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Uni oo es ing wi h slowly a ying ends
Jou nal o Time Se ies Analysis
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JOURNAL OF TIME SERIES ANALYSIS
J. Time Se . Anal. 42: 85–106 (2021)
Published online 20 Sep embe 2020 in Wiley Online Lib a y
(wileyonlinelib a y.com) DOI: 10.1111/j sa.12557
ORIGINAL ARTICLE
UNIT ROOT TESTING WITH SLOWLY VARYING TRENDS
SVEN OTTO
Uni e si y o Bonn, Ins i u e o Finance and S a is ics, Bonn, Ge many
A uni oo es is p oposed o ime se ies wi h a gene al nonlinea de e minis ic end componen . I is shown ha asymp o -
ically he pooled OLS es ima o o o e lapping blocks il e s ou any end componen ha sa is ies some Lipschi z condi ion.
Unde bo h ixed-band small-bblock asymp o ics, he limi ing dis ibu ion o he -s a is ic o he uni oo hypo hesis is
de i ed. Nuisance pa ame e co ec ions p o ide he e oskedas ici y- obus es s, and se ial co ela ion is accoun ed o by
p e-whi ening. A Mon e Ca lo s udy ha conside s slowly a ying ends yields bo h good size and imp o ed powe esul s
o he p oposed es s when compa ed o con en ional uni oo es s.
Recei ed 12 Ap il 2019; Accep ed 13 Augus 2020
Keywo ds: Uni oo es s; nonlinea ends; he e oskedas ici y
JEL. C12; C14; C22
MOS subjec classi ica ion: 62M10.
1. INTRODUCTION
I is widely deba ed in he ime se ies li e a u e whe he mac oeconomic a iables such as GDP, in la ion, and
in e es a es a e I(1)o I(0)a ound a de e minis ic end. Dickey–Fulle - ype uni oo es s o en ail o ejec he
null hypo hesis o hese ime se ies. The end componen o a ime se ies y is ypically ea ed as known up o
some pa ame e ec o . The mos commonly applied uni oo es s, such as hose de eloped by Dickey and Fulle
(1979), