A comparison of semiparametric tests for fractional cointegration
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Leschinski, Ch is ian; Voges, Michelle; Sibbe sen, Philipp
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A compa ison o semipa ame ic es s o ac ional
coin eg a ion
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REGULAR ARTICLE
A compa ison o semipa ame ic es s o ac ional
coin eg a ion
Ch is ian Leschinski1·Michelle Voges1·Philipp Sibbe sen1
Recei ed: 28 Janua y 2019 / Re ised: 11 Feb ua y 2020 / Published online: 24 Ma ch 2020
© The Au ho (s) 2020
Abs ac
The e a e a ious compe ing p ocedu es o de e mine whe he ac ional coin eg a ion
is p esen in a mul i a ia e ime se ies, bu no s anda d app oach has eme ged. We
p o ide a syn hesis o his li e a u e and conduc a de ailed compa a i e Mon e Ca lo
s udy o guide empi ical esea che s in hei choice o app op ia e me hodologies.
Special a en ion is paid on empi ically ele an issues such as assump ions abou he
o m o he unde lying p ocess and he abili y o he p ocedu es o dis inguish be ween
sho - un co ela ion and long- un equilib ia. I is ound ha se e al app oaches a e
se e elyo e sizedinp esenceo co ela edsho - uncomponen sand ha heme hods
show di e en pe o mance in e ms o powe when applied o common-componen
models ins ead o iangula sys ems.
Keywo ds Long memo y ·F ac ional coin eg a ion ·Semipa ame ic es ima ion and
es ing
1 In oduc ion
The concep o coin eg a ion de i es i s popula i y om he ac ha i allows o model
equilib ium ela ionshipsbe weennon-s a iona y imese ies.Themos popula es s in
he s anda d I(1)/I(0)se ing includes he wo-s ep p ocedu e by Engle and G ange
(1987), he ace es by Johansen (1988) and he p incipal componen es by Phillips
and Oulia is (1988) which a e subjec o se e al compa ai i e s udies like Reime s
(1992) and Höglund and Ös e ma k (2003). In p ac ice, howe e , s anda d coin eg a-
ion analysis can o en no be applied, since he I(1)/I(0) amewo k is oo es ic i e.
Elec onic supplemen a y ma e ial The online e sion o his a icle (h ps://doi.o g/10.1007/s00362-
020-01169-1) con ains supplemen a y ma e ial, which is a ailable o au ho ized use s.
BPhilipp Sibbe sen
[email p o ec ed]e .de
1Leibniz Uni e si ä Hanno e , Königswo he Pla z 1, 30167 Hanno e , Ge many
123
1998 C. Leschinski e al.
Fo example, he se ies o in e es may be pe sis en bu no ha e a uni oo , o he
de ia ions om he equilib ium may be mo e pe sis en han he I(0)model allows.
F ac ionalcoin eg a iono e comes hesesho comings,byallowing o non-in ege
in eg a ion o de s o he a iables in he sys em and any (possibly non-ze o) memo y
o de in he coin eg a ing esiduals as long as i is educed compa ed o he o iginal
sys em. Consequen ly, ac ional coin eg a ion p omises o acili a e he modeling
o a la ge numbe o equilib ium ela ionships compa ed o s anda d coin eg a ion.
This has led o he de elopmen o a ious es ing and ank es ima ion p ocedu es o
de e mine whe he ac ional coin eg a ion is p esen in a mul i a ia e ime se ies.
Pa ame ic app oaches include Johansen (2008), Łasak (2010), Johansen and
Nielsen (2012), Łasak and Velasco (2015), and Johansen and Nielsen (2019), among
o he s, who conside ac ional ex ensions o he coin eg a ed VAR model o Johansen
(1988). Fu he mo e, B ei ung and Hassle (2002) in oduce a ace es o de e -
mine he coin eg a ing ank, A a ucci and Velasco (2009) sugges ank es ima ion
in a eg ession amewo k, and Hassle and B ei ung (2006) de elop a ime domain
esidual-based es .
Semipa ame ic app oaches, on he o he hand, ha e he ad an age ha hey allow
he esea che o ocus on he long- un ela ionship be ween he se ies and do no
equi e he speci ica ion o sho - un dynamics. This li e a u e encompasses he
spec al-based ankes ima ionp ocedu eo RobinsonandYajima(2002)andi sex en-
sion by Nielsen and Shimo su (2007), a Hausmann- ype es based on he mul i a ia e
local Whi le es ima o in oduced by Robinson (2008a), a numbe o esidual-based
es s o he null hypo hesis o no ac ional coin eg a ion de eloped by Ma mol and
Velasco (2004), Chen and Hu ich (2006), Hualde and Velasco (2008), and Wang e al.
(2015), a a iance- a io es p oposed by Nielsen (2010), a es based on a GPH- ype
es ima e o he coin eg a ion s eng h in oduced by Souza e al. (2018) and a ank
es ima ion p ocedu e based on an eigenanalysis o he au oco a iance unc ion om
Zhang e al. (2019).
Un o una ely, he domain o applicabili y o mos o hese p ocedu es is much
mo e es ic i e han he de ini ion o ac ional coin eg a ion. Some a e only appli-
cable in s a iona y sys ems—some only in non-s a iona y sys ems. Some p ocedu es
equi e he educ ion in memo y o be mo e han 1/2—some equi e he memo y o
he coin eg a ing esiduals o be less han 1/2.
Fu he mo e, he e a e di e en assump ions abou he o m o he ac ionally coin-
eg a ed sys em. Some app oaches assume ha one o he obse ed se ies i sel is an
obse a ion o he common unde lying end. O he app oaches assume an unobse ed
common unde lying end. We e e o hese models as he iangula sys em and he
common-componen s model. Which o hese assump ions is mo e sui able in p ac ice
depends on he speci ic applica ion. On he one hand, i may be app op ia e o hink o
he isk- ee in e es a e as an obse ed common componen ha is pe u bed by isk
p emia in isky bonds so ha a iangula model can be used. Fo coin eg a ed pai s o
s ocks, on he o he hand, i is unclea why he p ice o one s ock should be in e p e ed
as a pe u bed e sion o ano he s ock p ice so ha a common-componen s model is
mo e app op ia e. Finally, e en hough he de elopmen o each o hese p ocedu es
o de e mine whe he ac ional coin eg a ion is p esen is a majo heo e ical con i-
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 1999
bu ion, ela i ely li le e o has been de o ed o analyze how hey pe o m compa ed
o each o he .
He e, we y o add ess hese issues by p o iding a su ey o all he ank es ima ion
and es ing p ocedu es discussed abo e. To s udy he ela i e pe o mance o he
compe ing app oaches, we conduc an ex ensi e Mon e Ca lo analysis o hei size
and powe p ope ies. I is ound ha se e al p ocedu es - namely hose o Nielsen
and Shimo su (2007) o Robinson and Yajima (2002), Ma mol and Velasco (2004),
and Hualde and Velasco (2008) show se e e ini e sample size dis o ions in sys ems
wi h co ela ed sho - un componen s. The ela i e pe o mance in e ms o powe
dependson he o mo hesys em.Fo iangula sys ems and non-s a iona y common-
componen s models he es o Souza e al. (2018) pe o ms bes o e all, whe eas he
es o Chen and Hu ich (2006) is p e e able o s a iona y common-componen s
models.
The es o he pape is s uc u ed as ollows. The nex sec ion gi es he de ini ion
and model o ac ional coin eg a ion we adop and b ie ly e iews he basic es ima ion
me hods equi ed by he es s. Sec ion 3is di ided in o wo subsec ions desc ibing wo
ypes o es s, 3.1 con aining he es s based on a spec al ma ix and 3.2 summa izing
he es s based on coin eg a ing esiduals, Sec . 4p esen s ini e sample esul s, and
Sec . 5concludes.
2 F ac ional coin eg a ion: models and de ini ions
Ap-dimensional ec o - alued ime se ies X has long memo y i i s spec al densi y
ul ills
X(λ) ∼Λj(d)GΛj(d), as λ→0+,(1)
whe e Gis a eal, symme ic, and non-nega i e de ini e ma ix, Λj(d)=
diag λ−d1eiπd1/2,...,λ
−dpeiπdp/2is a p×pdiagonal ma ix, Λj(d)is i s complex
conjuga e anspose and ‘∼’ implies ha o each elemen he a io o eal and imag-
ina y pa s on he le - and igh -hand side ends o one. The elemen in he a- h ow
and b- h columns o he spec al ma ix X(λ) is deno ed by ab(λ) ∼gabλ−2d o
a,b∈{1,...,p}whe e gab deno es he espec i e elemen o G. The pe iodog am
o X a he Fou ie equencies is gi en by
IX(λj)=wX(λ j)wX(λj), (2)
wi h wX(λ) =1
√2πTT
=1X eiλ , and λj=2πj/T, o j=1,...,T/2, whe e
· deno es he g ea es in ege smalle han he a gumen .
The e is a numbe o di e en de ini ions o ac ional coin eg a ion in he li e a u e.
The mos common one goes back o Engle and G ange (1987). Acco ding o his
de ini ion he p-dimensional ime se ies X is coin eg a ed o ank , i all componen s
o X a e in eg a ed o o de d(deno ed by I(d)), and he e exis s a non-singula ma ix
βso ha he linea combina ions =βX a e I(d−ba)=I(d a)wi h d>ba>0
123
2000 C. Leschinski e al.
o all a=1,..., . The ma ix βis called he coin eg a ing ma ix and each o i s
columns is a coin eg a ing ec o . The elemen s o he ec o a e he coin eg a ing
esiduals. O he de ini ions a e gi en by Johansen (1995), Flô es J and Sza a z (1996),
Ma inucci and Robinson (2001), and Robinson and Yajima (2002) who also p o ide
a discussion o he implica ions o he di e en de ini ions.
S anda d coin eg a ion is a special case o he de ini ion abo e whe e d=1 and
d a=0 o all a. In his se up he sys em is non-s a iona y, whe eas he coin eg a ing
esiduals a e s a iona y. In con as o ha , ac ional coin eg a ion allows o a mo e
lexiblemodel so ha se e alcases can be dis inguished: weak coin eg a ion(b<0.5),
s ong coin eg a ion (b>0.5), s a iona y coin eg a ion (0 <d <d<0.5), o non-
s a iona y coin eg a ion (0.5<d <d).
In gene al, ( ac ional) coin eg a ion is an equilib ium concep whe e he pe sis-
ence o he coin eg a ing esidual d de e mines he speed o adjus men owa ds
he coin eg a ion equilib ium βX , and shocks ha e no pe manen in luence on he
equilib ium as long as d <1 holds.
As an example, conside he ac ionally (co-)in eg a ed bi a ia e model wi h X =
(X1 ,X2 ), whe e
X1 =c1+ξ1Y +Δ−(d−b1)u1 1( >0)(3)
X2 =c2+ξ2Y +Δ−(d−b2)u2 1( >0)(4)
and Y =Δ−de 1( >0). (5)
He e, u =(u1 ,u2 )is a weakly-dependen ze o-mean p ocess wi h cons an co a i-
ance ma ix Ωuand spec al densi y ma ix u(λ),e (wi h a iance σ2
eand spec al
densi y e(λ)) is a uni a ia e weakly-dependen ze o-mean p ocess ha is allowed o
be co ela ed wi h u , and Ldeno es he lag-ope a o so ha LY =Y −1. The ac-
ional di e ence ope a o Δd=(1−L)dis de ined in e ms o he binomial expansion
so ha (1−L)d=∞
k=0d
k(−1)kLk, wi h d
k=d(d−1)(d−2)...(d−(k−1))
k!. Fu he -
mo e, 1(·)deno es he indica o unc ion ha akes he alue one i i s a gumen is
ue and is ze o, o he wise. Finally, i is assumed ha d≥b1,b2≥0.
The unca ed p ocesses Δ−(d−ba)ua 1( >0)a e ac ionally-in eg a ed p ocesses
o ype-II which means hey a e only asymp o ically s a iona y o d<1/2, bu in
con as o ype-I p ocesses hey a e s ill de ined o d>1/2. Fo a de ailed discussion
c . Ma inucci and Robinson (1999).
In his bi a ia e model he e can be a mos one coin eg a ing ela ionship. In his
case =1 and βi sel is a coin eg a ing ec o . Ob iously, i he linea combina ion
βX = has educed memo y, he same is ue o e e y scala mul iple o i .
To iden i y he coin eg a ing ec o , i is he e o e cus oma y o apply some kind o
no maliza ion such as se ing he i s elemen o he ec o o uni y. In Eqs. (3) o
(5), ac ional coin eg a ion a ises i ξ1,ξ
2= 0, and b1,b2>0. In his case he
no malized coin eg a ing ec o is β=1,−ξ1
ξ2=1,−˜
βand he coin eg a ing
esidual is I(d−b)=I(d ), whe e b=min(b1,b2). No e ha his model is a
common-componen s model, bu i also nes s a iangula sys em. This is ob ained as a
special case i Ωu,22 =0 so ha X2 is a di ec ( escaled) obse a ion o he unde lying
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 2001
common end and only X1 is pe u bed wi h a coin eg a ion e o so ha b=b1.
S anda d coin eg a ion in he I(1)/I(0) amewo k is ob ained as a special case i
d=1 and b1=b2=1. I is also possible o ha e ξ1,ξ
2= 0, so ha bo h X1 and
X2 con ain he common componen Y , bu hey a e no coin eg a ed i b1=b2=0.
3 Tes s o no ac ional coin eg a ion
In he ollowing, we p o ide a comp ehensi e e iew o semipa ame ic es s and
es ima ion p ocedu es ha can be used o de e mine he o de o ac ional coin e-
g a ion in a p-dimensional ec o - alued ime se ies X . Acco ding o he de ini ion
discussed abo e, his equi es ha he componen s o X a e in eg a ed o he same
o de . In p ac ice, his can ei he be assumed based on domain-speci ic knowledge,
o i can be es ed wi h es s o he equali y o memo y pa ame e s ha allow o
coin eg a ion in oduced by, o example, Robinson and Yajima (2002), Nielsen and
Shimo su (2007), Hualde (2013), and Wang and Chan (2016). In pa icula Robin-
son and Yajima (2002) discuss in de ail how o pa i ion a ec o - alued ime se ies
in o sub ec o s wi h equal memo y pa ame e s. These can hen be used o u he
coin eg a ion analysis.
In he ollowing, i will be assumed ha all componen s o X a e I(d), which
means we abs ac om hese p e- es ing issues o ocus on he ac ual es s o he null
o no ac ional coin eg a ion. Fo all es s he hypo heses a e de ined by
H0:X is no ac ionally coin eg a ed (d=d ),
H1:X is ac ionally coin eg a ed (d>d ).
In con as o s anda d I(1)/I(0)coin eg a ion, he memo y pa ame e dis
unknown in ac ionally coin eg a ed sys ems and has o be es ima ed. Since mul-
i a ia e memo y es ima ion becomes inconsis en unde coin eg a ion, he memo y
pa ame e s a e es ima ed uni a ia ely and, i no s a ed o he wise, we employ he
means o he uni a ia e memo y es ima es in he es s.
The es s p esen ed in his Sec ion apply he mos common es ima o s: he log-
pe iodog am es ima o
dGPH o Geweke and Po e -Hudak (1983) and Robinson
(1995b), he local Whi le es ima o
dLW o Künsch (1987) and Robinson (1995a), o
he exac local Whi le es ima o
dELW o Shimo su and Phillips (2005) and Shimo su
(2010). All o hese es ima o s a e pe iodog am-based and employ he i s mFou ie
equencies. The gene al equi emen is ha m<T/2 ends o in ini y mo e slowly
han Tso ha 1
m+m
T→0asT→∞and e en he la ges equency 2πm/Tis
asymp o ically local o he ze o equency.
To es ima e he coin eg a ing ela ionship βX = when =1, he ec o is
pa i ioned such ha X =(y ,x ), whe e y is a scala and x is (p−1)×1. By
doing so, he ocus is on one possible coin eg a ing ela ion y =˜
βx + whe e ˜
βis
(p−1)-dimensional.
As in s anda d coin g a ion analysis he ec o ˜
βcan be es ima ed wi h o dina y
leas squa es (OLS) as long as d>1/2 so ha he se ies emains non-s a iona y. In
s a iona y long-memo y ime se ies, OLS is inconsis en in p esence o co ela ion
be ween he s a iona y eg esso s and he inno a ion e m (c . Robinson (1994)).
123
2002 C. Leschinski e al.
Robinson (1994) and Robinson and Ma inucci (2001) in oduce an al e na i e
es ima o o he coin eg a ing ec o ha is based on he pe iodog am local o he
ze o equency. In con as o OLS, his na ow-band equency domain leas squa es
(NBLS) es ima o is consis en unde coin eg a ion o all alues o dand has a non-
no mal limi ing dis ibu ion in he non-s a iona y egion. Ch is ensen and Nielsen
(2006a) ex end he asymp o ic esul s o he s a iona y egion whe e he es ima e ol-
lows an asymp o ic no mal dis ibu ion and Nielsen and F ede iksen (2011) p o ide
a co ec ion o he asymp o ic bias unde weak ac ional coin eg a ion.
Es ima ing he linea coin eg a ing ela ionship wi h NBLS equi es calcula ing he
a e aged c oss-pe iodog am o x wi h i sel and y by Ia
xx (λj)=2π
Tm
j=1ωx(λj)
ωx(λj)and Ia
xy (λj)=2π
Tm
j=1ωx(λj)ωy(λ j). The NBLS es ima e o ˜
βis hen
de ined by
βm=Ia
xx (λj)−1Ia
xy (λj). (6)
The bandwid h mhas o ul ill he usual local- o-ze o condi ion as T→∞. I no
speci ied o he wise, we employ NBLS o es ima e he coin eg a ing ec o . O he
es ima o s sugges ed in he li e a u e include es ima ion based on he eigen ec o s
o a e sion o Ia
X(λj)(c . Chen and Hu ich (2006)) and join es ima ion wi h he
memo y pa ame e s in mul i a ia e local Whi le app oaches such as hose o Nielsen
(2007), Robinson (2008b) and Shimo su (2012).
The ollowing e iew is di ided in o es s based on he spec al densi y local o
he o igin (Sec . 3.1) and es s based on es ima es o he coin eg a ing esiduals (Sec .
3.2). O cou se, his dis inc ion is no clea cu , since some o he esidual-based
app oaches also use he spec al p ope ies o he po en ial coin eg a ing esiduals
and o example he es o Nielsen (2010) is p esen ed as a a iance- a io es . Many
di e en ca ego iza ions would be possible. He e, we e e o hose app oaches as
”spec al-based” ha ely on he p ope ies o he spec um o he obse ed se ies
X i sel , and hose ha ely on he spec um o he coin eg a ing esidual a e called
” esidual-based”.
3.1 Tes s based on he spec al ma ix
A numbe o p ocedu es o de e mine he ac ional coin eg a ing ank o he p-
dimensional ime se ies X a e based on p ope ies o he escaled spec al ma ix
local o he ze o equency. This is deno ed by Gin Eq. (1) and has educed ank i
and only i X is ac ionally coin eg a ed. I ac ional coin eg a ion is p esen , he
numbe o eigen alues ha a e equal o ze o co esponds o he coin eg a ing ank .
Mo e de ails on he connec ion be ween ac ional coin eg a ion, uni cohe ence and
singula i y o Ga e gi en in Velasco (2003b) and Nielsen (2004).
Based on his p ope y Robinson and Yajima (2002) in oduce an in o ma ion c i-
e ion o de e mine he ac ional coin eg a ion ank ha is ex ended o non-s a iona y
p ocesses by Nielsen and Shimo su (2007). To ob ain an es ima e
Go G, he i s
s ep consis s in applying he uni a ia e exac local Whi le es ima o o Shimo su and
Phillips (2005) and Shimo su (2010) o each componen o X sepa a ely, using band-
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 2003
wid h m, and pooling hem o he a i hme ic mean
dELW. The es ima e o
G(
dELW)
is hen de ined by
G(
dELW)=1
m1m1
j=1Re IΔd(λj), whe e IΔdis he pe iodog am
o Δ
dELW X . The bandwid hs ha e o ul ill m1
m→0 in o de o ensu e as e con e -
gence o
dELW han o
G(
dELW).1Deno e he empi ical eigen alues calcula ed om
G(
dELW)and so ed in descending o de by
δa,G o a=1,...,p. The coin eg a ing
ank can hen be es ima ed using a model selec ion c i e ion ha is based on he pa ial
sum o he so ed eigen alues
NS =a g min
k=0,...,p−1⎛
⎝n(T)(p−k)−
p−k
a=1
δa,G⎞
⎠,(7)
whe e n(T)is a unc ion which ul ills n(T)+1
√m1n(T)→0asT→∞so ha n(T)
goes o ze o mo e slowly han he es ima ion e o in he eigen alues ha is o o de
OPm−1/2
1. Asymp o ically, he exp ession is he e o e minimal i only es ima es
o non-ze o eigen alues a e included in he sum.
To deal wi h si ua ions in which he scales o he componen s in X a e di e en ,
Nielsen and Shimo su (2007) sugges o base he p ocedu e on he co ela ion ma ix
P(
dELW)=
R(
dELW)−1/2
G(
dELW)
R(
dELW)−1/2ins ead o
G, whe e
R(
dELW)=
diag(g11,...,gpp)con ains he diagonal elemen s o
G(
dELW). This is admissible
since he ank o
Pis he same as ha o
Gin he limi . Nielsen and Shimo su
(2007) poin ou ha his app oach wo ks be e in simula ions and also ecommend
o use he bandwid h n(T)=m−0.3
1. The coin eg a ing ank es ima e is consis en
o ∈{0,...,p−1}. I is applicable o sys ems o dimension p≥2, and i does
no impose es ic ions on dand b. A simila ank es ima ion p ocedu e based on he
a e age o ini ely many ape ed pe iodog am o dina es local o he o igin was also
p oposed by Chen and Hu ich (2003).
The inconsis ency o he mul i a ia e local Whi le es ima o unde ac ional coin-
eg a ion is he basis o a es p ocedu e o iginally p oposed by Ma inucci and
Robinson (2001). They sugges a Hausman- ype es ha compa es mul i a ia e and
uni a ia e local Whi le es ima es. Unde he null hypo hesis o no coin eg a ion he
mul i a ia e es ima o is e icien and bo h a e consis en , whe eas unde he al e na-
i e o ac ional coin eg a ion he uni a ia e es ima o emains consis en , while he
mul i a ia e one does no .
This idea is o malized by Robinson (2008a). The es s a is ic is based on he objec-
i e unc ion o he mul i a ia e local Whi le es ima o (c . Loba o (1999), Shimo su
(2007)) S(d)=log de
G∗(d)−2pd
mm
j=1log λjwi h
G∗(d)=1
mm
j=1IX(λj)λ2d
j
and i s de i a i e
s∗(d)=
G∗(d)−1
H∗(d)(8)
1We ollow he no a ion o Nielsen and Shimo su (2007) and use m1 o he bandwid h in he es ima ion
o G(d)and m o ha o d. No e ha Robinson and Yajima (2002) chose he opposi e no a ion.
123
2004 C. Leschinski e al.
wi h
H∗(d)=1
mm
j=1νjIX(λj)λ2d
jand νj=log j−1
mm
k=1log k. Simila o he
p e ious p ocedu e, he memo y pa ame e dis es ima ed by pooling he uni a ia e
es ima es ob ained by applying he local Whi le es ima o o each o he componen
se ies. The equally weigh ed a e age is deno ed by
dLW. To ob ain a es s a is ic, he
de i a i e s∗(d) om (8) is e alua ed a his a e aged uni a ia e es ima e:
W∗
Rob =ms∗(
dLW)2
N2 (
F∗2)−p(9)
wi h
F∗=
R∗−1/2
G∗(
dLW)
R∗−1/2and
R∗=diag(g∗
11,...,g∗
pp), whe e g∗
aa,a=
1,...,p, a e he diagonal elemen s o
G∗(
dLW). The scaled de i a i e m1/2s∗(
dLW)
is asymp o ically no mal so ha he es ollows a χ2
1-dis ibu ion i app op ia ely
s anda dized by he e m in he denomina o .
The es gene a es powe because G(d)is singula unde he al e na i e o ac-
ional coin eg a ion so ha he in e se
G∗(
dLW)−1o he es ima e and consequen ly
he ace s∗
dLWbecome la ge. This is a sco e- ype es ha a oids he calcula-
ion o he mul i a ia e local Whi le es ima o ha can be nume ically expansi e.
Since he e iciency o he mul i a ia e es ima e is ob ained wi h a single New on s ep
om he uni a ia e es ima e in di ec ion o he mul i a ia e one, s∗
dLWis di ec ly
p opo iona e o he di e ence be ween he e icien and he ine icien es ima e.
This es allows se ies o dimensions la ge han wo, bu i is es ic ed o p ocesses
wi h d∈(−1/2,1/2)and ocuses on he empi ically ele an ange d∈(0,1/2).A
non-s a iona y ex ension based on a immed e sion o he local Whi le es ima o is
p oposed, bu he size and powe p ope ies o his es in simula ions appea o depend
hea ily on he sample size.2
An al e na i e way o allow o non-s a iona y p ocesses would be o base he
es on he objec i e unc ion o he mul i a ia e exac local Whi le es ima o (as in
Shimo su (2012), bu wi hou allowing o ac ional coin eg a ion) and uni a ia e
ELW es ima es. Since he exac local Whi le es ima es ha e he same asymp o ic
p ope ies as he local Whi le es ima e o d∈(−1/2,1/2), he es would ha e he
same limi ing dis ibu ion.
Fo a bi a ia e p ocess wi h known d∈(0,1], Souza e al. (2018) p opose a
es based on an es ima e o bob ained om he de e minan o he immed and
unca ed spec al ma ix o he ac ionally di e enced p ocess ia a log-pe iodog am
eg ession. Deno e he ac ionally di e enced p ocess by ΔdX =(ΔdX1 ,Δ
dX2 )
wi h spec al densi y ma ix Δd(λ), hen he de e minan DΔd(λ) o Δd(λ) depends
on he memo y educ ion pa ame e b∈[0,d]and can be app oxima ed by
DΔd(λ) ∼˜g|1−e−iλ|2b,as λ→0+,(10)
whe e ˜gis a cons an and ini e scala . Unde coin eg a ion, Δd(λ) does no ha e ull
ank nea he o igin (like Gin (1)) so ha i s de e minan DΔd(λ) app oaches ze o as
λ→0+. The memo y educ ion bcan be es ima ed om he logged e sion o Eq.
(10) using a log-pe iodog am ype eg ession,
2These esul s a e a ailable om he au ho s upon eques .
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 2011
eigen alue is OP(Tmin{0,1−2(d−b+)}), so ha i goes o ze o mo e slowly han unde
he null hypo hesis.
The es is es ic i e in ha i equi es non-s a iona y p ocesses and, p e e ably,
s a iona y esidual p ocesses, bu as shown by his Mon e Ca lo simula ion he es s ill
exhibi s powe i d >0.5 and b>0. Fu he mo e, i is applicable o mul i a ia e
sys ems and is able o es ima e he numbe o coin eg a ing ela ions.
Wang e al. (2015) p opose a simple esidual-based es in a bi a ia e se ing whe e
X =(X1 ,X2 ). The es s a is ic is based on he pa ial sum o Δd X2 , which is he
demeaned second componen se ies ac ionally di e enced wi h he memo y o de
o he po en ial coin eg a ing esidual . I is gi en by
WWWC =T−1/2T
=1Δ
d Z2
2π
22(0)
,(16)
whe e 22 is he spec al densi y o ei he u2 o e in (3), depending on whe he a
iangula model o a common-componen s model is assumed.
Unde he null hypo hesis d =dso ha Δd Z2 is I(0)and he app op ia ely
escaled sum is asymp o ically s anda d no mal. Unde he al e na i e Δd Z2 is I(b),
so ha he es s a is ic di e ges wi h a e OP(Tb).
To make his es s a is ic easible he spec al densi y 22 can be es ima ed om
he pe iodog am o he ac ionally di e enced p ocess Δ
dZ2 ollowing he app oach
o Hualde (2013):
22(0)=1
(2m+1)m
j=−mIΔ
dZ2(λj), whe e IΔ
dZ2(λj)is he pe i-
odog am o Δ
dZ2 .
While Wang e al. (2015) a e agnos ic abou he me hod ha is used o he es i-
ma ion o he memo y pa ame e s dand d , hey assume ha d>1/2 so ha he
coin eg a ing ec o can be es ima ed using o dina y leas squa es. The memo y o de s
can be es ima ed om OLS
and Z2 using any o he common semipa ame ic es i-
ma es such as ELW wi h bandwid h mas in
22 ha ul ills he usual bandwid h
condi ions. The me hod does no impose any es ic ions on he ac ional coin e-
g a ing s eng h b. As he Mon e Ca lo simula ions below show, he non-s a iona i y
equi emen (d>1/2) can be ci cum en ed i he coin eg a ing esidual is based
on he NBLS es ima e o he coin eg a ing ec o ins ead o he OLS es ima e.
Zhang e al. (2019) p opose an al e na i e es ima o o he coin eg a ing space ha
is based on he eigen ec o s o he non-nega i e ma ix
M=j0
j=0
ΩZ(j)
ΩZ(j),
whe e
ΩZ(j)=1
TT−j
=1Z +jZ
is he au oco a iance ma ix a lag jand j0is
a ixed in ege . The ma ix
Mis hus he sum o he ou e p oduc s o he i s j0
au oco a iance ma ices wi h hemsel es. The ou e p oduc is used ins ead o he
co a iance ma ices
ΩZ(j) o ensu e ha he e is no in o ma ion cancella ion o e
di e en lags in
M. I is assumed ha d>0.5 and d <0.5.
The eigen alues o
Min descending o de a e deno ed by
δa,M o a=1,...,p
and he co esponding eigen ec o s a e deno ed by χa,M. Simila o he ma ix Gin
(1), he i s p− eigen alues o Ma e non-ze o, whe eas he emaining a e ze o.
Fo known he eigen ec o s co esponding o he smalles eigen alues p o ide a
consis en es ima e o he coin eg a ing space.
123
2012 C. Leschinski e al.
I is unknown, he ppo en ial coin eg a ing esiduals a e es ima ed using he
eigen ec o s so ha M
a =χ
a,MX . By he same a gumen as in he p ocedu e o
Chen and Hu ich (2006), he esidual co esponding o he smalles eigen alue is
mos likely a coin eg a ing esidual wi h educed memo y o d =d−band he
esidual co esponding o he la ges eigen alue is I(d).
The coin eg a ing ank can be es ima ed using a simple c i e ion based on he
summed au oco ela ions o he po en ial coin eg a ing esiduals. De ine
Qa(k0)=
k0
k=1ρa(k),
wi h ρa(k)=
1
T−kT−k
=1( M
a, +k− M
a )( a − M
a )
1
TT
=1( M
a − M
a )2,
whe e M
a is he mean o M
a . The coin eg a ing ank es ima o coun s he ins ances
when he a e aged au oco ela ion is smalle han a h eshold c0∈(0,1):
ZRY =
p
a=1
1Qa(k0)
k0
<c0.(17)
I he esidual M
a is s a iona y (d <1/2), he escaled sum o au oco ela ions
Qa(k0)/k0con e ges o ze o asymp o ically o k0→∞, since he au oco ela ions
a e asymp o ically p opo iona e o k2d −1. Unde ce ain egula i y condi ions his
es ima eis consis en .E en hough heconsis encyisonlyp o en o ≥1inTheo em
4.2 o Zhang e al. (2019), ou simula ions below show ha i also wo ks well in
disc imina ing be ween =0 and =1.
I should be no ed ha he au ho s de ine =pi all componen s o X a e I(0).
This leads o some abuse o no a ion and canno be in e p e ed as he coin eg a ing
ank in a na ow sense. Based on hei simula ions Zhang e al. (2019) ecommend o
use j0=5, k0=20 and c0=0.3. The es ima o is easy o implemen and applicable
o highe dimensional p ocesses. Howe e , he equi emen o d>0.5 and d <0.5
is es ic i e.
4 Mon e Ca lo S udy
Theasymp o icp ope ies o all es sand ank es ima es p esen edinSec .3a e de i ed
by he espec i e au ho s, and some o hem also p esen simula ions o explo e he
ini e sample esul s o he es s a is ics. This howe e is no he case o all es s and a
comp ehensi e compa a i e s udy sui ed o guide he choice o app op ia e me hods in
p ac ical applica ions is en i ely missing. To close his gap, we conduc an ex ensi e
Mon e Ca lo s udy. In addi ion o gene al esul s, we a e pa icula ly in e es ed in
answe ing wo empi ically mo i a ed ques ions.
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 2013
(i) How does co ela ion be ween he unde lying sho - un componen s in luence
he size o he es s? This ques ion is impo an , since applied esea che s will gene ally
wan o es o ac ional coin eg a ion i wo ela ed se ies seem o be co-mo ing.
Simila ajec o ies, howe e , can also be gene a ed by pe sis en p ocesses wi h highly
co ela ed inno a ions. Tes s o he null hypo hesis o no ac ional coin eg a ion
should he e o e be obus o a ela i ely high deg ee o co ela ion be ween he sho -
un componen s o he se ies.
(ii) Is he e a no able di e ence in he powe o he es s depending on whe he he
da a is gene a ed om a iangula model o om a common-componen s model? Bo h
models a e used in he li e a u e o mo i a e and cons uc es ing p ocedu es, bu o ou
knowledge simula ion esul s a e ypically based on he iangula ep esen a ion. In
p ac ice, ei he model could be jus i ied—depending on he applica ion. Fo example,
i one is conside ed wi h po en ial ac ional coin eg a ing ela ionships be ween s ock
p ices, i is no clea why one o he s ock p ices should be seen as a pe u bed e sion
o he o he one (as i is he case in he iangula model ha ea s he se ies in an
asymme ic way) so ha he common-componen s model is mo e sui able. In con as
o ha , in he case o he po en ial pa i y be ween implied ola ili y and he expec ed
a e age ealized ola ili y o e he nex mon h ( he so-called implied- ealized pa i y
analyzed by Ch is ensen and P abhala (1998), Ch is ensen and Nielsen (2006b), and
Nielsen (2007), among o he s), he e is heo e ical eason o assume ha he implied
ola ili y is a pe u bed e sion o he expec ed a e age u u e ealized ola ili y, since
i con ainsa a iance- iskp emium (c .Che no (2007)).The e o e,a iangula model
is mo e sui able.
We ocus on h ee da a gene a ing p ocesses (DGPs) based on he gene al model
om Eqs. (3) o(5). Fo simplici y we se c1=c2=0 and b=b1=b2so ha he
p ocesses a e mean ze o and ha e a common memo y educ ion pa ame e . A simple
bi a ia e model wi hou ac ional coin eg a ion is cons uc ed by se ing ξ1=ξ2=0.
This model— e e ed o as (size) DGP1—is gi en by
X1 =Δ−du1 1{ >0},(18)
X2 =Δ−du2 1{ >0},(19)
whe e co ela ion be ween u1 and u2 is allowed. This is ou size-DGP. Fo he powe
simula ions, we conside a iangula model and a common-componen s model. In
bo h cases we se ξ1=ξ2=1 which implies a coin eg a ing ec o o β=(1,−1).
The iangula model DGP2 is gi en by
X1 =Y +Δ−(d−b)u1 1{ >0},(20)
X2 =Y ,(21)
and he common-componen s model DGP3 is de ined by
X1 =Y +Δ−(d−b)u1 1{ >0},(22)
X2 =Y +Δ−(d−b)u2 1{ >0}.(23)
123
2014 C. Leschinski e al.
In bo h DGP2 and DGP3 we ha e Y =Δ−de 1{ >0}. The unde lying sho - un
componen s u1 and u2 ,o u1 and e —depending on he DGP—ha e uni a iance
and co ela ion ρ.
