Pe on, Pie e; Yamamo o, Yohei; Zhou, Jing
A icle
Tes ing join ly o s uc u al changes in he e o a iance
and coe icien s o a linea eg ession model
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: Pe on, Pie e; Yamamo o, Yohei; Zhou, Jing (2020) : Tes ing join ly o s uc u al
changes in he e o a iance and coe icien s o a linea eg ession model, Quan i a i e Economics,
ISSN 1759-7331, The Econome ic Socie y, New Ha en, CT, Vol. 11, Iss. 3, pp. 1019-1057,
h ps://doi.o g/10.3982/QE1332
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Quan i a i e Economics 11 (2020), 1019–1057 1759-7331/20201019
Tes ing join ly o s uc u al changes in he e o a iance and
coe icien s o a linea eg ession model
Pie e Pe on
Depa men o Economics, Bos on Uni e si y
Yohei Yamamo o
Depa men o Economics, Hi o subashi Uni e si y
Jing Zhou
Seeking Sense (Shanghai) In es men Managemen Co., L d.
We p o ide a comp ehensi e ea men o he p oblem o es ing join ly o s uc-
u al changes in bo h he eg ession coe icien s and he a iance o he e o s in
a single equa ion sys em in ol ing s a iona y eg esso s. Ou amewo k is qui e
gene al in ha we allow o gene al mixing- ype eg esso s and he assump ions
on he e o s a e qui e mild. Thei dis ibu ion can be nonno mal and condi-
ional he e oskedas ici y is pe mi ed. Ex ensions o he case wi h se ially co e-
la ed e o s a e also ea ed. We p o ide he equi ed ools o add ess he ollow-
ing es ing p oblems, among o he s: (a) es ing o gi en numbe s o changes in
eg ession coe icien s and a iance o he e o s; (b) es ing o some unknown
numbe o changes wi hin some p especi ied maximum; (c) es ing o changes
in a iance ( eg ession coe icien s) allowing o a gi en numbe o changes in
he eg ession coe icien s ( a iance); (d) a sequen ial p ocedu e o es ima e he
numbe o changes p esen . These es ing p oblems a e impo an o p ac ical
applica ions as wi nessed by in e es s in mac oeconomics and inance whe e doc-
umen ing s uc u al changes in he a iabili y o shocks o simple au o eg essions
o ec o au o eg essi e models ha e been a conce n.
Keywo ds. Change-poin , a iance shi , condi ional he e oskedas ici y, likeli-
hood a io es s.
JEL classi ica ion. C22.
1. In oduc ion
Bo h he s a is ics and econome ics li e a u e con ain a as amoun o wo k on issues
ela ed o s uc u al changes wi h unknown b eak da es, mos o i designed o a single
change ( o an ex ensi e e iew, see Pe on (2006)andCasini and Pe on (2019b)). The
Pie e Pe on: [email p o ec ed]
Yohei Yamamo o: [email p o ec ed]
Jing Zhou: [email p o ec ed]
Pe on acknowledges inancial suppo om he Na ional Science Founda ion unde G an SES-0649350.
We a e g a e ul o Zhongjun Qu o commen s and o poin ing ou an e o in a p e ious d a .
©2020 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h p://qeconomics.o g.h ps://doi.o g/10.3982/QE1332
1020 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
p oblem o mul iple s uc u al changes has ecei ed a en ion mos ly in he con ex o
a single eg ession. Bai and Pe on (1998,2003a) p o ided a comp ehensi e ea men :
consis ency o es ima es o he b eak da es, es s o s uc u al changes, con idence in-
e als o he b eak da es, me hods o selec he numbe o b eaks, and e icien algo-
i hms o compu e he es ima es; see also Hawkins (1976). Pe on and Qu (2006)ex-
ended his analysis o he case whe e a bi a y linea es ic ions a e imposed on he
coe icien s o he model. Also, Ku ozumi and Tu aando j (2011) p oposed an in o ma-
ion c i e ion o he selec ion o he numbe o changes; see also Liu, Wu, and Zidek
(1997). Bai, Lumsdaine, and S ock (1998) conside ed asymp o ically alid in e ence o
he es ima e o a single b eak da e in mul i a ia e ime se ies allowing s a iona y o in e-
g a ed eg esso s as well as ends wi h es ima ion ca ied using a quasi maximum like-
lihood (QML) p ocedu e. Also, Bai (2000) conside ed a segmen ed s a iona y VAR model
es ima ed again by QML when he b eak can occu in he pa ame e s o he condi ional
mean, he a iance o he e o e m o bo h. Kej iwal and Pe on (2008,2010) deal wi h
mul iple s uc u al changes in a single equa ion coin eg a ed model. Pe on and Ya-
mamo o (2014) de i ed he limi dis ibu ion o he es ima es o he b eak da es in mod-
els wi h endogenous eg esso s es ima ed ia an ins umen al a iable me hod, while
hey a gue in Pe on and Yamamo o (2015) ha using s anda d leas -squa es me hods
is p e e able bo h o es ima ion and es ing. Casini and Pe on (2019a)p o idedalimi
dis ibu ion o he leas -squa es es ima e o he b eak da e in a linea model based on a
con inuous- ime asymp o ic amewo k, which deli e s subs an ial imp o emen s wi h
espec o in e ence using he concep o highes densi y egions.
Wi h espec o es ing o changes in he a iance o he eg ession e o , he e-
sul s a e qui e spa se. Ho á h (1993) conside ed a change in he mean and a iance
(occu ing a he same ime) o a sequence o i.i.d. andom a iables wi h momen s co -
esponding o hose o a no mal dis ibu ion. Da is, Huang, and Yao (1995)ex ended he
analysis o an au o eg essi e p ocess unde simila condi ions. Aue, Ho mann, Ho á h,
and Reimhe (2009) p oposed nonpa ame ic es s o changes in he a iances o au-
oco a iances o mul i a ia e linea o nonlinea ime se ies models. Deng and Pe on
(2008) ex ended he CUSUM o squa es (o CUSQ) es o B own, Du bin, and E ans
(1975) allowing gene al condi ions on he eg esso s and he e o s (as sugges ed by In-
clán and Tiao (1994) o no mally dis ibu ed ime se ies). Xu (2013) p o ided a u he
ex ension wi h a obus es ima e o he long- un a iance o he squa ed e o s o close
ele ance o ou objec i es. And ews (1993) conside ed a one- ime s uc u al change
unde a Gene alized Me hod o Momen (GMM) se ing, he eby allowing o changes
in bo h coe icien s and a iance hough occu ing a he same da e; see McConnell
and Pé ez-Qui ós (2000) o a ela ed applica ion. Qu and Pe on (2007a)conside eda
mul i a ia e sys em es ima ed by quasi maximum likelihood, which p o ides me hods
o es ima e models wi h s uc u al changes in bo h he eg ession coe icien s and he
co a iance ma ix o he e o s. They p o ide a limi dis ibu ion heo y o in e ence
abou he b eak da es and also conside es ing o mul iple s uc u al changes, hough
es ic ed o no mally dis ibu ed e o s and b eaks in coe icien s and a iance occu -
ing a di e en da es.
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1021
We build on Qu and Pe on (2007a) o p o ide a comp ehensi e ea men o es -
ing join ly o s uc u al changes in bo h he eg ession coe icien s and he a iance o
he e o s in a single equa ion in ol ing s a iona y eg esso s, allowing he b eak da es
o be di e en o o e lap. Ou amewo k is gene al and allows o gene al mixing- ype
eg esso s. The assump ions on he e o s a e mild; hei dis ibu ion can be nonno -
mal and condi ional he e oskedas ici y is pe mi ed. Ex ensions o he case wi h se ially
co ela ed e o s a e also ea ed. We p o ide he equi ed ools o add ess he ollow-
ing es ing p oblems, among o he s: (a) es ing o gi en numbe s o changes in eg es-
sion coe icien s and a iance o he e o s; (b) es ing o some unknown numbe o
changes wi hin some p especi ied maximum; (c) es ing o changes in a iance ( eg es-
sion coe icien s) allowing o a gi en numbe o changes in he eg ession coe icien s
( a iance); (d) sequen ial p ocedu es o es ima e he numbe o changes p esen . No e
ha we adop a QML app oach ins ead o one based on GMM. Ei he could be used in
p inciple. The main ad an age o using he QML app oach based on no mal e o s is
i s ha i allows a na u al ex ension o Bai and Pe on (1998) widely used in p ac ice.
Second, and mo e impo an ly pe haps, we can use he e icien algo i hm de eloped
in Qu and Pe on (2007a). This is especially impo an in he cu en con ex since e en
only wo b eaks in coe icien s and a iance implies ou possible b eak da es. Hence a
compu a ionally e icien me hod o es ima e he b eak da es is needed.
These es ing p oblems a e impo an o p ac ical applica ions; o example, doc-
umen ing s uc u al changes in he a iabili y o shocks in au o eg essi e models; see
Blancha d and Simon (2001), He e a and Pesa en o (2005), Kim and Nelson (1999),
McConnell and Pé ez-Qui ós (2000), Sensie and an Dijk (2004), and S ock and Wa -
son (2002). Gi en he lack o p ope es ing p ocedu es, a common app oach is o apply
asup-Wald ype es s(e.g.,And ews (1993), Bai and Pe on (1998)) o changes in he
mean o he absolu e alue o he es ima ed esiduals, a a he ad hoc p ocedu e. To es
o a change in a iance only (imposing no change in he eg ession coe icien s), one
can apply a CUSUM o squa es es o he es ima ed esiduals, which is adequa e only
i no change in coe icien is p esen . O en, changes in bo h coe icien s and a iance
occu a possibly di e en da es. A common me hod is o i s es o changes in he e-
g ession coe icien s and condi ioning on he b eak da es ound, hen es o changes in
a iance. This is clea ly inapp op ia e as in he i s s ep he es s su e s o se e e size
dis o ions. Also, neglec ing changes in eg ession coe icien s when es ing o changes
in a iance induces bo h size dis o ions and a loss o powe ; o example, Pe on and
Yamamo o (2019a)andPi a akis (2004). Hence, wha is needed is a join app oach. To
do so, ou es ing p ocedu es a e based on quasi likelihood a io es s using a likelihood
unc ion o iden ically and independen ly dis ibu ed no mal e o s. We hen apply co -
ec ions o ha e limi dis ibu ions ee o nuisance pa ame e s wi h nonno mal dis i-
bu ion and condi ional he e oskedas ici y. We also conside ex ensions ha allow o
se ial co ela ion.
The empi ical use ulness o ou p oposed p ocedu e is pe haps bes explained ia
applica ions ela ed o changes in he a iance o many mac oeconomic a iables (i.e.,
he g ea mode a ion); see Gadea, Gómez-Loscos, and Pé ez-Qui ós (2018)andPe on
and Yamamo o (2019b). The es ing issues o in e es a e, among o he s: (a) es ing o a
1022 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
change in a iance in 1984 ( he commonly accep ed da e o he s a o he g ea mode -
a ion); (b) es ing o an addi ional change in a iance, say ollowing he g ea ecession
o 2007; (c) es ima ing he o al numbe o changes; (d) es ing whe he any changes a e
p esen ; (e) pe o ming all hese es s allowing o changes in he pa ame e s o a con-
di ional eg ession model (e.g., a change in slope in 1973 o GDP as a gued in Pe on
(1989)); ( ) pe o ming all he co esponding es s when es ing o changes in he e-
g ession pa ame e s allowing o changes in he a iance o he e o s. Fo ins ance, an
issue o in e es in mac oeconomics is whe he he g ea mode a ion was due o changes
in he pe sis ence pa ame e s ( he sum o he au o eg essi e coe icien s) as sugges ed
by he “imp o ed policy” hypo hesis o in he e o a iance as sugges ed by he “good
luck” hypo hesis o in bo h. Ou es s allow o disen angle hese e ec s, including cases
wi h mul iple b eaks. Sec ion 7p o ides empi ical examples ela ed o in la ion and eal
in e es a e se ies. To each he igh conclusion abou he numbe and na u e o he
changes, we use all es s p oposed in his pape in a ca e ul way. Ob iously, he num-
be o po en ial o he applica ions abound. One could a gue ha i is su icien o ha e
es s o changes in pa ame e s ha a e obus o unknown pa e ns o changes in a i-
ance. An example is he wo k o Gó ecki, Ho á h, and Kokoszka (2018). Howe e , hei
es s a e based on a wo-s ep app oach; i s es ima ing he e o p ocess assuming no
coe icien b eaks and subsequen ly es ing o changes in he coe icien s using his es-
ima e. Acco dingly, he es s can su e om se e e powe losses as he es ima ed e o
p ocess can be con amina ed when s uc u al changes a e ac ually p esen in he coe -
icien s. Indeed, un epo ed simula ions show hei es s o ha e nonmono onic powe ,
ha is, powe ha dec eases as he magni ude o he change in he eg ession pa ame-
e s inc eases. This es ing p oblem is easily co e ed ia ou sup LR3T and UDmaxLR3T
es s, which main ain good powe p ope ies. Simila ly, one could be con en wi h only
es ing o a change in a iance allowing o unspeci ied changes in he eg ession pa-
ame e s. The only es s we know ha ackle his issue a e based on he supLR2T and
UDmax LR2T es s ha we p opose.
The pape is s uc u ed as ollows. Sec ion 2p esen s he models and es ing p ob-
lems, wi h he quasi-likelihood es s s a ed in Sec ion 3.Sec ion4discusses he assump-
ions needed on he eg esso s and e o s, de i es he ele an limi dis ibu ions unde
he a ious null hypo heses and p oposes co ec ed e sions o he es s ha ha e limi
dis ibu ions- ee o nuisance pa ame e s. Sec ion 4.1 deals wi h he case o ma ingale
di e ence e o s, Sec ion 4.2 ex ends he analysis o se ially co ela ed e o s, Sec ion 4.3
co e s he case wi h an unknown numbe o b eaks. Sec ion 4.4 discusses es s o an
addi ional b eak in ei he he eg ession coe icien s o he a iance. Sec ion 5p o ides
simula ion esul s o assess he adequacy o he sugges ed p ocedu es in e ms o hei
ini e sample size and powe and p o ides some p ac ical guidelines. Sec ion 6discusses
me hods o es ima e he numbe o b eaks in he eg ession coe icien s and he a i-
ance. Sec ion 7p o ides empi ical applica ions and Sec ion 8b ie concluding ema ks.
An Appendix con ains some echnical de i a ions. Addi ional ma e ial can be ound in
he Online Supplemen al Ma e ial in he eplica ion ile (Pe on, Yamamo o, and Zhou
(2020)).

Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1023
2. Model and es ing p oblems
We s a wi h a desc ip ion o he mos gene al speci ica ion o he model conside ed
whe e mul iple b eaks occu in bo h he coe icien s o he condi ional mean and he
a iance o he e o s, a possibly di e en imes. This will allow us o se up he no a ion
used h oughou he pape . The main amewo k o analysis can be desc ibed by he
ollowing mul iple linea eg ession wi h mb eaks (o m+1 egimes) in he condi ional
mean equa ion:
y =x
β+z
δj+u  =Tc
j−1+1Tc
j(1)
o j=1m+1.In hismodel,y is he obse ed dependen a iable a ime ;bo h
x (p×1)andz (q×1)a e ec o so co a ia esandβand δj(j=1m+1)a e
he co esponding ec o s o coe icien s; u is he dis u bance a ime . The b eak
da es (Tc
1Tc
m)a e explici ly ea ed as unknown (wi h he con en ion Tc
0=0and
Tc
m+1=Tused). This is a pa ial s uc u al change model since he pa ame e ec o
βis no subjec o shi s and is es ima ed using he en i e sample. When p=0,we
ob ain a pu e s uc u al change model wi h all coe icien s subjec o change. We also
allow o nb eaks (o n+1 egimes) o he a iance o he e o s occu ing a un-
known da es (T
1T
n). Acco dingly, E(u )=0and E(u2
)=σ2
i o T
i−1+1≤ ≤T
i
(i=1n +1), whe e again we use he con en ion ha T
0=0and T
n+1=T.We
allow he b eaks in he a iance and in he eg ession coe icien s o happen a di -
e en imes, hence he m- ec o (Tc
1Tc
m)and he n- ec o (T
1T
n)can ha e
all dis inc elemen s o hey can o e lap pa ly o comple ely. We le Kdeno e he
o al numbe o b eak da es and max[m n]≤K≤m+n. When he he b eaks o e -
lap comple ely, m=n=K. The mul iple linea eg ession sys em (1) may be ex-
p essed in ma ix o m as Y=Xβ +¯
Zδ +U,whe eY=(y1yT),X=(x1xT),
U=(u1uT),δ=(δ
1δ
m+1),and ¯
Zdiagonally pa i ions Za (T c
1Tc
m),
ha is, ¯
Z=diag(Z1Zm+1)wi h Zj=(zTc
j−1+1zTc
j). The ue alue o he pa-
ame e s a e δ0=(δ0
1δ0
m+1)and (Tc0
1Tc0
m)and ¯
Z0diagonally pa i ions Z
a (Tc0
1Tc0
m). Hence, he da a-gene a ing p ocess (DGP) is Y=Xβ0+¯
Z0δ0+U
wi h E(UU)=Ω0, whe e he diagonal elemen s o Ω0a e σ2
i0 o T 0
i−1+1≤ ≤T 0
i
(i=1n+1). We also conside cases wi h se ial co ela ion in he e o s o which
he o -diagonal elemen s o Ω0need no be 0. This is a special case o he class o mod-
els conside ed by Qu and Pe on (2007a). Thei me hod o es ima ion is quasi maximum
likelihood (QML) assuming se ially unco ela ed Gaussian e o s. They p o e consis-
ency o he es ima es o he b eak ac ions (λ0
1λ0
K)≡(T0
1/TT0
K/T),whe e
T0
i(i=1K) deno es he union o he elemen s o (Tc0
1Tc0
m)and (T 0
1T 0
n).
This is done unde gene al condi ions on he eg esso s and he e o s; see Sec ion 4.Im-
po an ly, om a p ac ical pe spec i e, hey p o ide an e icien es ima ion algo i hm,
which we build upon.
The es ing p oblems a e he ollowing: TP-1: H0:{m=n=0} e sus H1:{m=0
n=na};TP-2:H0:{m=man=0} e sus H1:{m=man=na};TP-3:H0:{m=0n=na}
e sus H1:{m=man=na};TP-4:H0:{m=n=0} e sus H1:{m=man=na},whe e
maand naa e some posi i e numbe s selec ed a p io i. We shall also conside es -
ing p oblems whe e he al e na i es speci y some unknown numbe s o b eaks, up o
1024 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
some maximum. These a e: TP-5: H0:{m=n=0} e sus H1:{m=01≤n≤N};TP-6:
H0:{m=man=0} e sus H1:{m=ma1≤n≤N};TP-7:H0:{m=0n=na} e sus
H1:{1≤m≤Mn =na};TP-8:H0:{m=n=0} e sus H1:{1≤m≤M1≤n≤N}.
We shall deal wi h: TP-9: {m=man =na} e sus H1:{m=ma+1n =na}; TP-10:
{m=man=na} e sus H1:{m=man=na+1},whe emaand nanonnega i e in ege s.
These a e use ul o e i y he adequacy o a model wi h some numbe o b eaks assess-
ing whe he including one mo e is wa an ed. In Sec ion 6, we also conside sequen ial
es ing p ocedu es ha allow es ima ing he numbe o b eaks in bo h δand σ2.
3. The quasi-likelihood a io es s
We conside he likelihood a io (LR) es s ob ained assuming no mally dis ibu ed and
se ially unco ela ed e o s, o TP-1 o TP-4. We es ima e he model using he quasi-
maximum likelihood es ima ion me hod (QMLE). Conside TP-1 wi h no change in δ
(m=q=0) and es ing o nachanges in σ2.Unde H0, he log-likelihood unc ion is
log ˜
LT=−(T/2)(log 2π+1)−(T/2)log ˜σ2(2)
whe e ˜σ2=T−1T
=1(y −x
˜
β)2and ˜
β=(T
=1x x
)−1(T
=1x y ).Unde H1, o agi en
pa i ion {T
1T
n}, he log-likelihood alue is gi en by
log ˆ
LTT
1T
n=−(T/2)(log 2π+1)−
na+1

i=1T
i−T
i−1/2log ˆσ2
i(3)
whe e he QMLE join ly sol es ˆ
β=(na+1
i=1T
i
=T
i−1+1x x
/ˆσ2
i)−1(na+1
i=1T
i
=T
i−1+1x y /
ˆσ2
i)and ˆσ2
i=(T
i−T
i−1)−1T
i
=T
i−1+1(y −x
ˆ
β)2, o i=1na+1.Hence, heSup-LR
es is
supLR1T (naε|m=n=0)=sup
(λ
1λ
na)∈Λ ε
2log ˆ
LTT
1T
na−log ˜
LT
=2log ˆ
LTˆ
T
1 ˆ
T
na−log ˜
LT
whe e (ˆ
T
1 ˆ
T
na)a e he QMLE ob ained imposing he es ic ion o no change in
he coe icien s and Λ ε ={(λ
1λ
na);|λ
i+1−λ
i|≥ε(i=1na−1)λ
1≥ε λ
na≤
1−ε},wi hεa unca ion imposing a minimal leng h o each segmen . Fo TP-2,
he e a e mab eaks in δunde bo h H0and H1, so he es pe ains o assess whe he
he e a e 0o nab eaks in a iance. Fo a gi en pa i ion {Tc
1Tc
ma}, he likelihood
unc ion unde H0is log 
LT(Tc
1Tc
ma)=−(T/2)(log 2π+1)−(T/2)log ˜σ2,whe e
˜σ2=T−1T
=1(y −x
˜
β−z
˜
δ j)2,˜
β=(XM¯
ZX)−1XM¯
ZYand 
δ j =(Z
jZj)−1Zj(Yj−
Xj˜
β) o Tc
j−1< ≤Tc
j,wi hM¯
Z=I−¯
Z( ¯
Z¯
Z)−1¯
Z,¯
Z=diag(Z1Zma+1),andZj=
(zTc
j−1+1zTc
j),Yj=(yTc
j−1+1yTc
j),Xj=(xTc
j−1+1xTc
j) o Tc
j−1< ≤Tc
j(j=
1ma+1). The log-likelihood alue unde H1is, o gi en pa i ions {Tc
1Tc
ma}
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1025
and {T
1T
na},
log ˆ
LTTc
1Tc
ma;T
1T
na=−(T/2)(log 2π+1)−
na+1

i=1T
i−T
i−1/2log ˆσ2
i(4)
whe e he QMLE sol es he ollowing equa ions: ˆσ2
i=(T
i−T
i−1)−1T
i
=T
i−1+1(y −
x
ˆ
β−z
ˆ
δ j)2(i=1na+1)and ˆ
β=(XσM¯
ZσXσ)−1XσM¯
ZσYσ,whe eM¯
Zσ=
I−¯
Zσ(¯
Z
σ¯
Zσ)−1¯
Z
σwi h ¯
Zσ=diag(Zσ
1Zσ
ma+1),Zσ
j=(zσ
Tc
j−1+1zσ
Tc
j
),andzσ
=
(z /ˆσi), o T
i−1< ≤T
i(i=1na+1). Also, ˆ
δ j =(Zσ
jZσ
j)−1Zσ
j(Yσ
j−Xσ
jˆ
β) o
Tc
j−1< ≤Tc
j,whe eYσ
j=(yσ
Tc
j−1+1yσ
Tc
j
),Xσ
j=(xσ
Tc
j−1+1xσ
Tc
j
)wi h xσ
=(x /ˆσi)
and yσ
=(y /ˆσi).Hence,
supLR2T (manaε|n=0ma)
=2sup
(λc
1λc
ma;λ
1λ
na)∈Λε
log ˆ
LTTc
1Tc
ma;T
1T
na
−sup
(λc
1λc
ma)∈Λcε
log ˜
LTTc
1Tc
ma
=2log ˆ
LT˜
Tc
1 ˜
Tc
ma;˜
T
1 ˜
T
na−log ˜
LTˆ
Tc
1 ˆ
Tc
ma
whe e Λcε ={(λc
1λc
m);|λc
j+1−λc
j|≥ε(j=1ma−1)λc
1≥ε λc
ma≤1−ε}and
Λε=λc
1λc
mλ
1λ
n; o (λ1λK)=λc
1λc
m∪λ
1λ
n
|λj+1−λj|≥ε(j=1K−1) λ1≥ε λK≤1−ε(5)
No e ha we deno e he es ima es o he b eak da es in coe icien s and a iance by a
“∼” when hese a e ob ained join ly, and by a “^” when ob ained sepa a ely.
The se Λεwhich de ines he possible alues o he b eak ac ions in δ(λc
1λc
m)
and in σ2(λ
1λ
m) allows hem o ha e some (o all) common elemen s o be com-
ple ely di e en . Wha is impo an is ha each b eak ac ion be sepa a ed by some
ε>0. This does complica e in e ence since many cases need o be conside ed. To illus-
a e, conside ma=na=1.Wecanha eK=1, a one b eak model wi h bo h δand σ2
changing a he same da e. On he o he hand, i K=2, he b eak da e o he change in
δis di e en om ha o he change in σ2. This leads o wo addi ional possible cases:
(a) λc
1≤λ
1−ε( he b eak in δis be o e ha in σ2), (b) λc
1≥λ
1+ε( he b eak in δis a -
e ha in σ2). The maximized likelihood unc ion o hese wo cases can be e alua ed
using he algo i hm o Qu and Pe on (2007a) since i pe mi s imposing es ic ions. Fo
example, i λc
1≤λ
1−ε, we ha e a wo b eak model and he es ic ions a e ha he e o
a iances in he i s and second egimes a e iden ical, and he coe icien s a e he same
in he second and hi d egimes. Hence, o he case ma=na=1, he e a e h ee maxi-
mized likelihood alues o cons uc and he es co esponds o he maximal alue o e
hese h ee cases. When mao naa e g ea e han one, mo e cases need o be conside ed,
bu he p inciple is he same.
1026 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
Fo TP-3, H0speci ies nab eaks in σ2and none in δ. Fo a pa i ion {T
1T
na},
he likelihood unc ion is log 
LT(T
1T
na)=−(T/2)(log 2π+1)−na+1
i=1[(T
i−
T
i−1)/2]log ˜σ2
i,whe e ˜σ2
i=(T
i−T
i−1)−1T
i
=T
i−1+1(y −x
˜
β−z
˜
δ)2 o i=1na+1,
wi h (˜
β˜
δ)=(W σWσ)−1WσYσ,Wσ=(wσ
1wσ
T)and wσ
=(xσ
zσ
).Unde H1,
he e a e mab eaks in δand nab eaks in σ2and he likelihood unc ion is (4). The sup-
LR es is
supLR3T (manaε|m=0na)
=2sup
(λc
1λc
ma;λ
1λ
na)∈Λε
log ˆ
LTTc
1Tc
ma;T
1T
na
−sup
(λ
1λ
na)∈Λ ε
log ˜
LTT
1T
na
=2log ˆ
LT˜
Tc
1 ˜
Tc
ma;˜
T
1 ˜
T
na−log ˜
LTˆ
T
1 ˆ
T
na
Fo TP-4, unde H0we ha e no b eak and he log-likelihood unc ion is (2). H1speci ies
mab eaks in δand nab eaks in σ2and he log likelihood is (4). Hence, he Sup-LR es is
supLR4T (manaε|n=m=0)
=2sup
(λc
1λc
ma;λ
1λ
na)∈Λε
log ˆ
LTTc
1Tc
ma;T
1T
na−log ˜
LT
=2log ˆ
LT˜
Tc
1 ˜
Tc
ma;˜
T
1 ˜
T
na−log ˜
LT(6)
4. The limi ing dis ibu ions o he es s
The limi dis ibu ion o he es s o ma ingale di e ence e o s is p esen ed in Sec-
ion 4.1 wi h ex ensions o se ially co ela ed e o s in Sec ion 4.2.Sec ion4.3 deals wi h
double maximum es s and Sec ion 4.4 wi h es s o an addi ional b eak; “→p” deno es
con e gence in p obabili y, “⇒” weak con e gence unde he Sko ohod opology and
·is he Euclidean no m.
4.1 The case wi h ma ingale di e ence e o s
When σ2is cons an unde H0bu allowed o change unde H1(TP-1, 2, 4), we speci y:
•Assump ion A1. The e o s {u } o m an a ay o ma ingale di e ences ela i e o
F =σ- ield{z −1z x −1x u −2u −1},E(u2
)=σ2
0 o all and T−1/2×
[Ts]
=1(u2
/σ2
0−1)⇒ψW (s),whe e W(s)is a Wiene p ocess and ψ=limT→∞ a (T−1/2×
T
=1(u2
/σ2
0−1)).
Assump ion A1 ules ou ins abili y in he e o p ocess and s a es ha a basic unc-
ional cen al limi heo em holds o he pa ial sums o he squa ed e o s. When
changes in he coe icien s a e es ed (TP-3 and TP-4), we assume, wi h w =(x
z
):
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1033
(b) Fo TP-6, unde Assump ions A1 and A3:
UDmax LR2T =max
1≤na≤Nn−1
asupLR∗
2T (manaε|n=0ma)
⇒max
1≤na≤Nn−1
asup
(λ
1λ
na)∈Λc
ε
na

i=1λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
≤max
1≤na≤Nn−1
asup
(λ
1λ
na)∈Λ ε
na

i=1λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
(c) Fo TP-7, unde Assump ions A2 and A3:
UDmax LR3T =max
1≤ma≤Mm−1
asupLR3T (manaε|m=0na)
⇒max
1≤ma≤Mm−1
asup
(λc
1λc
ma)∈Λ
cε
ma

j=1λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
≤max
1≤ma≤Mm−1
asup
(λc
1λc
ma)∈Λcε
ma

j=1λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
(d) Fo TP-8, unde Assump ions A1 and A2:
UDmax LR4T =max
1≤na≤Nmax
1≤ma≤M(na+ma)−1supLR∗
4T (manaε|n=m=0)
⇒max
1≤na≤Nmax
1≤ma≤M(na+ma)−1
×sup
(λc
1λc
ma;λ
1λ
na)∈Λεma

j=1λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
+
na

i=1λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
≤max
1≤na≤Nmax
1≤ma≤M(na+ma)−1
×sup
(λc
1λc
ma;λ
1λ
na)∈Λc εma

j=1λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
+
na

i=1λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
Fo TP-5 o TP-7, he c i ical alues o he limi dis ibu ions a e a ailable in Bai and
Pe on (1998,2003b) o No Mequal o 5. Fo TP-5 and TP-6, he esul s a e alid o
ma ingale di e ences o se ially co ela ed e o s. This is no he case o TP-7 and TP-8

1034 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
o easons discussed abo e. We hen conside he maximum o he Wald- ype es s dis-
cussed Sec ion 4.2. The limi dis ibu ion applicable o TP-8 is new. Table 1p esen s c i -
ical alues ob ained using simula ions as discussed abo e o he case o a ixed numbe
o b eaks unde H1, o ε=01015and 020,and alueso Mand Nup o 2;seePe on
and Yamamo o (2019b) o addi ional c i ical alues wi h MN =234.
4.4 Tes ing o an addi ional b eak
We now conside TP-9 and TP-10, which assess whe he including an addi ional b eak
is wa an ed. Le (˜
Tc
1 ˜
Tc
m;˜
T
1 ˜
T
n)be he es ima es o he b eak da es in δand
σ2ob ained join ly by maximizing he quasi-likelihood unc ion assuming mb eaks in
δand nb eaks in σ2. Fo TP-9, he issue is whe he an addi ional b eak in δis p esen .
The es is
supSeq9T (m +1n|m n) =max
1≤j≤m+1sup
τ∈Λc
jε
log ˆ
LT˜
Tc
1 ˜
Tc
j−1τ ˜
Tc
j ˜
Tc
m;˜
T
1 ˜
T
n
−log ˆ
LT˜
Tc
1 ˜
Tc
m;˜
T
1 ˜
T
n
whe e Λc
jε ={τ;˜
Tc
j−1+(˜
Tc
j−˜
Tc
j−1)ε ≤τ≤˜
Tc
j−(˜
Tc
j−˜
Tc
j−1)ε}. This amoun s o pe o m-
ing m+1 es s o a single b eak in δ o each o he m+1 egimes de ined by he pa -
i ion {˜
Tc
1 ˜
Tc
m}. No e ha he e a e di e en scena ios when allowing b eaks in δ
and in σ2 o happen a di e en da es, since (˜
Tc
1 ˜
Tc
m)and (˜
T
1 ˜
T
n)can pa ly
o comple ely o e lap o be al oge he di e en . This implies wo possible cases: (1) i
he nb eak da es in σ2a e a subse o he mb eak da es in δ, he e is no a iance
b eak be ween ˜
Tc
j−1and ˜
Tc
j; (2) o he wise, he e is one o mo e a iance b eaks be-
ween ˜
Tc
j−1and ˜
Tc
j. In ei he cases, one can appeal o he esul s o Theo em 1(c) wi h
ma=1since any alue o nais allowed, including 0. I is hen easy o deduce ha , in
he case o ma ingale e o s, he limi dis ibu ion o he es is, unde Assump ions
A2 and A3,limT→∞P(supSeq9T (m +1n|mn) ≤x) =Gqε(x)m+1,whe eGqε(x) is he
cumula i e dis ibu ion unc ion o he andom a iable supλ∈Λ1ε (Wq(λ) −λWq(1))2/
(λ(1−λ)),whe eΛ1ε ={λ;ε<λ<1−ε}. The c i ical alues o he dis ibu ion unc ion
Gqε(x)m+1can be ound in Bai and Pe on (1998,2003b). Wi h se ial co ela ion in he
e o s, he p inciple is he same excep ha he s a is ic is based on he obus Wald es
supF3T as de ined by (13) applied o a one b eak es o each segmen . Fo TP-10, sim-
ila conside a ions apply. He e, he issue is whe he an addi ional b eak in he a iance
is p esen . The es s a is ic is
supSeq10T (m n +1|mn)
=(2/ˆ
ψ) max
1≤i≤n+1sup
τ∈Λ
iε
log ˆ
LT˜
Tc
1 ˜
Tc
m;˜
T
1 ˜
T
i−1τ ˜
T
i ˜
T
m
−log ˆ
LT˜
Tc
1 ˜
Tc
m;˜
T
1 ˜
T
n
whe e Λ
iε ={τ;˜
T
i−1+(˜
T
i−˜
T
i−1)ε ≤τ≤˜
T
i−(˜
T
i−˜
T
i−1)ε}. The co ec ion ac o (2/ˆ
ψ)
is needed o ensu e ha he limi dis ibu ion o he es is ee o nuisance pa ame e s
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1035
when he e o s a e allowed o be nonno mal, se ially co ela ed and condi ionally he -
e oskedas ic. One can hen use pa (b) o Theo em 1 o deduce ha , unde Assump ions
A1 and A3 applied o each segmen s unde H0:limT→∞ P(supSeq10T (mn +1|m n) ≤
x) =G1ε(x)n+1.
4.5 Local asymp o ic powe
Supplemen D con ains de ails abou he local asymp o ic powe unc ion o selec ed
es s. We b ie ly summa ize he ele an esul s. We conside model (1) ocusing on he
case o n=m=1wi h he ollowing assump ions.
•Assump ion L1. Assump ions A1 and A3 hold wi h σ20 −σ10 =σ∗/√T.We also
ha e T−1/2[Ts]
=1[(uσ
)2−1]⇒ψW (s) wi h ψ=limT→∞ a (T −1/2T
=1[(uσ
)2−1])and
T−1[Ts]
=1(uσ
)2p
→suni o mly in s.
•Assump ion L2. Assump ions A2 and A3 hold wi h δ0
2−δ0
1=δ∗/√T.
We de i e he local asymp o ic powe o he es s supLR2T (n =1m =1ε|n=0
m=1)and supLR3T (m =1n =1ε|m=0n =1)and he co esponding es s wi h
no nuisance b eaks accoun ed o , ha is, supLR1T and he s anda d supLRT es .
Lemma S.1 shows ha he local asymp o ic powe o he sup LR2T es coincides wi h
ha o sup LR1T excep ha he se o pe missible b eak da es Λc
 is smalle han
Λ , which has no p ac ical e ec . Lemma S.2 shows ha he local asymp o ic powe
o supLR3T is he same as ha o sup LRTde i ed in And ews (1993, Theo em 4),
again excep ha he se o pe missible b eak da es is Λ
c ins ead o Λc.Hence,when
es ing o changes in a iance ( esp., coe icien s) allowing o changes in coe icien s
( esp., a iance), we ha e he same local asymp o ic powe unc ion as when es ing o
changes in a iance ( esp., coe icien s) when no change in coe icien ( esp., a iance)
is p esen . Hence, he e is no loss in local asymp o ic powe adop ing ou mo e gene al
app oach.
We also de i ed he local asymp o ic powe unc ion o he CUSQ es (see (14)below
o i s de ini ion) and compa ed i o ha o he sup LR1T and sup LR2T es s. Figu e S.1
shows he asymp o ic local powe unc ions o he sup LR1T and CUSQ es s when a
b eak in a iance occu s a λ 0=0305and 07and no b eak occu s in he coe i-
cien s. They show he local asymp o ic powe unc ions o be almos iden ical. Figu e S.2
p esen s he local asymp o ic powe unc ions o he supLR2T es when i accoun s o
a coe icien b eak a λc0=0305o 07. I also shows ha he local asymp o ic powe
unc ions o he CUSQ es unde he assump ion o no b eak in he coe icien s. This
simula ion design gi es an ad an age o he CUSQ. Indeed, he powe o he sup LR2T
es is sligh ly lowe when he a iance and he coe icien b eak da es coincide. This
is because he pe missible b eak da es a ound he ue b eak da e a e no conside ed
due o he concu en nuisance b eak. Howe e , he powe loss o he sup LR2T es is
e y mino . The powe o bo h es s a e almos iden ical e en hough he sup LR2T es
conside s a single nuisance b eak when he wo b eaks a e a apa . ha is, he case o
(λ 0λc0)=(0307)and (0703).
1036 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
5. Mon e Ca lo expe imen s
We p o ide simula ion esul s o assess he size and powe p ope ies o some es s
p oposed; Sec ion 5.1 o a iance b eaks, Sec ion 5.2 o condi ional es s, Sec ion 5.3
o he sup LR∗
4T and UDmax es s. Supplemen E p o ides addi ional esul s o he
supLR1T and sup LR2T es s wi h nonno mal e o s. Following Bai and Ng (2005), we
use: (a) he dis ibu ion wi h 5 deg ees o eedom, (b) a mix u e o wo no mal dis i-
bu ions: 1I(z ≤05)+ 2I(z > 05),whe ez∼U[01], 1∼N(−11)and 2∼N(11),
(c) he χ2dis ibu ion wi h 5 deg ees o eedom and (d) an exponen ial dis ibu ion
−ln( ), ∼U[01]. The esul s show ha he exac size o he es s is simila ly close o
he nominal size. As expec ed, powe is lowe o all dis ibu ions, hough he ex en o
he powe loss is mino and he es s emain in o ma i e. Ou es s o changes in a i-
ance e ain hei powe ad an age o e he CUSQ es .
5.1 Tes ing o a iance b eaks only
We now conside he case o es ing only o a iance b eaks assuming no change in δ.
We in es iga e he p ope ies o he ollowing es s: he supLR∗
1T (naε|m=n=0),ab-
b e ia ed sup LR∗
1T (naε)and he UD max LR1T o an unknown numbe o b eaks up
o N=5. We also conside a co ec ed e sion o he CUSUM o squa es es o B own,
Du bin, and E ans (1975), as ex ended by Deng and Pe on (2008), gi en by
CUSQ =sup
λ∈[01]T−1/2[Tλ]

=1˜
2
−[Tλ]/TT

=1˜
2
/ˆϕ1/2
a(14)
wi h ˆϕa=T−1(T−1)
j=−(T−1)ω(jbT)T
=|j|+1ˆη ˆη −j,whe e ˆη =˜
2
−ˆσ,ˆσ2=T−1T
=1˜
2
and ˜
deno es he ecu si e esiduals. Also ω(jbT)is he quad a ic spec al ke nel
and he bandwid h bTis selec ed using And ews’(1991)me hodwi hanAR(1)ap-
p oxima ion. The aim o he design is o add ess he ollowing issues: (a) he size o
he supLR∗
1T (naε) and UDmax LR1T es s; (b) he ela i e powe o he h ee es s;
(c) he powe losses ob ained when unde speci ying he numbe o b eaks; (d) he el-
a i e powe o UDmax LR1T compa ed o sup LR∗
1T (naε) wi h naspeci ied o be he
ue numbe o b eaks. We conside a dynamic model wi h GARCH e o s, o which
he DGP is gi en by y =c+αy −1+e ,e =u √h ,u ∼i.i.d. N(01),h =τ1+τ21( >
[05T])+γe2
−1+ρh −1,whe ewese h0=τ1/(1−γ−ρ),c=05,τ1=01,andε=015.
We conside α=0207and he GARCH(11)coe icien s a e se o γ=010305,and
ρ=02. The size and powe o 5% nominal size es s a e e alua ed a T=100200.The
magni ude o he change τ2 a ies be ween 0(size) and 03. The esul s a e p esen ed in
Table 2.Thesup LR∗
1T (1ε)and UD maxLR1T es s show size dis o ions when γ=05
wi h T=100 bu he size is close o 5% when T=200. The CUSQ es is sligh ly unde -
sized. The UD maxLR1T es has powe close o ha o sup LR∗
1T (1ε), despi e ha ing a
b oade ange o al e na i es. The powe o he la e wo es s domina es ha o CUSQ
especially when T=100. Supplemen F shows he esul s o be obus o a s a ic mean
model wi h no mal e o s.
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1037
Table 2. Size and powe o he supLR∗
1T (na=1ε),UDmaxLR1T and CUSQ es s in a dynamic
model wi h GARCH(11)e o s.
T=100
α=02α=07
γ=01γ=03γ=05γ=01γ=03γ=05
τ2LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ
00059 0059 0029 0083 0086 0039 0098 0099 0042 0066 0061 0029 0078 0084 0038 0097 0092 0039
005 0171 0167 0158 0165 0171 0103 0151 0155 0082 0164 0158 0149 0147 0149 0100 0137 0140 0080
010396 0373 0354 0307 0307 0232 0224 0228 0136 0383 0367 0356 0300 0297 0232 0218 0224 0138
015 0593 0575 0574 0432 0409 0349 0312 0312 0199 0591 0573 0564 0425 0414 0330 0307 0308 0201
020744 0725 0693 0542 0542 0446 0415 0408 0270 0741 0723 0684 0534 0534 0441 0384 0385 0259
030902 0888 0851 0741 0738 0626 0535 0540 0370 0897 0887 0856 0724 0724 0624 0534 0534 0376
T=200
α=02α=07
γ=01γ=03γ=05γ=01γ=03γ=05
τ2LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ LR UDmax CUSQ
00049 0044 0034 0058 0060 0035 0064 0063 0045 0055 0056 0036 0061 0064 0034 0060 0061 0040
005 0315 0311 0335 0217 0202 0203 0129 0123 0110 0311 0303 0332 0208 0202 0205 0122 0115 0100
010709 0692 0751 0446 0431 0455 0263 0249 0225 0702 0682 0734 0442 0428 0448 0257 0241 0222
015 0918 0910 0928 0672 0648 0649 0404 0384 0345 0918 0912 0923 0648 0641 0643 0386 0370 0335
020980 0977 0979 0780 0764 0764 0510 0497 0456 0981 0980 0981 0777 0766 0763 0496 0489 0441
030997 0996 0997 0910 0903 0878 0682 0662 0601 0997 0997 0998 0903 0898 0877 0676 0654 0606
No e:DGP:y =c+αy −1+e ,e =u √h ,wi hu ∼i.i.d. N(01),h =τ1+τ21( > [05T])+γe2
−1+ρh −1,h0=τ1/(1−
γ−ρ),c=05,τ1=01,ρ=02;ε=015.
We now u n o a case wi h wo b eaks in a iance. The DGP is y =e ;e ∼
i.i.d. N(01+θ1(T
1< ≤T
2)), ha is, he a iance inc eases a T
1and e u ns o i s
o iginal le el a T
2. We conside wo scena ios: {T
1=[03T]T
2=[06T]} and {T
1=
[02T]T
2=[08T]}.Wese T=200 and ε=010015. The magni ude o he b eak in
σ2 a ies be ween θ=0(size) and θ=3. We again conside he UDmaxLR1T es wi h
N=5bu include bo h he sup LR∗
1T (1ε) es o a single b eak and he sup LR∗
1T (2ε)
es o wo b eaks o assess he ex en o powe gains when speci ying he co ec num-
be o b eaks. The esul s a e p esen ed in Table 3. Conside i s he size o he es s. The
supLR∗
1T (1ε),supLR∗
1T (2ε)and UDmaxLR1T es s a e sligh ly conse a i e and he
CUSQ e en mo e so wi h an exac size o 0025. As expec ed, powe inc eases as εin-
c eases since he ange o al e na i es is smalle . When compa ing he supLR∗
1T (1ε)
and supLR∗
1T (2ε) es s, he la e is mo e powe ul, indica ing ha allowing o he
co ec numbe o b eaks imp o es powe . The UDmax LR1T es has powe be ween
hose o he sup LR∗
1T (1ε)and supLR∗
1T (2ε) es s. These es s a e conside ably mo e
powe ul han he CUSQ, which has li le powe .
5.2 Condi ional es s
We now conside he p ope ies o he es s ha condi ion on ei he b eaks in coe i-
cien s ( esp., a iance) when es ing o changes in a iance ( esp., coe icien s). Con-
side i s he size and powe o supLR∗
2T (manaε|n=0ma)which es s o nachanges
in σ2condi ional on machanges in δwi h ε=0102.Wese ma=na=1and he DGP is
1038 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
Table 3. Size and powe o he supLR∗
1T (naε),UDmax LR1T and CUSQ es s wi h no mal
e o s and wo a iance b eaks.
T
1=[03T],T
2=[06T]T
1=[02T],T
2=[08T]
ε=010 ε=015 ε=010 ε=015
θn
a=1na=2UDmax na=1na=2UDmax CUSQ na=1na=2UDmax na=1na=2UDmax CUSQ
00035 0034 0036 0033 0025 0030 0025 0035 0034 0036 0033 0025 0030 0025
025 0049 0040 0045 0066 0054 0064 0031 0067 0043 0062 0063 0052 0064 0035
050111 0120 0103 0117 0159 0121 0059 0158 0138 0139 0166 0170 0165 0036
075 0164 0260 0195 0171 0294 0209 0085 0263 0283 0265 0276 0360 0287 0044
10213 0418 0289 0239 0493 0340 0124 0390 0472 0390 0428 0520 0442 0061
125 0291 0575 0404 0328 0674 0495 0147 0538 0647 0558 0563 0707 0606 0053
150356 0703 0513 0405 0778 0613 0197 0647 0780 0676 0706 0837 0731 0065
20456 0835 0701 0530 0893 0761 0276 0798 0915 0841 0828 0946 0868 0083
250621 0935 0848 0686 0959 0882 0375 0907 0971 0931 0930 0986 0950 0133
30693 0959 0895 0728 0983 0919 0430 0943 0987 0961 0963 0993 0977 0120
No e:DGP:y =e ;e ∼i.i.d. N(01+θ1(T
1< ≤T
2)),T=200.
a simple mean shi model wi h a change o magni ude μ2a mid-sample wi h i.i.d. no -
mal e o s ha ing a change in a iance o magni ude θ(unde H1) ha occu s a [025T].
The esul s o size a e p esen ed in Table 4. The es is sligh ly conse a i e and mo e
so as he imming is la ge . This is due o he ac ha he limi dis ibu ion used is an
uppe bound. The esul s o powe a e p esen ed in Table 5. I inc eases apidly wi h
he magni ude o he a iance b eak θand wi h T. I also ma ginally inc eases wi h he
alue o he imming ε.
We nex in es iga e he size and powe o sup LR∗
3T (manaε|m=0na)which es s
o machanges in δcondi ional on nachanges in σ2wi h ε=0102.Weagainse
ma=na=1and conside he mean model in which σ2changes a mid-sample. We also
conside an AR(1)model y =c+αy −1+e wi h c=05,α=05and e being i.i.d. no mal
Table 4. Size o he sup LR∗
2T (ma=1na=1ε|n=0ma=1) es wi h di e en imming pa-
ame e εin he case o no mal e o s.
T=100 T=200
μ2 ε01015 02025 01015 02025
00045 0042 0030 0023 0039 0032 0030 0031
010038 0028 0033 0030 0045 0046 0036 0037
025 0037 0039 0034 0030 0034 0034 0035 0030
050037 0035 0036 0033 0031 0025 0029 0027
075 0043 0047 0046 0041 0044 0033 0035 0031
10034 0031 0031 0031 0034 0027 0020 0017
20030 0023 0028 0028 0041 0029 0028 0029
40034 0032 0031 0027 0034 0026 0024 0026
10 0038 0033 0032 0031 0038 0033 0025 0022
20 0031 0030 0035 0027 0040 0034 0023 0021
No e:DGP:y =μ1+μ21( > [05T])+e ,e ∼i.i.d. N(01),μ1=0.

Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1039
Table 5. Powe o he supLR∗
2T (ma=1na=1ε|n=0ma=1) es wi h di e en imming
pa ame e εin he case o no mal e o s.
T=100
ε=01ε=02
θ μ2001052 41020001052 41020
025 0063 0046 0047 0056 0065 0063 0053 0056 0040 0043 0049 0045 0044 0047
050101 0094 0089 0090 0099 0096 0101 0091 0092 0097 0077 0096 0091 0101
075 0150 0162 0133 0168 0177 0181 0178 0168 0174 0160 0176 0177 0176 0171
10237 0233 0218 0212 0222 0244 0242 0270 0285 0226 0225 0231 0236 0235
125 0270 0300 0319 0293 0353 0362 0327 0318 0323 0335 0316 0375 0383 0321
150388 0379 0378 0419 0417 0448 0398 0443 0431 0435 0425 0448 0462 0445
20533 0519 0496 0557 0556 0598 0559 0592 0586 0558 0588 0602 0620 0594
30760 0771 0771 0779 0830 0843 0802 0827 0823 0825 0822 0857 0863 0838
40887 0876 0865 0892 0908 0909 0916 0921 0910 0920 0924 0927 0943 0940
T=200
ε=01ε=02
θ μ2001052 41020001052 41020
025 0052 0066 0066 0077 0084 0092 0090 0063 0067 0059 0073 0074 0067 0071
050175 0177 0153 0204 0178 0207 0219 0205 0188 0165 0216 0185 0199 0212
075 0311 0352 0340 0361 0382 0369 0365 0383 0385 0364 0376 0384 0385 0381
10485 0506 0469 0518 0553 0529 0567 0551 0566 0529 0542 0585 0574 0599
125 0648 0643 0660 0716 0716 0717 0741 0695 0685 0694 0729 0745 0760 0770
150771 0771 0773 0821 0827 0842 0821 0834 0813 0824 0852 0851 0871 0851
20918 0907 0928 0933 0962 0942 0955 0943 0943 0953 0950 0972 0961 0973
30990 0996 0992 0996 0999 0998 0996 0997 0998 0996 0996 0999 0999 0998
40998 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
No e:DGP:y =μ1+μ21( > [05T])+e ,e ∼i.i.d. N(01+θ1( > [025T])).
e o s ha ing a change in a iance a [05T]wi h magni ude θ. This is done o in es i-
ga e po en ial size dis o ions due o la ge a iance changes. As discussed in Sec ion 4.1,
a change in a iance induces a change in he ma ginal dis ibu ion o he eg esso s
when lagged dependen a iables a e included. The esul s o he size o he es s a e
p esen ed in Table 6. The size unde he mean model is close o he nominal le el bu
he es becomes conse a i e as εinc eases since he limi ing dis ibu ion used is a
bound. The size unde he AR(1)model is e y simila wi h he dis o ions being e en
smalle . This indica es ha he sh inking a iance assump ion is no binding. The e-
sul s o powe a e p esen ed in Table 7 o he mean model wi h a coe icien change
a [025T]. The powe quickly inc eases as he b eak magni ude θand Tinc ease. The
powe again ma ginally inc eases wi h ε.
5.3 Size and powe o he supLR∗
4T and UDmax LR4T es s
We now conside he sup LR∗
4T and UDmax LR4T (simply labeled UDmax) es s.To his
end, we use a model wi h GARCH(11)e o s so ha he DGP is y =e wi h e =u √h ,
whe e u ∼i.i.d. N(01),h =τ1+γe2
−1+ρh −1,h0=τ1/(1−γ−ρ),τ1=1,ρ=02and γ
1040 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
Table 6. Size o he sup LR∗
3T (ma=1na=1ε|m=0na=1) es wi h di e en imming pa-
ame e εin he case o no mal e o s.
T=100 T=200
θ ε01015 02025 01015 02025
Panel (a)
00.043 0.053 0.051 0.031 0.042 0.041 0.039 0.036
010.050 0.053 0.033 0.037 0.027 0.035 0.033 0.026
025 0.042 0.042 0.042 0.023 0.034 0.044 0.039 0.040
050.044 0.024 0.038 0.038 0.036 0.035 0.035 0.028
075 0.039 0.039 0.037 0.033 0.043 0.038 0.040 0.034
10.033 0.043 0.045 0.027 0.029 0.044 0.042 0.029
20.046 0.045 0.039 0.022 0.038 0.032 0.029 0.013
40.030 0.054 0.035 0.020 0.038 0.032 0.030 0.014
10 0.034 0.043 0.030 0.027 0.037 0.035 0.031 0.015
20 0.046 0.039 0.027 0.027 0.032 0.039 0.030 0.012
Panel (b)
00.069 0.066 0.066 0.055 0.049 0.043 0.050 0.042
010.057 0.060 0.062 0.056 0.044 0.047 0.048 0.039
025 0.057 0.055 0.055 0.049 0.039 0.044 0.053 0.035
050.050 0.058 0.048 0.043 0.051 0.044 0.050 0.035
075 0.055 0.055 0.057 0.046 0.043 0.036 0.036 0.034
10.065 0.055 0.051 0.042 0.044 0.053 0.045 0.028
20.047 0.066 0.062 0.045 0.043 0.040 0.040 0.027
40.052 0.053 0.039 0.025 0.030 0.051 0.031 0.017
10 0.050 0.063 0.050 0.026 0.043 0.038 0.034 0.018
20 0.040 0.065 0.059 0.024 0.048 0.038 0.034 0.025
No e: Panel (a): DGP: y =μ1+e ,e ∼i.i.d. N(01+θ1( > [05T])),μ1=0; panel (b): DGP: y =c+αy −1+e ,e ∼
i.i.d. N(01+θ1( > [05T])),c=0,α=05.
akes alues 010305.Also,ε=0102.Fo heUDmax es , M=N=2and o he
supLR∗
4T es , we conside he ollowing combina ions: (a) ma=na=1,(b)ma=1,
na=2,(c)ma=2,na=1.Wese T=100200. The esul s, p esen ed in Table 8,show
ha he size is close o o sligh ly lowe han he nominal 5% le el (some cases ha e
sligh libe al size dis o ions when T=100, which, howe e , dec ease when T=200).
Supplemen G shows ha he es s ha e good sizes wi h i.i.d. no mal e o s.
We now conside he powe o hese es s. Since some pa ial esul s o he one b eak
case a e a ailable in Tables S.6–S.7 o he sup LR∗
4T es , we concen a e on he case wi h
a di e en numbe o b eaks in coe icien s and in a iance. We also only conside i.i.d.
no mal e o s hough he hyb id- ype co ec ion is s ill applied. Table 9p esen s he e-
sul s o he case wi h one b eak in coe icien and wo b eaks in a iance, in which case
he DGP is y =μ1+μ21( > Tc)+e ,e ∼i.i.d. N(01+θ1(T
1< ≤T
2)) wi h μ1=0,
μ2=θand ε=01. Fi e di e en con igu a ions o b eak da es a e conside ed. We ana-
lyze wo o ms o he sup LR∗
4T es : (a) one es ing o a single b eak in bo h mean and
a iance, (b) one co ec ly es ing o wo changes in a iance and one change in mean.
This is done o in es iga e he ex en o he powe di e ences when unde speci ying he
numbe o b eaks. As expec ed, he powe inc eases apidly wi h θand wi h T.Wi h he
DGP used, he powe is simila whe he accoun ing o one o (co ec ly) wo b eaks in
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1041
Table 7. Powe o he supLR∗
3T (ma=1na=1ε|m=0na=1) es wi h di e en imming
pa ame e εin he case o no mal e o s.
T=100
ε=01ε=02
μ2 θ001052 4 1020 001052 4 1020
010050 0050 0055 0058 0059 0057 0059 0050 0049 0043 0034 0031 0037 0030
025 0096 0092 0092 0082 0078 0074 0080 0117 0115 0110 0088 0077 0077 0077
050349 0351 0340 0300 0263 0255 0245 0353 0350 0334 0305 0283 0283 0243
075 0670 0663 0651 0580 0538 0503 0485 0702 0696 0692 0625 0586 0586 0544
10901 0899 0892 0853 0821 0799 0785 0930 0929 0929 0901 0866 0866 0811
41000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
T=200
ε=01ε=02
μ2 θ001052 4 1020 001052 4 1020
010059 0062 0054 0046 0043 0045 0049 0059 0056 0044 0058 0055 0053 0042
025 0175 0170 0178 0140 0136 0136 0138 0192 0179 0183 0158 0142 0132 0135
050650 0609 0585 0556 0518 0494 0466 0681 0655 0673 0583 0542 0506 0482
075 0939 0959 0934 0913 0901 0882 0847 0963 0965 0963 0913 0909 0878 0883
11000 0999 0997 0995 0989 0988 0987 1000 0998 0999 0998 0998 0996 0995
41000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
No e:DGP:y =μ1+μ21( > [025T])+e ,e ∼i.i.d. N(01+θ1( > [05T])),μ1=0.
a iance and he powe o he UDmax es is also simila o he powe o bo h e sions
o he sup LR∗
4T es . This may, howe e , be DGP speci ic. Table 10 p esen s he esul s
o he case wi h wo b eaks in coe icien and one b eak in a iance, wi h he DGP gi en
by y =μ1+μ21(Tc
1< ≤Tc
2)+e ,e ∼i.i.d. N(01+θ1( > T )) wi h μ1=0and μ2=θ.
Table 8. Size o he sup LR∗
4T (mana)and UD max LR4T es s in he case o GARCH(11)e -
o s.
T=100
ε=01ε=02
γm
a=na=1ma=1,na=2ma=2,na=1UDmax ma=na=1ma=1,na=2ma=2,na=1UDmax
010044 0046 0047 0050 0037 0040 0035 0046
030048 0065 0051 0073 0041 0052 0042 0055
050072 0083 0075 0085 0065 0069 0059 0061
T=200
ε=01ε=02
γm
a=na=1ma=1,na=2ma=2,na=1UDmax ma=na=1ma=1,na=2ma=2,na=1UDmax
010034 0035 0034 0041 0036 0034 0037 0037
030032 0041 0035 0043 0036 0037 0031 0040
050039 0044 0041 0051 0040 0040 0024 0040
No e:DGP:y =e ,e =u √h ,wi hu ∼i.i.d. N(01),h =τ1+γe2
−1+ρh −1,τ1=1,ρ=02,h0=τ1/(1−γ−ρ).
1042 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
Table 9. Powe o he supLR∗
4T (mana)and UD max LR4T es s o DGPs wi h one b eak in coe icien s and wo b eaks in a iance.
T=100
Tc=T
1=[03T],
T
2=[06T]
Tc=T
2=[06T],
T
1=[03T]
Tc=[03T],T
1=[05T],
T
2=[06T]
Tc=[05T],T
1=[03T],
T
2=[06T]
Tc=[06T],T
1=[03T],
T
2=[05T]
ma=1ma=1ma=1ma=1ma=1ma=1ma=1ma=1ma=1ma=1
θn
a=1na=2UDmax na=1na=2UDmax na=1na=2UDmax na=1na=2UDmax na=1na=2UDmax
025 0081 0069 0090 0091 0082 0097 0083 0069 0086 0089 0085 0097 0092 0079 0100
050263 0263 0280 0314 0280 0313 0262 0233 0269 0320 0294 0326 0318 0281 0315
075 0576 0560 0586 0655 0631 0643 0592 0570 0583 0687 0661 0691 0648 0628 0650
10854 0860 0857 0892 0902 0896 0874 0861 0877 0895 0906 0918 0890 0886 0888
125 0980 0974 0976 0988 0985 0984 0982 0974 0982 0986 0983 0987 0983 0987 0987
151000 1000 0997 0998 0999 1000 0999 0997 0998 1000 1000 1000 1000 0999 0999
T=200
Tc=T
1=[03T],
T
2=[06T]
Tc=T
2=[06T],
T
1=[03T]
Tc=[03T],T
1=[05T],
T
2=[06T]
Tc=[05T],T
1=[03T],
T
2=[06T]
Tc=[06T],T
1=[03T],
T
2=[05T]
ma=1ma=1ma=1ma=1ma=1ma=1ma=1ma=1ma=1ma=1
θn
a=1na=2UDmax na=1na=2UDmax na=1na=2UDmax na=1na=2UDmax na=1na=2UDmax
025 0119 0124 0129 0156 0138 0159 0128 0109 0125 0152 0153 0158 0142 0134 0149
050552 0561 0569 0633 0622 0637 0547 0515 0545 0642 0645 0656 0628 0593 0624
075 0925 0929 0925 0961 0958 0955 0935 0927 0931 0968 0976 0971 0966 0956 0962
11000 1000 1000 1000 1000 1000 1000 0999 1000 1000 1000 1000 1000 0999 1000
125 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
151000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
No e:DGP:y =μ1+μ21( > Tc)+e ,e ∼i.i.d. N(01+θ1(T
1< ≤T
2)),μ1=0,μ2=θ,ε=01.
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1049
8. Conclusion
This pape p o ided ools o es ing o mul iple s uc u al b eaks in he e o a iance
o a linea eg ession model wi h o wi hou he p esence o b eaks in he eg ession co-
e icien s. An inno a ion is ha we do no impose any es ic ions on he b eak da es,
ha is, he b eaks in he eg ession coe icien s and in he a iance can happen a he
same ime o a di e en imes. We p oposed s a is ics wi h asymp o ic dis ibu ions
in a ian o nuisance pa ame e s and alid wi h nonno mal e o s and condi ional he -
e oskedas ici y, as well as se ial co ela ion. Ex ensi e simula ions o he ini e sample
p ope ies show ha ou p ocedu es pe o m well in e ms o size and powe . A speci ic
o gene al p ocedu e o es ima e he numbe and ype o b eaks based on a p oposed
sequen ial es is shown o pe o m well in selec ing he numbe and ypes o b eaks.
Appendix
P oo o Theo em 1. Pa (a) ollows om Qu and Pe on (2007a, Theo em 5) unde
Assump ion A1. Fo pa (b),
supLR2T (manaε|n=0ma)
=2log ˆ
LT˜
Tc
1 ˜
Tc
ma;˜
T
1 ˜
T
na−log ˜
LTˆ
Tc
1 ˆ
Tc
ma
=Tlog ˜σ2−
na+1

i=1˜
T
i−˜
T
i−1log ˆσ2
i
=
na

i=1˜
T
i+1log ˜σ2
1i+1−˜
T
ilog ˜σ2
1i −˜
T
i+1−˜
T
ilog ˆσ2
i+1+˜
T
1log ˜σ2
11−log ˆσ2
1
whe e ˜σ2
1i =(˜
T
i)−1˜
T
i
=1(y −x
˜
β−z
˜
δ j)2wi h ˜
δ j =˜
δj o ˆ
Tc
j−1< ≤ˆ
Tc
j(also le δ0
j =
δ0
j o Tc0
j−1< ≤Tc0
j)(j=1ma+1)and ˆσ2
i=(˜
T
i−˜
T
i−1)−1˜
T
i
=˜
T
i−1+1(y −x
ˆ
β−
z
ˆ
δ j)2. Applying a Taylo expansion o log ˜σ2
1i+1,log ˜σ2
1i and log ˆσ2
i+1a ound logσ2
0,we
ob ain
supLR2T (manaε|n=0ma)=
na

i=1Fi
1T +Fi
2T +op(1)
whe e
Fi
1T =σ2
0−1˜
T
i+1˜σ2
1i+1−˜
T
i˜σ2
1i −˜
T
i+1−˜
T
iˆσ2
i+1
=σ2
0−1˜
T
i+1

