕ∗and he new EWC es s ha a e ini ially designed o be (nea ly)
op imal a 1(as in Sec ion 3) bu a e now a ˜
1 o a ious (ρ1,ρ2)’s. In con as , panel (b)
le e ages Co olla y 4.4 o endogenize ˜
1in ob aining he e iciency bound and in selec -
ing q o he op imal EWC es . I no e ha o he exis ence o he op imal es in hose
cases I es ic he anges o ρ1and ρ2such ha Assump ion 4.2(c) is sa is ied, bu he
co esponding spec al densi y ˜
1may s ill be nonmono one. Displayed esul s in panel
(a) sugges ha he new EWC es is nea ly as powe ul as he es ϕ∗a AR(2) al e na i es,
e en i bo h a e de i ed unde di e en a ionales and nei he possesses a well-de ined
Figu e 6. Boxplo s o weigh ed a e age powe di e ences be ween ϕ∗and EWC es s. No es:
The “uni o mly maximal” unc ion o Fco esponds o an AR(1) wi h coe icien ρ0∈[0.8, 0.9].
Powe s a e di ec ed a AR(2) al e na i es wi h oo s ρ1and ρ2(−0.8 ≤ρ1(ρ2)≤0.8 in panel (a);
0≤ρ1≤0.8 and −0.8 ≤ρ2≤0 in panel (b)). Sample size Tis 100, and he le el o signi icance is
5%.

Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1127
Figu e 7. Boxplo s o weigh ed a e age powe o weigh ed cosine es s. No es: The “uni o mly
maximal” unc ion co esponds o an AR(1) wi h oo ρ0. Powe s a e di ec ed a AR(2) al e na-
i es wi h oo s ρ1and ρ2(−0.7 ≤ρ1(ρ2)≤0.7). 5% es s using qweigh ed cosines a e consid-
e ed, in which 1000 se s o andom posi i e weigh s a e used o each (ρ0,ρ1,ρ2).SamplesizeT
is 100.
sense o op imali y by cons uc ion. Panel (b) co obo a es he nea op imali y ea u e
o he op imal EWC es when ϕ∗is by design op imal acco ding o Co olla y 4.4 a a
es ic ed ye conside ably la ge se o AR(2) al e na i es. I no e ha he appea ance o
nega i e alues in Figu e 6migh be a ibu ed o nume ical e o s in calcula ing he
c i ical alues o bo h es s.13
I is no ed in Commen 3 o Sec ion 3.1 ha bo h he op imal and EWC es s belong o
he so-called weigh ed cosine es s. I in es iga e whe he he nea op imali y is a unique
ea u e o EWC es s in ha class. Speci ically, I conside such es s using qweigh ed
cosines bu wi h 1000 se s o andom posi i e weigh s ha a e no malized o sum o
one. The c i ical alues a e ob ained such ha hese es s exac ly con ol size a AR(1)
wi h coe icien ρ0. Figu e 7displays boxplo s o he esul ing weigh ed a e age powe
a AR(2) al e na i es wi h oo s ρ1and ρ2.No e ha q∗is by cons uc ion 7 and 5 when
powe is di ec ed a 1in panels (a) and (b), espec i ely. I is ound ha he dispe sion o
weigh ed a e age powe is ela i ely small a q∗ac oss all es s and all AR(2) al e na i es,
sugges ing ha he e may exis a se o weigh ed cosine es s, including he EWC es ,
ha nea ly achie e he e iciency bounds. In ac , he EWC es included is no e en he
13Fo nume ical s abili y, I se he ole ance le el o be 0.003 a ound 0.05 in o de o ob ain he c i ical
alues, so i is no su p ising o ha e nume ical e o s in app oxima ing ejec ion p obabili ies o be o
he o de 10−3. Wi h a smalle ole ance le el, he nume ical in eg a ion o (12) using 2000-poin Gaussian
quad a u e may s ill p oduce complex- alued numbe s. Mo eo e , in ob aining c q∗ o ϕ∗,because he
posi i eness condi ions, and hus he in eg al exp ession o (12) may no hold a p io i o a gi en ˜
q,I i s
simula e 100,000 ϕ∗unde o de e mine a p elimina y q∗and hen use a bisec ion me hod o ob ain
amo ep ecisec q∗while checking he posi i eness condi ions. In he abo e senses, he plo s in Figu e 6
a e bound by small nume ical and simula ion e o s, and nega i e alues a e no e idence o dismiss he
heo ies.
1128 Liyu Dou Quan i a i e Economics 15 (2024)
bes one, as i s qis no op imized ye . Fu he mo e, he compa isons o boxplo s ac oss
di e en qhin ha one migh need o choose weigh s ( es s a is ic) mo e judiciously
when a ela i ely la ge qis used, and he heo e ical insigh s so a ecommend he EWC
es as a good candida e so long as he adjus ed c i ical alue is adop ed.
4.3 Relaxa ion o he Whi le- ype app oxima ion
The heo e ical discussions so a a e en i ely based on he Whi le- ype diagonal s uc-
u e. Fo bo h heo e ical in e es and p ac ical ele ance, i is na u al o ask whe he
he abo e insigh s on op imal HAR in e ence con inue o hold wi hou ha s uc u e. To
ha end, I main ain he c i e ia o weigh ed a e age powe maximizing and scale in a i-
ance and s ill di ec powe a he la spec um 1. The goal is o seek powe ul es s as
unc ions o Ys=Y/√YYin he p oblem o
He
0:Y∼N0, 0( ), ∈F(16)
agains He
1, 1:Y∼N0, κT−1diag1, κ−1,,κ−1,
whe e he supe sc ip edeno es he exac model. No e ha He
1, 1is iden ical o Hd
1, 1,
because 1exac ly becomes diagonal unde 1.
Fi s o all, I no e ha i is, in gene al, di icul o de i e he op imal es o (16). This
is mainly due o he complica ed manne by which en e s 0( ). In his case, e en i i
is ue ha he leas a o able dis ibu ion pu s a poin mass on some unc ion ∗∈F,
i s de e mina ion seems a he di icul . Despi e so, one s ill can ob ain bounds on he
powe o any size-con olling es by using he bounding app oach o Ellio , Mülle , and
Wa son (2015). Recall om Sec ion 2 ha o any p obabili y dis ibu ion o e F, he
likelihood a io es o H
0,agains H
1, 1yields such a powe bound. I he powe o a
alid ad hoc es ϕah is close o he powe bound o some , henϕah is known o be
close o op imal, as no subs an ially mo e powe ul es exis s. I u ns ou ha he in-
sigh s om he diagonal model a e use ul in guessing a good and in sugges ing he
nea op imali y o he EWC es in he exac model. In pa icula , o a gi en ain [0, 1],
le abe a poin mass dis ibu ion on he kinked unc ion a(φ), as was de ined in As-
sump ion 3.2(c). Fo e e y a, he likelihood a io es o H
0,aagains H
1, 1yields a powe
bound. I nume ically sea ch o asuch ha he esul ing powe bound is minimized.
Deno e his aby a†and he esul ing by †. The powe bound I employ o gauge he
e iciency o ad hoc es s is hen he powe o
ϕ†, 1=1Y0( a†)Y−1Y1( 1)Y>c , (17)
o some c such ha E[ϕ†, 1]=αunde H
0,a. I u ns ou ha he EWC es essen ially
achie es his bound, a e app op ia e c i ical alue adjus men and op imally choos-
ing q.
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1129
Fo an EWC es wi h a gi en q, he null ejec ion p obabili y a a gi en and c i ical
alue c now becomes
P
Y0




q

j=1
Y2
j/q
≥c =PZ2
0
q

j=1
λj( )Z2
j
≥1, (18)
in which {λj( )}q
j=1a e posi i e eigen alues o M(c ,q)0,q( )(no malized by he mag-
ni ude o he only nega i e eigen alue),14 whe e 0,q( )is he uppe le (q+1)×(q+1)
block ma ix o 0( )and M(c ,q)=diag(−1, c 2/q,c 2/q,,c 2/q).By hesamea -
gumen s in Sec ion 3,(18) is maximized a such ha all λj( )’s a e join ly minimized.
The opaque mapping om λj( )back o , howe e , p e en s us om explici ly iden i-
ying he null ejec ion p obabili y maximize (s) like unde Assump ion 3.2, ende ing
he c i ical alue adjus men gene ally in easible. In spa ial se ings, Mülle and Wa -
son (2022) conside a pa ame ic Fwi h a well-de ined bound on he pa ame e as a
benchma k model o ob ain easible adjus men and hen obus i y he size-con olling
p ope y o he esul ing EWC- ype es in mo e gene al model classes using he abo e
eigen alue insigh s.
I, howe e , ake a nume ical app oach: I app oxima e as a linea combina ion ˆ
o basis unc ions, nume ically sea ch he weigh s such ha esul ing ˆ
maximizes (18)
unde an addi ional assump ion ha is noninc easing o e [0, π],15 and ob ain he
c i ical alue c a,e
qacco dingly in he same way as in Sec ion 3. See Appendix B o he
compu a ional de ails. I hen p oceed as in Sec ion 3.2 o selec qop imally. In he con-
ex o Tables 3and 4(AR(1) ), i is ound ha he di e ence be ween c a
qand c a,e
qis
conside ably small and he la ges size dis o ion o 5% le el EWC es using c a
qis o he
o de 10−3in he exac model. These nume ical indings a e conside ably obus when
di e en F’s a e conside ed.16
Tables 6and 7summa ize he weigh ed a e age powe o he op imal EWC es and
he weigh ed a e age powe bound induced by (17), pa alleling he exe cises in Tables 4
and 5, espec i ely. As can be seen, o mos Fconside ed, he op imal EWC es essen-
ially achie es he co esponding weigh ed a e age powe bound, and hus possesses a
no ion o nea op imali y in he spi i o Lemma 1 in Ellio , Mülle , and Wa son (2015).
I no e ha he ela i ely la ge di e ence be ween he weigh ed a e age powe o he
op imal EWC es and he co esponding bound (e.g., unde la ge ρin Table 6and unde
la ge Cin Table 7) is no in o ma i e abou he e iciency o he op imal EWC es , since
i can a ise ei he because he bound is a om he leas uppe bound, o because he
ad hoc es is ine icien .
The p ac ical implica ions o using he EWC es om Sec ion 3.2.2 emain. In he
exac model and in he con ex o Figu e 5, he con en ional wisdom and he op imal
14See Lemma 1(i) in Mülle and Wa son (2022) o he p oo ha he e is only one nega i e eigen alue.
No e he sign di e ence be ween M(c ,q)he e and D(c )in hei con ex .
15The benchma k models conside ed in Mülle and Wa son (2022) all sa is y his shape es ic ion.
16I e e in e es ed eade s o Tables 7, 12, 13, 14, and 15 in an ea lie e sion o his pape (c . Dou (2020))
o nume ical de ails. I choose no o include hem in his e sion o he sake o space.
1130 Liyu Dou Quan i a i e Economics 15 (2024)
Table 6. Weigh ed a e age powe (WAP) bound and he WAP o he op imal EWC es .
ρ0.50 0.60 0.70 0.80 0.90 0.95 0.98 0.99
WAP o op imal EWC 0.501 0.486 0.465 0.432 0.345 0.228 0.095 0.067
WAP bound 0.505 0.493 0.475 0.440 0.359 0.254 0.133 0.087
No e:The unc ion o Fco esponds o an AR(1) wi h coe icien ρ.All in Fa e noninc easing o e [0, π].Nominal
le el is 5%. Sample size Tis 100.
EWC es con inue o sugges using 3 and 6 o q, espec i ely, bu he co esponding
S uden - c i ical alue has o be enla ged by a sligh ly highe ac o o 1.14 o exac size
con ol. In e ms o weigh ed a e age powe , he e is a 14% gain by using he op imal
EWC es . This e iciency ad an age is u he e iden when he local al e na i e μ=
δT−1/2(1−ρ1)−1is conside ed wi h powe di ec ed a ρ1=0 and e en wi h c a
q,asin
Figu e 1. I ei e a e he gene al akeaway he e: One should use he EWC es wi h a la ge
qand app op ia ely enla ged c i ical alues o mo e powe ul HAR in e ence.
5. P ac ical implemen a ion
In his sec ion, I discuss he p ac ical implemen a ion o he op imal EWC es in he
loca ion model and ex end i o in e ence abou a scala pa ame e in eg ession models.
5.1 Loca ion model
Recall ha he abo e heo e ical discussions sugges he ad an age o using a la ge q
and adjus ed c i ical alue when implemen ing he EWC es , and his new es pos-
sesses a no ion o nea op imali y unde p e-speci ied e iciency c i e ia and a smoo h-
ness class F. As a p ac ical ma e , one migh like o es ima e he smoo hness class F,in
pa icula he associa ed , om da a. Un o una ely, he a emp is no use ul in heo y.
This is because he (nea ly) op imal es s depend on F,anda“la ge ”Finco po a ing
sampling unce ain ies po en ially leads o a lowe powe . Pu di e en ly, one canno
es ima e Fands illcon olsize(c .Pö sche (2002)).
Bu how o de e mine a easonable in ac ual implemen a ions? Gi en he analogy
be ween he op imal EWC es and he es conside ed in Sun (2014a)when is pa ame-
e ized in he local- o-uni y o m (see Commen 4 in Sec ion 3), I ollow Sun (2014a)and
sugges he p ac i ione s calib a e in he ollowing way when es ing abou he popula-
ion mean o an obse ed scala ime se ies {y }T
=1. One i s compu es he OLS es ima o
Table 7. Weigh ed a e age powe (WAP) bound and he WAP o he op imal EWC es .
C10.0 5.6 3.2 1.8 1.0 0.6 0.2 0.1
WAP o op imal EWC 0.307 0.368 0.425 0.460 0.486 0.501 0.524 0.530
WAP bound 0.321 0.381 0.431 0.466 0.488 0.505 0.527 0.534
No e:The unc ion o Fis (φ)=exp(−Cφ).Nominalle elis5%. Sample size T=100.
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1137
o a iable u=√1−s. The exp ession (22)a n=1 becomes
J1(ζ1)=2
π1
0
du
1−u2+ζ1=2
πa csin 1
1+ζ1
,
which ollows om a change o a iable =u/√1+ζ1and he ac ha he an ide i a-
i e o (1− 2)−1/2is a csin .
Lemma A.5. Fo 0<α<1,
(a) c 1exis s i and only i m(π/T )=κ−1.m(π/T )≶κ−1i and only i 1≶c 1.
(b) cond1holds i m(π/T )=κ−1.
P oo .(a)In(20)a ˜
q=1, i m(π/T )=κ−1, he edoesno exis ac 1and αsuch ha
(20) holds. On he o he hand, a ea angemen o he e en in (20)a ˜
q=1gi es
PZ2
0+Z2
1>Z2
0+κm(π/T )Z2
1c 1=α.
I ollows ha m(π/T )≶κ−1i and only i 1 ≶c 1.Mo eo e ,i m(π/T )>κ
−1, hesec-
ond pa o Lemma A.4 in conjunc ion wi h (20)a ˜
q=1gi es
c 1=1
κm(π/T )sin2(απ/2)+cos2(απ/2),
which always exis s o e e y 0 <α<1. In a simila ein, i m(π/T )<κ
−1,weha e
c 1=1
κm(π/T )cos2(απ/2)+sin2(απ/2),
which always exis s. Thus, c 1exis s i and only i m(π/T )= κ−1. (b) ollows om he
abo e ha cond1holds i m(π/T )=κ−1.
In wha ollows, I old mos o he pa s co esponding o c q>1 in he p oo s. Pa ly,
his is because hey can be wo ked ou by exac ly symme ic a gumen s. Also, he mos
ele an esul s om hese auxilia y lemmas in es ablishing pa (2) o Theo ems 3.3,
4.3, and Co olla y 4.4 a e when κ>m
(π/T )−1, which is equi alen o c q<1 o allqby
Lemmas A.5 and A.10.
Lemma A.6. Fo m(π/T )=κ−1and 0<α<1, i cond˜
qis iola ed o some 1<˜
q≤T−2,
hen condqis also iola ed o any ˜
q+1≤q≤T−1.
P oo . Suppose cond˜
qis iola ed while cond˜
q+1holds. We ha e
max
j=1,2,,˜
qc ˜
qκm(πj/T )−1≥0and min
j=1,2,,˜
qc ˜
qκm(πj/T )−1≤0. (23)
Conside minj=1,2,,˜
q+1{c ˜
q+1κm(πj/T )−1}>0. We mus ha e c ˜
q+1<1; o he -
wise, (20) does no hold a ˜
q+1. On he o he hand, 0 <minj=1,2,,˜
q+1{c ˜
q+1κm(πj/T )−

1138 Liyu Dou Quan i a i e Economics 15 (2024)
1}≤minj=1,2,,˜
q{c ˜
q+1κm(πj/T )−1}. This, in conjunc ion wi h he second pa o (23),
implies ha c ˜
q<c ˜
q+1<1, which we nex show is impossible. Suppose c ˜
q<c ˜
q+1<1
is ue. Deno e A+
˜
q={j|1≤j≤˜
q,c ˜
qκm(πj/T )−1>0}(A+
˜
q=∅;o he wise,(20)is io-
la ed o ˜
q). Now (20)a ˜
qgi es
α=P(1−c ˜
q)Z2
0>˜
q

j=1c ˜
qκm(πj/T )−1Z2
j
=PZ2
0>1
1−c ˜
q
j∈A+
˜
qc ˜
qκm(πj/T )−1Z2
j+1
1−c ˜
q
j/∈A+
˜
qc ˜
qκm(πj/T )−1Z2
j
≥PZ2
0>1
1−c ˜
q
j∈A+
˜
qc ˜
qκm(πj/T )−1Z2
j(24)
>PZ2
0>1
1−c ˜
q+1
j∈A+
˜
qc ˜
q+1κm(πj/T )−1Z2
j(25)
>PZ2
0>1
1−c ˜
q+1
j∈A+
˜
qc ˜
q+1κm(πj/T )−1Z2
j
+1
1−c ˜
q+1
j/∈A+
˜
qc ˜
q+1κm(πj/T )−1Z2
j(26)
>PZ2
0>1
1−c ˜
q+1
˜
q+1

j=1c ˜
q+1κm(πj/T )−1Z2
j=α,
whe e (24) is due o he ac ha P(A≥C+B)≥P(A≥C)when A,B,Ca e indepen-
den andom a iables and B≤0 almos su ely. The inequali y (25) is due o Lemma A.1
and he ac ha o any j∈A+
˜
q,1
1−c ˜
q[c ˜
qκm(πj/T )−1]<1
1−c ˜
q+1[c ˜
q+1κm(πj/T )−1]
unde c ˜
q<c ˜
q+1<1. The inequali y (26) is due o Lemma A.1.
The o he hal o (23)co esponds oc ˜
q+1>1 and he p oo is exac ly symme ic o
he abo e. O e all, we ha e o m(π/T )=κand 0 <α<1, i cond˜
qis iola ed o some
1<˜
q≤T−2, hen condqis also iola ed o any ˜
q+1≤q≤T−1 by induc ions.
Co olla y A.7. Fo m(π/T )=κ−1and 0<α<1, i cond˜
qholds o some 3≤˜
q≤T−1,
hen condqalso holds o any 2≤q≤˜
q−1.
P oo . This is he con aposi i e s a emen o Lemma A.6.
Co olla y A.8. Fo m(π/T )=κ−1and 0<α<1, ei he one o he ollowing will hold:
(a) he e exis s a unique 1≤q∗≤T−2such ha condqis sa is ied o all 1≤q≤q∗
and iola ed o all q∗+1≤q≤T−1;
(b) condqis sa is ied o all 1≤q≤T−1. In his case,de ine q∗=T−1.
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1139
P oo .I cond
T−1holds, by Co olla y A.7, (b) is ue. O he wise, i condT−2holds, hen
(a) is ue wi h q∗=T−2. O he wise, gi en ha cond1always holds by Lemma A.5,
backwa d induc ions lead (a) o be ue o a unique 1 ≤q∗≤T−3.
Co olla y A.9. I mequals 1and 0<α<1, hen condqholds o all 1≤q≤T−1.
P oo .In hiscase,
F=G={m}={ 1}and condT−1is i ially sa is ied. I ollows ha
condqholds o all 1 ≤q≤T−1 by Co olla y A.7.
Lemma A.10. Fo m(π/T )=κ−1,0<α<1, and q∗as de ined in Co olla y A.8,ei he one
o he ollowing will hold:
(a) c q>1 o all 1≤q≤q∗,and i q∗≥2, c q+1>c q,q=1, 2, ,q∗−1;
(b) c q<1 o all 1≤q≤q∗,and i q∗≥2, c q+1<c q,q=1, 2, ,q∗−1.
P oo . Lemma A.5 leads o he conclusions o q∗=1. We now ocus on q∗≥2. Sup-
pose κm(π/T )>1, hen Lemma A.5 implies ha c 1<1. Suppose he e exis s a ˜
q=
min{q|2≤q≤q∗,c q>1}.(Ino e ha c qcanno be 1 o any q≤q∗;o he wise,(20)
canno hold a he co esponding q.) Then we mus ha e
max
j=1,2,,˜
q−1c ˜
q−1κm(πj/T )−1<max
j=1,2,,˜
q−1c ˜
qκm(πj/T )−1
≤max
j=1,2,,˜
qc ˜
qκm(πj/T )−1<0.
This is a con adic ion, because minj=1,2,,˜
q−1{c ˜
q−1κm(πj/T )−1}>0. I subse-
quen ly implies ha c q<1 o all1≤q≤q∗. Mo eo e , o each j=1, ,q∗,Rj(x)=
[κm(πj/T )x−1]/(1−x)is mono onically inc easing in (0, 1). (To see his, no e ha
κm(πj/T )−1>κm
(πj/T )c q∗−1>0 o e e y 1 ≤j≤q∗.) Fo (20) o hold sequen ially,
we necessa ily need c q+1<c q,q=1, 2, ,q∗−1. (O he wise, he LHS o (20)would
always be below αby Lemma A.1.) Pa (b) is p o ed, and pa (a) holds by symme ic
a gumen s.
Lemma A.11. Fo m(π/T )= κ−1,0<α<1, and q∗as de ined in Co olla y A.8,i
addi ionally m(πj/T )≥m(π(j+1)/T),j=0, 1, ,T−2, and q∗<T−1, hen
κ−1(m(πj/T ))−1≥c q∗ o j>q
∗.
P oo . De ine Q(x,q)=P((1−x)Z2
0>q
j=1[xκm(πj/T )−1]Z2
j).Gi enm(πj/T )≥
m(π(j+1)/T),j=q∗+1, ,T−2, i su ices o show κ−1(m(π(q∗+1)/T))−1≥c q∗.
Suppose no ; hen we mus ha e κ−1(m(π(q∗+1)/T))−1<c q∗. Suppose κm(π/T )>1,
hen c q∗<1 by Lemma A.10:
Qc q∗,q∗+1=P[1−c q∗]Z2
0>
q∗+1

j=1c q∗κm(πj/T )−1Z2
j
<P[1−c q∗]Z2
0>
q∗

j=1c q∗κm(πj/T )−1Z2
j=Qc q∗,q∗=α.
1140 Liyu Dou Quan i a i e Economics 15 (2024)
On he o he hand, 0 <κ
−1(m(π(q∗+1)/T))−1<c q∗<1. Then
Qκ−1mπq∗+1/T−1,q∗+1
=P1−κ−1mπq∗+1/T−1Z2
0>
q∗+1

j=1mπq∗+1/T−1m(πj/T )−1Z2
j
=P1−κ−1mπq∗+1/T−1Z2
0>
q∗

j=1mπq∗+1/T−1m(πj/T )−1Z2
j
=Qκ−1mπq∗+1/T−1,q∗>Q
c q∗,q∗=α,
whe e he las bu one inequali y ollows om he ac ha Q(·,q∗)is mono oni-
cally dec easing in (0, c q∗)unde κm(π/T )>1. By he con inui y o Q(·,q∗+1)
and he in e media e alue heo em, he e mus exis a numbe , deno ed by c q∗+1,
such ha Q(c q∗+1,q∗+1)=α. The e is a con adic ion, because condq∗+1now
holds, iola ing Co olla y A.8. Suppose κm(π/T )<1 ins ead, he p oo ollows by
almos symme ic a gumen s as abo e. O e all, we ha e κ−1(m(π(q∗+1)/T))−1=
minq∗+1≤j≤T−1κ−1(m(πj/T ))−1≥c q∗.
A.2 P oo o Theo em 4.3
Pa 1 holds by he de ini ion o β0,1and by simply ecognizing ha he al e na-
i e Hd
1, (de ined analogous o Hd
1, 1) is included in he null Hd
0. Any non i ial size-
con olling es hus canno be mo e powe ul han he i ial andomized es , which
does no depend on any 0no 1.
In pa 2, m(π/T )>κ
−1. Unde Assump ion 4.2(a,b,c) and o 0 <α<1, Co ol-
la y A.8 shows ha he e exis s a unique q∗such ha ei he (i) condqholds o
1≤q≤q∗and is iola ed o q∗+1≤q≤T−1, o (ii) condqholds o all 1 ≤
q≤T−1,whe ewede ineq∗=T−1. I conjec u e ha he pai o leas a o -
able dis ibu ions (∗
0,∗
1) pu p obabili y masses on unc ions { ∗}⊂Fand {g∗}⊂
G,inwhich ∗(φ)= (φ)1[|φ|≤πq∗/T]+a∗(φ) (φ)1[|φ|>πq
∗/T]and g∗(φ)=
(φ)1[|φ|≤πq∗/T]+b∗(φ) (φ)1[|φ|>πq
∗/T], o a∗(φ)≥1andb∗(φ)≤1such ha
a∗(φ) (φ)/(b∗(φ) (φ)) =(κc q∗)−1 o |φ|>πq
∗/T. Assump ion 4.2(d,e) ensu es ha
such wo se s o unc ions a e nonemp y as long as he pai o unc ions (a∗,b∗)exis s.
This ue by Lemma A.11 (m(φ)≤(κc q∗)−1 o |φ|>πq
∗/T).
Now i s le b∗(φ)=1 o allφand a∗(φ)=(κc q∗m(φ))−1. The bes le el α es o
Hd
0, ∗agains Hd
1,g∗is
ϕ ∗,g∗=1Y2
0+
T−1

j=1
Y2
j/ ∗(πj/T )
Y2
0+κ
T−1

j=1
Y2
j/g∗(πj/T )
>c ,
Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1141
o some c ≥0such ha EPY, ∗[ϕ ∗,g∗(Ys)] =α,whe ePY,˜
deno es he join dis ibu-
ion o Ya =˜
unde Hd
0. I ollows ha
α=PY, ∗Y2
0+
T−1