We conside sample sizes o T∈{100,500,1000,2500}and alues o d∈
{0.4,0.7,1}in he s a iona y and non-s a iona y egion. Unde ac ional coin e-
g a ion, he memo y educ ion bis linked o he alue o dso ha b∈{d/3,d}.
Consequen ly, he e is ei he a memo y educ ion o 0 i b=do a weake o m o
coin eg a ion i b=d/3. In o de o examine he impac o co ela ion be ween he
sho - un componen s, we conside ρ∈{0,0.45,0.9,0.99}. No e ha he esul s
o b=d/3, δm=0.55 and u he obus ness es s (o he size DGPs and p=3-
dimensional p ocesses) a e a ailable online as supplemen a y ma e ial.
The semipa ame ic na u e o he es s and ank es ima es equi es se e al band-
wid h choices. The memo y es ima ion wi h (E)LW es ima o s in ol ed in all me hods
is based on he bandwid h m ha de e mines he numbe o equencies included in
he es ima ion. We use m=Tδmwi h δm={0.55,0.75} o accoun o sensi i i ies
ega ding bandwid h choice. Wi h ega d o he o he bandwid h choices, we ollow
he ecommenda ions by he au ho s: m1=Tδm−0.1and p(T)=m−0.3
1 o Nielsen
and Shimo su (2007) o Robinson and Yajima (2002), l=1 o Souza e al. (2018),
m3=25 o Chen and Hu ich (2006), c0=0.3, j0=5 and k0=20 o Zhang e al.
(2019), and o Ma mol and Velasco (2004)wese m=T2/3and m2=Tδm.All
es s a e ca ied ou allowing o a non-ze o mean.
The esul s p esen ed a e based on 5000 eplica ions and a nominal signi icance
le el o α=0.05. Since he es s impose di e en condi ions on dand d ,wema k he
cells in he ables in bold whe e he me hods ha e well-de ined asymp o ic p ope ies
and a e supposed o deli e good esul s. In some cases he me hods gi e sa is ac o y
esul s beyond hese limi a ions. Fo example, we implemen he me hod o Wang e al.
(2015) using a NBLS es ima e o he coin eg a ing ec o ins ead o he OLS es ima e.
This makes he es applicable in s a iona y ime se ies as well as in non-s a iona y
ones.
Since he limi ing dis ibu ions o he non-pi o al es s a is ics o Ma mol and
Velasco (2004) and Nielsen (2010) depend on dand i is assumed ha d>1/2, i is
unclea which c i ical alues would be used in he s a iona y egion. The espec i e
ields a e he e o e le blank.
Fu he , i should be no ed ha he me hods o Nielsen and Shimo su (2007)(o
Robinson and Yajima (2002)) and Zhang e al. (2019) a e no es s bu ank es ima es.
Ins ead o he ejec ion equency, we he e o e epo he a io o co ec ly es ima ed
coin eg a ing anks. The e o e, he esul s canno be in e p e ed as size o powe , and
in he size able and g aphs he es ima es should yield 0 ins ead o 0.05, since he
es ima es do no in ol e any signi icance le el.
Table 1displays size esul s based on DGP 1 wi h δm=0.75. The me hods ha
ha e well de ined asymp o ic p ope ies ac oss all pa ame e cons ella ions co e ed
in he able a e hose o Nielsen and Shimo su (2007), Chen and Hu ich (2006),
Hualde and Velasco (2008), and Souza e al. (2018). I can be obse ed ha all o
hese me hods achie e good size p ope ies o ρ=0, excep o he es o Chen and
Hu ich (2006), when d=0.4. Among hose ou p ocedu es, only he es s o Souza
e al. (2018) (dis ega ding he smalles sample) and Chen and Hu ich (2006) do no
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 2015
Table 1 Size (* ank es ima ion) based on DGP1 wi h δm=0.75
Me hod ρ0 0.45 0.9 0.99
T/d0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1
NS07* 100 0.000 0.000 0.014 0.130 0.138 0.241 1.000 1.000 0.998 1.000 1.000 1.000
500 0.000 0.000 0.001 0.000 0.000 0.049 1.000 1.000 0.994 1.000 1.000 1.000
1000 0.000 0.000 0.000 0.000 0.000 0.021 1.000 1.000 0.989 1.000 1.000 1.000
2500 0.000 0.000 0.000 0.000 0.000 0.006 1.000 1.000 0.976 1.000 1.000 1.000
CH06 100 0.219 0.119 0.021 0.105 0.073 0.029 0.077 0.040 0.032 0.075 0.033 0.033
500 0.177 0.058 0.032 0.076 0.031 0.021 0.060 0.026 0.017 0.064 0.027 0.020
1000 0.136 0.051 0.031 0.064 0.025 0.018 0.045 0.018 0.018 0.046 0.020 0.017
2500 0.129 0.044 0.023 0.059 0.023 0.015 0.039 0.018 0.012 0.040 0.017 0.012
HV08 100 0.001 0.008 0.015 0.017 0.022 0.024 0.128 0.098 0.058 0.312 0.263 0.130
500 0.002 0.018 0.020 0.027 0.029 0.022 0.309 0.141 0.028 0.764 0.440 0.098
1000 0.003 0.023 0.022 0.039 0.029 0.021 0.399 0.128 0.028 0.805 0.501 0.076
2500 0.003 0.027 0.022 0.060 0.035 0.021 0.494 0.140 0.024 0.835 0.535 0.057
SRFB18 100 0.114 0.117 0.101 0.107 0.111 0.110 0.112 0.121 0.114 0.109 0.115 0.105
500 0.054 0.054 0.049 0.049 0.052 0.052 0.046 0.055 0.046 0.056 0.052 0.045
1000 0.041 0.047 0.042 0.043 0.043 0.040 0.044 0.047 0.048 0.045 0.043 0.044
2500 0.037 0.039 0.029 0.035 0.037 0.037 0.036 0.036 0.034 0.033 0.035 0.035
123
2016 C. Leschinski e al.
Table 1 con inued
Me hod ρ0 0.45 0.9 0.99
T/d0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1
R08 100 0.174 0.183 0.092 0.050 0.059 0.066 0.036 0.048 0.039 0.042 0.045 0.039
500 0.233 0.254 0.104 0.049 0.080 0.063 0.052 0.066 0.041 0.049 0.062 0.043
1000 0.239 0.275 0.090 0.053 0.080 0.064 0.051 0.076 0.046 0.057 0.074 0.042
2500 0.265 0.304 0.084 0.055 0.094 0.056 0.054 0.084 0.039 0.050 0.084 0.040
WWC15 100 0.080 0.095 0.090 0.079 0.087 0.094 0.081 0.096 0.098 0.078 0.091 0.094
500 0.069 0.068 0.075 0.068 0.068 0.074 0.065 0.074 0.074 0.072 0.072 0.068
1000 0.066 0.064 0.067 0.069 0.062 0.066 0.065 0.056 0.067 0.055 0.060 0.068
2500 0.057 0.059 0.056 0.059 0.052 0.061 0.060 0.057 0.059 0.055 0.049 0.061
ZRY19* 100 0.101 0.644 0.652 0.071 0.597 0.627 0.057 0.560 0.573 0.062 0.565 0.576
500 0.401 0.058 0.000 0.288 0.043 0.000 0.270 0.036 0.000 0.281 0.030 0.000
1000 0.548 0.000 0.000 0.410 0.000 0.000 0.408 0.000 0.000 0.401 0.000 0.000
2500 0.677 0.000 0.000 0.517 0.000 0.000 0.505 0.000 0.000 0.510 0.000 0.000
N10 100 0.035 0.059 0.041 0.061 0.053 0.053 0.058 0.063
500 0.048 0.055 0.048 0.052 0.064 0.057 0.060 0.055
1000 0.057 0.056 0.056 0.055 0.070 0.060 0.064 0.055
2500 0.067 0.054 0.078 0.056 0.078 0.056 0.076 0.053
MV04 100 0.034 0.057 0.036 0.066 0.104 0.101 0.332 0.219
500 0.041 0.052 0.047 0.056 0.158 0.092 0.496 0.234
1000 0.039 0.052 0.047 0.055 0.141 0.081 0.490 0.224
2500 0.036 0.052 0.045 0.056 0.127 0.069 0.442 0.197
We abb e ia e he me hods wi h he ini ial le e s o he au ho s’ names and he yea o publica ion
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 2017
o e - ejec i ρinc eases.3Fo low alues o d he es o Hualde and Velasco (2008)
al eady becomes o e sized o ρ=0.45 and as ρinc eases i becomes o e sized o
highe alues o d, oo. The ank es ima ion p ocedu e o Robinson and Yajima (2002)
and Nielsen and Shimo su (2007) is e en mo e a ec ed and es ima es a coin eg a ing
ank o one in nea ly all cases i ρ≥0.9.