=˜
T
i+1y −x
˜
β−z
˜
δ j2−y −x
ˆ
β−z
ˆ
δ j2
and
Fi
2T =−(1/2)˜
T
i+1˜σ2
1i+1−σ2
0
σ2
02
−˜
T
i˜σ2
1i −σ2
0
σ2
02
−˜
T
i+1−˜
T
iˆσ2
i+1−σ2
0
σ2
02
=(1/2)(I +II +III) (A.1)

1050 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
We i s show ha Fi
1T =op(1). We can exp ess Fi
1T as
σ2
0−1Ui+1+Xi+1β0−˜
β
+Zi+1δ0
j −˜
δ jUi+1+Xi+1β0−˜
β+Zi+1δ0
j −˜
δ j
−Ui+1+Xi+1β0−ˆ
β
+Zi+1δ0
j −ˆ
δ jUi+1+Xi+1β0−ˆ
β+Zi+1δ0
j −ˆ
δ j
=σ2
0−1(ˆ
β−˜
β)X
i+1Xi+1(ˆ
β−˜
β) +(ˆ
δ j −˜
δ j)Z
i+1Zi+1(ˆ
δ j −˜
δ j)
+(ˆ
β−˜
β)X
i+1Zi+1(ˆ
δ j −˜
δ j)+2(β −ˆ
β)X
i+1Xi+1(ˆ
β−˜
β)
+2δ0
j −ˆ
δ jZ
i+1Zi+1(ˆ
δ j −˜
δ j)+2(ˆ
β−˜
β)X
i+1Zi+1δ0
j −ˆ
δ j
+2(β −ˆ
β)X
i+1Zi+1(ˆ
δ j −˜
δ j)+2(ˆ
β−˜
β)X
i+1Ui+1+2(ˆ
δ j −˜
δ j)Z
i+1Ui+1
The esul ollows using he ac s ha X
i+1Xi+1=Op(T),Z
i+1Zi+1=Op(T),X
i+1Zi+1=
Op(T),X
i+1Ui+1=Op(T1/2)and Z
i+1Ui+1=Op(T1/2).Also,unde H0wi h Assump-
ion A1, he es ima es o he b eak ac ions con e ge o he ue b eak ac ions a a
as enough a e so ha he es ima es o he pa ame e s o he models a e consis en
and ha e he same limi dis ibu ion as when he b eak da es a e known. We ha e:
β0−ˆ
β=Op(T−1/2),δ0
j −ˆ
δ j =Op(T −1/2),ˆ
β−˜
β=op(T−1/2)and ˆ
δ j −˜
δ j =op(T −1/2).
The las wo quan i ies a e op(T −1/2)since √T( ˆ
β−β0)and √T(˜
β−β0)ha e he same
limi dis ibu ion unde H0, and likewise o √T(ˆ
δ j −δ0
j)and √T(˜
δ j −δ0
j).Fo Fi
2T ,
√I=˜
T
i+1−1/2˜
T
i+1

=1y −x
˜
β−z
˜
δ j/σ02−1
=˜
T
i+1−1/2˜
T
i+1

=1(u /σ0)2−1+op(1)
⇒ψW λ
i+1/λ
i+1
by Assump ion A1. Simila ly, √II ⇒√ψW (λ
i)/λ
iand
√III =˜
T
i+1−˜
T
i/T−1/2T−1/2
T
i+1

=T
i+1(u /σ0)2−1+op(1)
=˜
T
i+1−˜
T
i/T−1/2T−1/2
T
i+1

=1(u /σ0)2−1−T−1/2
T
i

=1(u /σ0)2−1+op(1)
⇒ψWλ
i+1−Wλ
i/λ
i+1−λ
i
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1051
The e o e,
Fi
2T ⇒−(ψ/2)W2λ
i+1
λ
i+1−W2λ
i
λ
i−Wλ
i+1−Wλ
i2
λ
i+1−λ
i
=(ψ/2)λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
This yields
supLR2T (manaε|n=0ma)⇒sup
(λ
1λ
na)∈Λc
ε
na

i=1
ψ
2λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
≤sup
(λ
1λ
na)∈Λ ε
na

i=1
ψ
2λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
because Λc
ε ⊆Λ ε. Fo pa (c),
supLR3T (manaε|m=0na)
=2log ˆ
LT˜
Tc
1 ˜
Tc
ma;˜
T
1 ˜
T
na−log ˜
LTˆ
T
1 ˆ
T
na
=
na+1

i=1ˆ
T
i−ˆ
T
i−1log ˜σ2
i−
na+1

i=1˜
T
i−˜
T
i−1log ˆσ2
i
whe e ˜σ2
i=(ˆ
T
i−ˆ
T
i−1)−1ˆ
T
i
=ˆ
T
i−1+1(y −x
˜
β−z
˜
δ)2and ˆσ2
i=(˜
T
i−˜
T
i−1)−1×
˜
T
i
=˜
T
i−1+1(y −x
ˆ
β−z
ˆ
δ j)2. Applying a Taylo expansion on log ˜σ2
iand log ˆσ2
i
a ound logσ2
i0,weob ain
supLR3T (manaε|m=0na)=
na+1

i=1Fi
1T +Fi
2T +op(1)
whe e Fi
1T =(ˆ
T
i−ˆ
T
i−1)( ˜σ2
i/σ2
i0)−(˜
T
i−˜
T
i−1)( ˆσ2
i/σ2
i0)and
Fi
2T =−(1/2)ˆ
T
i−ˆ
T
i−1˜σ2
i−σ2
i0/σ2
i02−˜
T
i−˜
T
i−1ˆσ2
i−σ2
i0/σ2
i02
We i s show ha Fi
2T =op(1)as ollows. We ha e:
Fi
2T =−(1/2)ˆ
T
i−ˆ
T
i−1˜σ2
i−σ2
i0
σ2
i02
−˜
T
i−˜
T
i−1ˆσ2
i−σ2
i0
σ2
i02
=−(1/2)T−1ˆ
T
i−ˆ
T
i−1T1/2˜σ2
i−σ2
i0
σ2
i02
−T−1˜
T
i−˜
T
i−1T1/2ˆσ2
i−σ2
i0
σ2
i02
1052 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
whe e [(ˆ
T
i−ˆ
T
i−1)/T][√T(˜σ2
i−σ2
i0)/σ2
i0]2and [(˜
T
i−˜
T
i−1)/T][√T(ˆσ2
i−σ2
i0)/σ2
i0]2ha e
he same limi dis ibu ion unde Assump ion A3.Fo Fi
1T ,le σ0=σ10 wi hou loss o
gene ali y, hen
na+1

i=1
Fi
1T =σ2
0−1na+1

i=1ˆ
T
i−ˆ
T
i−1˜σ2
i−˜
T
i−˜
T
i−1ˆσ2
i
+σ2
0−1na+1

i=1σ2
i0−σ2
0/σ2
i0ˆ
T
i−ˆ
T
i−1˜σ2
i−˜
T
i−˜
T
i−1ˆσ2
i
The i s e m becomes
σ2
0−1na+1

i=1ˆ
T
i−ˆ
T
i−1˜σ2
i−˜
T
i−˜
T
i−1ˆσ2
i
=σ2
0−1T

=1y −x
˜
β−z
˜
δ2−y −x
ˆ
β−z
ˆ
δ j2
=σ2
0−1ma

j=1˜
Tc
j+1

=1y −x
˜
β−z
˜
δ2−
˜
Tc
j

=1y −x
˜
β−z
˜
δ2
−
˜
Tc
j+1

=˜
Tc
j+1y −x
ˆ
β−z
ˆ
δj+12
+σ2
0−1˜
Tc
1

=1y −x
˜
β−z
˜
δ2−σ2
0−1˜
Tc
1

=1y −x
ˆ
β−z
ˆ
δ12
=σ2
0−1ma

j=1D (1j+1)−D (1j)−Du(j +1)+D (11)−Du(1)(A.2)
whe e D (1j) =˜
Tc
j
=1(y −x
˜
β−z
˜
δ)2and Du(j) =˜
Tc
j
=˜
Tc
j−1+1(y −x
ˆ
β−z
ˆ
δj)2.The
second e m is op(1)by Assump ion A3. Using simila de i a ions as in Qu and Pe on
(2007b), we ob ain
D (1j+1)−D (1j)−Du(j +1)
=−U
1:j+1Z1:j+1Z
1:j+1Z1:j+1−1Z
1:j+1U1:j+1+U
1:jZ1:jZ
1:jZ1:j−1Z
1:jU1:j
+U
j+1Zj+1Z
j+1Zj+1−1Z
j+1Uj+1+op(1)
⇒λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1053
by Assump ion A2. This yields
supLR3T (manaε|m=0na)⇒sup
(λc
1λc
ma)∈Λ
cε
ma

j=1λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
≤sup
(λc
1λc
ma)∈Λcε
ma

j=1λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
because Λ
cε ⊆Λcε. Fo pa (d), we ha e:
supLR4T (manaε|m=n=0)
=2sup
(λc
1λc
ma;λ
1λ
na)∈Λε
log ˆ
LTTc
1Tc
ma;T
1T
na−log ˜
LT
=2log ˆ
LT˜
Tc
1 ˜
Tc
ma;˜
T
1 ˜
T
na−log ˜
LT
=Tlog ˜σ2−
na+1

i=1˜
T
i−˜
T
i−1log ˆσ2
i
=
na

i=1˜
T
i+1log ˜σ2
1i+1−˜
T
ilog ˜σ2
1i −˜
T
i+1−˜
T
ilog ˆσ2
i+1+˜
T
1log ˜σ2
11−log ˆσ2
1
whe e ˜σ2
1i =(˜
T
i)−1˜
T
i
=1(y −x
˜
β−z
˜
δ)2. Applying a Taylo expansion o log ˜σ2
1i+1,
log ˜σ2
1i and log ˆσ2
i+1a ound logσ2
0,weob ain
supLR4T (manaε|m=n=0)=
na

i=1Fi
1T +Fi
2T +op(1)
whe e he i s e mis hesameasin(A.2), so ha
na

i=1
Fi
1T =
na

i=1σ2
0−1˜
T
i+1˜σ2
1i+1−˜
T
i˜σ2
1i −˜
T
i+1−˜
T
iˆσ2
i+1+σ2
0−1˜
T
1˜σ2
11−ˆσ2
1
=σ2
0−1T