j=1
Y2
j/ ∗(πj/T )
Y2
0+κ
T−1

j=1
Y2
j/g∗(πj/T )
>c 
=PY, ∗Y2
0+
T−1

j=1
Y2
j/ ∗(πj/T )>c Y2
0+κ
T−1

j=1
Y2
j/g∗(πj/T )
=P(1−c )Z2
0>
T−1

j=1c κ ∗(πj/T )/g∗(πj/T )−1Z2
j
=P(1−c )Z2
0>
q∗

j=1c κm(πj/T )−1Z2
j+
T−1

j=q∗+1
[c /c q∗−1]Z2
j, (27)
whe e he las equali y ollows om he de ini ion o ∗and g∗. Because Yis a
con inuous andom ec o , he c i ical alue c is unique. By ma ching (27)wi h
(20)a ˜
q=q∗,weha ec =c q∗.Also, hee en s{Y2
0+T−1
j=1Y2
j/ ∗(πj/T )
Y2
0+κT−1
j=1Y2
j/g∗(πj/T )>c q∗}and
{Y2
0+q∗
j=1Y2
j/ (πj/T )
Y2
0+κq∗
j=1Y2
j/ (πj/T )>c q∗}a e equi alen PY,˜
-almos su ely, uni o mly in ˜
∈F.The
ejec ion egions de ined by ϕ ∗,g∗and he op imal es s a is ic in (14) a e hus iden i-
cal.
I emains o check he ollowing condi ions: (1) ϕ ∗,g∗is also he bes le el α es
o Hd
0,∗
0agains Hd
1,∗
1;(2)ϕ ∗,g∗uni o mly con ols size Hd
0and has i s la ges size
dis o ion a ;(3)ϕ ∗,g∗a ains β∗
0,∗
1a ; and (4) o any o he (0,1),β∗
0,∗
1≤
β0,1.
Fo (1), no e ha ϕ ∗,g∗is o exac size αunde H0,∗
0:
ϕ ∗,g∗(ys)h0, (ys)d∗
0( )dys=ϕ ∗,g∗(ys)h0, ∗(ys)dysh0, d∗
0( )
=EPY, ∗[ϕ ∗,g∗(Ys)] =α,
whe e he second equali y holds because he se o dis ibu ions o ϕ ∗,g∗is degene -
a e unde ∗
0. By he same logic, he ejec ion p obabili ies o ϕ ∗,g∗unde H1,g∗and
1142 Liyu Dou Quan i a i e Economics 15 (2024)
H1,∗
1a e iden ical. Because he bes le el α es o Hd
0,∗
0agains Hd
1,∗
1is unique, (1)
holds.
Fo (2),conside agi en ˜
∈F,
EPY,˜
ϕ ∗,g∗(Y)=PY,˜
Y2
0+
q∗

j=1
Y2
j/ (πj/T )
Y2
0+κ
q∗

j=1
Y2
j/ (πj/T )
>c q∗
=P[1−c q∗]Z2
0>
q∗

j=1c q∗κm(πj/T )−1˜
(πj/T )
(πj/T )Z2
j
=PZ2
0>1
1−c q∗
q∗

j=1c q∗κm(πj/T )−1˜
(πj/T )
(πj/T )Z2
j(28)
≤PZ2
0>1
1−c q∗
q∗

j=1c q∗κm(πj/T )−1Z2
j=α,
whe e (28) ollows om(b)inLemmaA.10 unde he condi ion m(π/T )>κ
−1,
and he inequali y ollows om he de ini ion o q∗and Lemma A.1 unde Assump-
ion 4.2(a).
Fo (3),conside agi en ˜
g∈Gand le P1
Y,˜
gdeno e he join dis ibu ion o Yunde
Hd
1, ˜
g,
EP1
Y,˜
gϕ ∗,g∗(Y)=P1
Y,˜
gY2
0+
q∗

j=1
Y2
j/ (πj/T )
Y2
0+κ
q∗

j=1
Y2
j/ (πj/T )
>c q∗
=P[1−c q∗]Z2
0>
q∗

j=1c q∗−κ−1m(πj/T )−1˜
g(πj/T )
(πj/T )Z2
j
=PZ2
0>1
1−c q∗
q∗

j=1c q∗−κ−1m(πj/T )−1˜
g(πj/T )
(πj/T )Z2
j
≥PZ2
0>1
1−c q∗
q∗

j=1c q∗−κ−1m(πj/T )−1Z2
j,
whe e he inequali y ollows om Co olla y A.2 unde Assump ion 4.2(b). In his sense,
β∗
0,∗
1=EP1
Y,
[ϕ ∗,g∗(Y)].

Quan i a i e Economics 15 (2024) Op imal HAR in e ence 1143
Fo (4), because ϕ ∗,g∗uni o mly con ols size unde Hd
0, i also con ols size unde
Hd
0,0. Then by he de ini ion o β0,1,
β0,1≥ϕ ∗,g∗ysh1, ˜
gysd1(˜
g)dys≥in
˜
g∈GEP1
Y,˜
gϕ ∗,g∗(Y)=β∗
0,∗
1.
A.3 P oo s o Theo em 3.3 and Co olla y 4.5
P oo o Co olla y 4.5 ollows immedia ely om hose in Sec ion A.2 wi h = 1.Theo-
em 3.3 is a special case o Co olla y 4.5 wi h G={ 1}, and i s esul s ollow by ealizing
ha β∗
0,∗
1, in ha case, is simply he weigh ed a e age powe o es (8)a 1.
A.4 P oo o Co olla y 4.4
Co olla y 4.4 is a special case o hose conside ed in Theo em 4.3 wi h G={˜
1}. The p oo
hus ollows di ec ly om hose in Sec ion A.2.
Appendix B: Compu a ional de ails in Sec ion 4.3
In his sec ion, I explain in de ail how o nume ically iden i y he null ejec ion p oba-
bili y maximize o he EWC es in es ing (16).
Le he n+1 node poin s {xi}n
i=0de ine a pa i ion o he in e al I=[0, π]in o
nsubin e als Ii=[xi−1,xi],i=1, 2, ,n, each o leng h hi=xi−xi−1,andx0=
0, xn=π.Le C0(I)deno e he space o con inuous unc ions on I,andP1(Ii)de-
no e he space o linea unc ions on Ii.Le {ςi}n
i=0be a se o basis unc ions o he
space Fho con inuous piecewise linea unc ions de ined by Fh={ : ∈C0(I), |Ii∈
P1(Ii)}. The basis unc ions {ςi}n
i=0a e no malized such ha ςj(xi)=1[i=j],i,j=
0, 1, ,n. By app oxima ing ia ˆ
=n
i=0 (xi)ςiand by (12), I app oxima e (18)
by
PZ2
0
q