In addi ion o he es s o Souza e al. (2018) and Chen and Hu ich (2006), he
modi ied e sion o he es by Wang e al. (2015) ha is based on he NBLS es ima o
ins ead o OLS also main ains sa is ac o y size p ope ies ac oss all alues o ρand
d.
Theg oupo p ocedu es ha is only applicable o non-s a iona ysys emsconsis so
Ma moland Velasco(2004),Nielsen (2010),andZhang e al.(2019). I canbeobse ed
ha he p ocedu e o Ma mol and Velasco (2004) beha es simila o ha o Hualde
and Velasco (2008) in he sense ha i is e y libe al o highe alues o ρand lowe
alues o d. Fo non-s a iona y se ies and la ge sample sizes he p ocedu e by Zhang
e al. (2019) es ima es co ec ly he coin eg a ing ank o be ze o—independen ly o
he deg ee o co ela ion. The a iance- a io s a is ic o Nielsen (2010) u ns ou o be
sligh ly libe al o d=0.7 in la ge samples, bu holds he nominal size o d=1
e en in small samples. In pa icula , he pe o mance is independen o he deg ee o
co ela ion.
Finally, he es o Robinson (2008a) is only applicable o s a iona y sys ems.
He e, i can be obse ed ha he es does no hold i s size o ρ=0. This is because
he Hausman- es ing p inciple equi es one o he es ima es o he memo y pa ame e
o be mo e e icien han he o he one, bu he mul i a ia e es ima e is no mo e
e icien in absence o co ela ion. Fo o he alues o ρ, howe e , he es has good
size p ope ies. In e es ingly, he es also has good size p ope ies i d=1, e en
hough i assumes s a iona i y. The in e media e alue o d=0.7, on he o he hand,
leads o a mode a ely o e sized es .
Figu e 1analyzes he in e ac ion be ween he deg ee o co ela ion ρand he choice
o he bandwid h δm. I shows he size o he es s in sca e plo s whe e he esul s wi h
no co ela ion (ρ=0) a e plo ed agains esul s wi h high co ela ion (ρ=0.99).
In he uppe panel, es s ha allow o s a iona y p ocesses (d=0.4) and in he
lowe panel (d=1) he non-s a iona i y- obus es s, i.e. all excep ha o Robinson
(2008a), a e displayed. The dashed lines ma k he nominal size le el o 0.05 so ha
ideally all poin s would lie on he in e sec ion be ween hese wo lines. The do ed
line is he bisec o implying ha me hods abo e he bisec o do be e wi h co ela ion
and me hods below he bisec o do be e wi hou . Black symbols gi e esul s wi h a
bandwid h pa ame e o δm=0.75 and g ay symbols wi h δm=0.55.
I can be obse ed ha he p ocedu es by Ma mol and Velasco (2004), Nielsen and
Shimo su (2007) and Hualde and Velasco (2008) lie below he bisec o and a e hus
nega i ely a ec ed by high co ela ion, whe eas he es s by Chen and Hu ich (2006)
and Robinson (2008a) lie abo e he bisec o . The emaining es s lie on he bisec o
indica ing obus ness o co ela ion. Rega ding bandwid h choice, he es s by Ma mol
and Velasco (2004), Chen and Hu ich (2006), Hualde and Velasco (2008) and Wang
e al. (2015) a e mo e libe al wi h a small bandwid h, whe eas Nielsen and Shimo su
3The es o Chen and Hu ich (2006) is conse a i e by cons uc ion as discussed in he p e ious sec ion.
123
2018 C. Leschinski e al.
CH06
NS07*
R08
SRFB18
WWC15
HV08
CH06
WWC15
HV08
0.00
0.05
0.10
0.15
0.20
0.25
0.00 0.25 0.50 0.75 1.00
High co ela ion
Ze o co ela ion
Bandwid h
a
a
δm=0.75
δm=0.55
Size wi h a ying co ela ion and bandwid h, d=0.4
CH06
MV04
NS07*
SRFB18
WWC15
HV08
N10
ZRY18*
WWC15
HV08
0.00
0.05
0.10
0.15
0.00 0.25 0.50 0.75 1.00
High co ela ion
Ze o co ela ion
Bandwid h
a
a
δm=0.75
δm=0.55
Size wi h a ying co ela ion and bandwid h, d=1
Fig. 1 Size (* ank es ima ion) based on DGP1 depending on co ela ion ρ∈{0,0.99}and bandwid h
δm∈{0.55,0.75}wi h T=1000
(2007), Robinson (2008a), Nielsen (2010), Souza e al. (2018) and Zhang e al. (2019)
a e ela i ely obus in e ms o size. In gene al, co ela ion in he unde lying sho -
un componen is mis aken o coin eg a ion mo e o en in s a iona y sys ems han in
non-s a iona y ones.
O e all, in e ms o size o bi a ia e sys ems and aking he ange o admissible
pa ame e alues in o accoun , we ind ha he es o Souza e al. (2018) has he bes
pe o mance, ollowed by hose o Chen and Hu ich (2006) and Wang e al. (2015).
Conside ing p ocedu es only applicable o non-s a iona y sys ems, Nielsen (2010) and
Zhang e al. (2019) a e e y eliable op ions as well.
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 2019
Table 2 Powe (* ank es ima ion), b=dand δm=0.75 o he iangula model (DGP2)
Me hod ρ0 0.45 0.9 0.99
T/d0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1
d 000000000000
NS07* 100 0.989 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
CH06 100 0.556 0.349 0.337 0.317 0.291 0.110 0.046 0.205 0.020 0.007 0.179 0.010
500 0.998 0.443 1.000 0.991 0.179 1.000 0.999 0.081 1.000 1.000 0.071 1.000
1000 1.000 0.811 1.000 1.000 0.518 1.000 1.000 0.277 1.000 1.000 0.241 1.000
2500 1.000 0.998 1.000 1.000 0.972 1.000 1.000 0.869 1.000 1.000 0.833 1.000
HV08 100 0.788 0.989 1.000 0.882 0.998 1.000 0.958 1.000 1.000 0.959 1.000 1.000
500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SRFB18 100 0.720 0.976 0.999 0.709 0.982 0.999 0.642 0.928 0.993 0.259 0.586 0.914
500 0.993 1.000 1.000 0.994 1.000 1.000 0.983 1.000 1.000 0.794 0.961 0.993
1000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.949 0.994 0.999
2500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 1.000 1.000
123
2020 C. Leschinski e al.
Table 2 con inued
Me hod ρ0 0.45 0.9 0.99
T/d0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1
d 000000000000
R08 100 0.044 0.245 0.572 0.009 0.023 0.200 0.049 0.060 0.065 0.671 0.463 0.147
500 0.870 1.000 1.000 0.368 0.869 0.998 0.202 0.239 0.454 1.000 0.957 0.599
1000 0.997 1.000 1.000 0.799 0.999 1.000 0.347 0.343 0.573 1.000 0.991 0.709
2500 1.000 1.000 1.000 0.999 1.000 1.000 0.576 0.486 0.707 1.000 1.000 0.804
WWC15 100 0.634 0.896 0.966 0.618 0.887 0.970 0.426 0.855 0.965 0.307 0.832 0.959
500 0.829 0.967 0.993 0.807 0.968 0.994 0.694 0.961 0.993 0.630 0.956 0.995
1000 0.867 0.982 0.998 0.850 0.979 0.998 0.763 0.977 0.997 0.695 0.978 0.997
2500 0.917 0.992 0.998 0.896 0.988 0.999 0.817 0.990 0.999 0.787 0.990 0.999
ZRY19* 100 0.020 0.403 0.797 0.007 0.339 0.780 0.006 0.289 0.770 0.008 0.289 0.764
500 0.082 0.983 1.000 0.032 0.974 1.000 0.016 0.953 1.000 0.010 0.954 1.000
1000 0.119 1.000 1.000 0.042 1.000 1.000 0.011 0.998 1.000 0.011 0.998 1.000
2500 0.200 1.000 1.000 0.049 1.000 1.000 0.013 1.000 1.000 0.007 1.000 1.000
N10 100 0.431 0.978 0.356 0.965 0.270 0.950 0.262 0.954
500 0.996 1.000 0.988 1.000 0.981 1.000 0.974 1.000
1000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
MV04 100 0.481 0.974 0.748 0.994 0.854 0.998 0.866 0.999
500 0.985 1.000 0.993 1.000 0.997 1.000 0.997 1.000
1000 0.997 1.000 0.999 1.000 1.000 1.000 1.000 1.000
2500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
123
A compa ison o semipa ame ic es s o ac ional coin eg a ion 2027
Table 4 con inued
Me hod size powe
φ−0.5 0.5 −0.5 0.5
T/d0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1
d 0.40.710.40.71000000
R08 100 0.133 0.147 0.131 0.146 0.179 0.410 0.757 0.955 0.994 0.684 0.961 1.000
500 0.053 0.067 0.068 0.088 0.093 0.351 0.991 1.000 1.000 0.993 1.000 1.000
1000 0.051 0.048 0.051 0.069 0.065 0.332 1.000 1.000 1.000 1.000 1.000 1.000
2500 0.029 0.038 0.048 0.053 0.049 0.294 1.000 1.000 1.000 1.000 1.000 1.000
WWC15 100 0.188 0.181 0.190 0.004 0.004 0.004 0.736 0.920 0.973 0.291 0.729 0.914
500 0.121 0.113 0.120 0.001 0.001 0.001 0.846 0.973 0.996 0.619 0.930 0.988
1000 0.109 0.098 0.100 0.001 0.001 0.002 0.873 0.985 0.997 0.746 0.963 0.996
2500 0.077 0.081 0.087 0.001 0.002 0.002 0.904 0.990 0.999 0.829 0.985 0.998
ZRY19* 100 0.041 0.107 0.134 0.013 0.008 0.004 0.930 1.000 1.000 0.924 0.996 1.000
500 0.039 0.082 0.096 0.032 0.010 0.001 1.000 1.000 1.000 1.000 1.000 1.000
1000 0.056 0.078 0.088 0.038 0.009 0.002 1.000 1.000 1.000 1.000 1.000 1.000
2500 0.063 0.073 0.077 0.046 0.008 0.002 1.000 1.000 1.000 1.000 1.000 1.000
N10 100 0.017 0.036 0.267 0.138 0.141 0.781 0.932 0.998
500 0.084 0.048 0.292 0.153 0.999 1.000 1.000 1.000
1000 0.110 0.046 0.282 0.134 1.000 1.000 1.000 1.000
2500 0.130 0.053 0.234 0.124 1.000 1.000 1.000 1.000
MV04 100 0.446 0.667 0.634 0.576 0.136 0.700 0.526 0.834
500 0.156 0.000 0.008 0.000 0.796 1.000 0.997 1.000
1000 0.007 0.000 0.000 0.000 0.966 1.000 1.000 1.000
2500 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000
123
2028 C. Leschinski e al.
Based on ou Mon e Ca lo s udies, we ind ha some o he p oposed app oaches
ha e weaknesses in hei ini e sample beha io in some empi ically ele an
scena ios—especially in p esence o co ela ed sho - un componen s. This conce ns
mos ly he me hods o Nielsen and Shimo su (2007) (o Robinson and Yajima (2002)),
Ma mol and Velasco (2004), and Hualde and Velasco (2008) ha ha e he highes
powe bu ha e size issues in case o s ongly co ela ed sho - un componen s. Wi h
ega d o iii.), we ind ha he size p ope ies o he es s in he iangula case and he
common-componen s model is gene ally compa able (see online ma e ial). Fo he
powe o he es s, howe e , he e a e impo an di e ences be ween he wo cases. In
pa icula , he es o Chen and Hu ich (2006) has much be e powe o s a iona y
sys ems unde he common componen s speci ica ion, whe eas he me hods o Robin-
son and Yajima (2002) and Hualde and Velasco (2008) become wo se in hei abili y
o de ec ac ional coin eg a ion.
Al hough he me hods o Robinson (2008a), Nielsen (2010), and Zhang e al. (2019)
u n ou o be obus o sho - un co ela ion and a e appealing due o hei simplici y,
hey impose p ac ically ele an es ic ions on he pe missible ange o dand b.
Howe e , i he e is p io knowledge abou he (non-) s a iona i y o he da a, hose
p ocedu es a e e y good op ions.
O e all, we conclude ha he es o Souza e al. (2018) o bi a ia e sys ems
has he bes p ope ies, bo h heo e ically and empi ically, and is a good choice o
he applied econome ician. I allows o he whole empi ically ele an ange o d
and b, i is obus o co ela ion and sho - un dynamics wi h posi i e coe icien s,
and i p o ides compa able pe o mance in bo h— iangula sys ems and common-
componen s models.
In highe dimensional sys ems, howe e , he es o Souza e al. (2018) is no longe
applicable and ha o Chen and Hu ich (2006) u ns ou o be libe al in ini e samples
om s a iona y p ocesses. He e, he p ocedu e o Robinson (2008a) can be ecom-
mended o s a iona y p ocesses and he ank es ima ion by Nielsen (2010) and Zhang
e al. (2019) should be p e e ed o non-s a iona y sys ems i he coin eg a ing esid-
uals can be expec ed o be s a iona y.
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and ins i u ional a ilia ions.
123