=1y −x
˜
β−z
˜
δ2−y −x
ˆ
β−z
ˆ
δ j2
=σ2
0−1ma

j=1D (1j+1)−D (1j)−Du(j +1)+D (11)−Du(1)
as shown in pa (c). The second e m is he same as (A.1) bu wi h no changes in δ o
cons uc ˜σ2
1i, ha is,LR de ined by (11). Hence,
Fi
2T =−(1/2)˜
T
i+1˜σ2
1i+1−σ2
0
σ2
02
−˜
T
i˜σ2
1i −σ2
0
σ2
02
−˜
T
i+1−˜
T
iˆσ2
i+1−σ2
0
σ2
02
1054 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
as shown in pa (b). F om he p oo o pa (c),
na

i=1
Fi
1T ⇒
ma

j=1λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
unde Assump ion A2 and om ha o pa (b),
Fi
2T ⇒ψ
2λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
unde Assump ion A1.Hence,weob ain
supLR4T (manaε|m=n=0)⇒sup
(λc
1λc
ma;λ
1λ
na)∈Λεma

j=1λc
jWqλc
j+1−λc
j+1Wqλc
j2
λc
j+1λc
jλc
j+1−λc
j
+ψ
2
na

i=1λ
iWλ
i+1−λ
i+1Wλ
i2
λ
i+1λ
iλ
i+1−λ
i
Re e ences
And ews, D. W. K. (1991), “He e oskedas ici y and au oco ela ion consis en co a iance
ma ix es ima ion.” Econome ica, 59, 817–858. [1029,1036]
And ews, D. W. K. (1993), “Tes s o pa ame e ins abili y and s uc u al change wi h
unknown change poin .” Econome ica, 61, 821–856. [1020,1021,1035]
Aue, A., S. Ho mann, L. Ho á h, and M. Reimhe (2009), “B eak de ec ion in he co-
a iance s uc u e o mul i a ia e ime se ies models.” The Annals o S a is ics, 37, 4046–
4087. [1020]
Bai, J. (2000), “Vec o au o eg essi e models wi h s uc u al changes in eg ession coe -
icien s and in a iance-co a iance ma ices.” The Annals o Economics and Finance,1,
303–339. [1020]
Bai, J., R. L. Lumsdaine, and J. H. S ock (1998), “Tes ing o and da ing b eaks in mul i-
a ia e ime se ies.” Re iew o Economic S udies, 65, 395–432. [1020]
Bai, J. and S. Ng (2005), “Tes s o skewness, ku osis, and no mali y o ime se ies da a.”
Jou nal o Business and Economic S a is ics, 23, 49–60. [1036]
Bai, J. and P. Pe on (1998), “Es ima ing and es ing linea models wi h mul iple s uc-
u al changes.” Econome ica, 66, 47–78. [1020,1021,1029,1032,1033,1034,1047]
Bai, J. and P. Pe on (2003a), “Compu a ion and analysis o mul iple s uc u al change
models.” Jou nal o Applied Econome ics, 18, 1–22. [1020,1047,1048]
Bai, J. and P. Pe on (2003b), “C i ical alues o mul iple s uc u al change es s.” Econo-
me ics Jou nal, 6, 72–78. [1029,1033,1034]

Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1055
Bai, J. and P. Pe on (2006), “Mul iple s uc u al change models: A simula ion analysis.”
In Econome ic Theo y and P ac ice: F on ie s o Analysis and Applied Resea ch (D. Co -
bae, S. Du lau , and B. E. Hansen, eds.), 212–237, Camb idge Uni e si y P ess, Cam-
b idge. [1032,1044]
Blancha d, O. and J. Simon (2001), “The long and la ge decline in U.S. ou pu ola ili y.”
B ookings Pape s on Economic Ac i i y, 1, 135–173. [1021]
B own, R. L., J. Du bin, and J. M. E ans (1975), “Techniques o es ing he cons ancy o
eg ession ela ionships o e ime.” Jou nal o he Royal S a is ical Socie y, Se ies B, 37,
149–163. [1020,1036]
Casini, A. and P. Pe on (2019a), “Con inuous eco d asymp o ics o s uc u al change
models.” Unpublished manusc ip , Depa men o Economics, Bos on Uni e si y. [1020]
Casini, A. and P. Pe on (2019b), “S uc u al b eaks in ime se ies.” In Ox o d Resea ch
Encyclopedia o Economics and Finance.[1019]
Ca alie e, G. and I. Geo gie (2019), “In e ence unde andom limi boo s ap mea-
su es.” Unpublished manusc ip , Uni e si y o Bologna. [1027]
Da is, R. A., D. Huang, and Y. C. Yao (1995), “Tes ing o a change in he pa ame e alues
and o de o an au o eg essi e model.” The Annals o S a is ics, 23, 282–304. [1020]
Deng, A. and P. Pe on (2008), “The limi dis ibu ion o he CUSUM o squa es es un-
de gene al mixing condi ions.” Econome ic Theo y, 24, 809–822. [1020,1036]
Gadea, M. D., A. Gómez-Loscos, and G. Pé ez-Qui ós (2018), “G ea mode a ion and
g ea ecession: F om plain sailing o s o my seas?” In e na ional Economic Re iew, 59,
2297–2321. [1021]
Ga cia, R. and P. Pe on (1996), “An analysis o he eal in e es a e unde egime shi s.”
Re iew o Economics and S a is ics, 78, 111–125. [1047]
Gó ecki, T., L. Ho á h, and P. Kokoszka (2018), “Change poin de ec ion in he -
e oscedas ic ime se ies.” Econome ics and S a is ics, 7, 63–88. [1022]
Hansen, B. E. (2000), “Tes ing o s uc u al change in condi ional models.” Jou nal o
Econome ics, 97, 93–115. [1027]
Hawkins, D. M. (1976), “Poin es ima ion o he pa ame e s o piecewise eg ession mod-
els.” Applied S a is ics, 25, 51–57. [1020]
He e a, A. M. and E. Pesa en o (2005), “The decline in U.S. ou pu ola ili y: S uc u al
changes and in en o y in es men .” Jou nal o Business and Economic S a is ics, 23, 462–
472. [1021]
Ho á h, L. (1993), “The maximum likelihood me hod o es ing changes in he pa am-
e e s o no mal obse a ions.” The Annals o S a is ics,21, 671–680. [1020]
Inclán, C. and G. C. Tiao (1994), “Use o cumula i e sums o squa es o e ospec i e
de ec ion o changes o a iance.” Jou nal o he Ame ican S a is ical Associa ion, 89,
913–923. [1020]
1056 Pe on, Yamamo o, and Zhou Quan i a i e Economics 11 (2020)
Kej iwal, M. (2009), “Tes s o a mean shi wi h good size and mono onic powe .” Eco-
nomics Le e s, 102, 78–82. [1029]
Kej iwal, M. and P. Pe on (2008), “The limi dis ibu ion o he es ima es in coin e-
g a ed eg ession models wi h mul iple s uc u al changes.” Jou nal o Econome ics,
146, 59–73. [1020]
Kej iwal, M. and P. Pe on (2010), “Tes ing o mul iple s uc u al changes in coin e-
g a ed eg ession models.” Jou nal o Business and Economic S a is ics, 28, 503–522.
[1020,1029]
Kim, C.-J. and C. R. Nelson (1999), “Has he U.S. economy become mo e s able?
A Bayesian app oach based on a Ma ko -swi ching model o he business cycle.” Re iew
o Economics and S a is ics, 81, 608–616. [1021]
Ku ozumi, E. and P. Tu aando j (2011), “Model selec ion c i e ia in mul i a ia e models
wi h mul iple s uc u al changes.” Jou nal o Econome ics, 164, 218–238. [1020]
Liu, J., S. Wu, and J. V. Zidek (1997), “On segmen ed mul i a ia e eg essions.” S a is ica
Sinica, 7, 497–525. [1020]
McConnell, M. M. and G. Pé ez-Qui ós (2000), “Ou pu luc ua ions in he Uni ed S a es:
Wha has changed since he ea ly 1980’s?” Ame ican Economic Re iew, 90, 1464–1476.
[1020,1021]
Pe on, P. (1989), “The g ea c ash, he oil p ice shock, and he uni oo hypo hesis.”
Econome ica, 57, 1361–1401. [1022]
Pe on, P. (2006), “Dealing wi h s uc u al b eaks.” In Palg a e Handbook o Econome -
ics,Vol.1:Econome icTheo y(K. Pa e son and T. C. Mills, eds.), 278–352, Palg a e
Macmillan, Basings oke. [1019]
Pe on, P. and Z. Qu (2006), “Es ima ing es ic ed s uc u al change models.” Jou nal o
Econome ics, 134, 373–399. [1020]
Pe on, P. and Y. Yamamo o (2014), “A no e on es ima ing and es ing o mul iple s uc-
u al changes in models wi h endogenous eg esso s ia 2SLS.” Econome ic Theo y, 30,
491–507. [1020]
Pe on, P. and Y. Yamamo o (2015), “Using OLS o es ima e and es o s uc u al
changes in models wi h endogenous eg esso s.” Jou nal o Applied Econome ics, 30,
119–144. [1020]
Pe on, P. and Y. Yamamo o (2019a), “Pi alls o wo s eps es ing o changes in he e o
a iance and coe icien s o a linea eg ession models.” Econome ics, 7, 22. [1021]
Pe on, P. and Y. Yamamo o (2019b), “The g ea mode a ion: Upda ed e idence wi h
join es s o mul iple s uc u al changes in a iance and pe sis ence.” Unpublished
manusc ip , Depa men o Economics, Bos on Uni e si y. [1021,1034,1047]
Pe on, P., Y. Yamamo o, J. Zhou (2020), “Supplemen o ‘Tes ing join ly o s uc u al
changes in he e o a iance and coe icien s o a linea eg ession model’.” Quan i a-
i e Economics Supplemen al Ma e ial, 11, h ps://doi.o g/10.3982/QE1332.[1022]
Quan i a i e Economics 11 (2020) Tes ing join ly o s uc u al changes 1057
Pi a akis, J.-Y. (2004), “Leas -squa es es ima ion and es s o b eaks in mean and a i-
ance unde misspeci ica ion.” Econome ics Jou nal, 7, 32–54. [1021]
Qu, Z. and P. Pe on (2007a), “Es ima ing and es ing mul iple s uc u al changes in
mul i a ia e eg essions.” Econome ica, 75, 459–502. [1020,1021,1023,1025,1027,
1029,1049]
Qu, Z. and P. Pe on (2007b), “Supplemen o ‘Es ima ing and es ing mul iple s uc-
u al changes in mul i a ia e eg essions’.” Econome ica Supplemen al Ma e ial, 75,
h ps://www.econome icsocie y.o g/si es/de aul / iles/ECTA5733SUPP_0.pd .[1052]
Sensie , M. and D. an Dijk (2004), “Tes ing o ola ili y changes in U.S. mac oeconomic
ime se ies.” Re iew o Economics and S a is ics, 86, 833–839. [1021]
S ock, J. H. and M. W. Wa son (2002), “Has he business cycle changed and why?” In
NBER Mac oeconomics Annual, Vol. 17 (M. Ge le and K. Rogo , eds.), 159–218, MIT
P ess, Camb idge, MA. [1021,1045,1047]
Xu, K.-L. (2013), “Powe ul es o s uc u al changes in ola ili y.” Jou nal o Econome -
ics, 173, 126–142. [1020]
Co-edi o Tao Zha handled his manusc ip .
Manusc ip ecei ed 16 Ap il, 2019; inal e sion accep ed 8 Feb ua y, 2020; a ailable on-
line 11 Feb ua y, 2020.