j=1
λj(ˆ
)Z2
j
≥1=2
π1
01−u2(q−1)/2du




q

j=11−u2+λj(ˆ
)
, (29)
which is a unc ion o he n-dimensional ec o ( (x1), (x2),, (xn)).(Byno mal-
iza ion, (x0)=1.) Wi h p e-compu ed {0(ςi)}n
i=0, hecompu a iono (29) akes e y
li le compu ing ime o each ˆ
, and i is easible o ob ain a global maximize o (29)
subjec o implied cons ain s on ( (x1), (x2),, (xn)) om a gi en F. I addi ionally
assume ha he unde lying spec um is noninc easing o e [0, π].
In ac ual implemen a ions, I choose n=50, and {xi}50
i=0a e log-spaced nodes in
[0, π]. The basis unc ions {ςi}n
i=0a e chosen o be he ha unc ions
ςi(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
(x−xi−1)/hii x∈Ii,
(xi+1−x)/hi+1i x∈Ii+1,
0o he wise.
(30)
1144 Liyu Dou Quan i a i e Economics 15 (2024)
I p e-compu e {0(ςi)}n
i=0wi h a 5000-poin Gaussian quad a u e o each nonze o ele-
men . Since each ςiis compac ly suppo ed, hese nume ical in eg a ions a e nea ly p e-
cise. Fo e e y ( (x1), (x2),, (xn)),0(ˆ
)is simply a linea combina ion o hese
p e-compu ed co a iance ma ices. No e, howe e , ha he ul ima e objec i e unc ion
I will op imize is (29), which in ol es 0( )implici ly h ough λj(ˆ
).Inanun epo ed
exe cise, o a gi en EWC es and a pa ame ic AR(1) class Fwi h coe icien a y-
ing o e a ine g id, I compa e he ejec ion p obabili ies ollowing he desc ibed ap-
p oxima e p ocedu e and an “exac ” p ocedu e in which each en y o 0( )is e alu-
a ed by nume ical in eg a ions ia Ma hema ica. The di e ences in he ejec ion p ob-
abili ies a e a mos o he o de 0.0001. I hus hold on o he abo e choice o nand
{xi}50
i=0.
I p oceed in h ee s eps o iden i y he null ejec ion p obabili y maximize o a ixed
EWC es o (16), in e ms o ( (x1), (x2),, (xn)):
(i) P og am up he null ejec ion p obabili y a a gi en ( (x1), (x2),, (xn))∈
Rn
+, and wi h a gi en qand easonable c (e.g., c a
q) om(29), whe e 0(ˆ
)=
n
i=0 (xi)0(ςi)wi h p e-compu ed {0(ςi)}n
i=0.
(ii) Randomly d aw 100 n-dimensional ec o s ( (x1), (x2),, (xn))such ha
each ec o co esponds o some ∈F. This is, in gene al, a challenging ask
since he numbe o nume ical cons ain s o be checked inc eases exponen-
ially wi h n o highe -o de smoo hness cons ain s. Fo easibili y, I ocus on
wo ypes o smoo hness classes: he class Fin which co esponds is AR(1)
wi h coe icien ρand ∈Fis noninc easing o e [0, π];and heclassFin
which ∈Fis Lipschi z con inuous in logs wi h Lipschi z cons an C.I is
no ha d o see ha o he i s ype, i su ices o check he mono onici y
cons ain consecu i ely and he lowe boundedness condi ion. Fo he second
ype, by he esul o Beliako (2006), he complexi y o checking he global Lip-
schi z condi ion is educed o consecu i e checking o local Lipschi z condi-
ions.
(iii) Use e e y n-dimensional ec o d awn in S ep (ii) as he ini ial condi ion o
op imize he null ejec ion p obabili y unc ion p og ammed in S ep (i), sub-
jec o linea cons ain s induced by smoo hness class F(as desc ibed in S ep
(ii)).
Unde he abo e speci ica ions, i akes abou 1 o 2 minu es o comple e he op imiza-
ion using mincon in MATLAB ia pa allel compu ing in 12 co es.
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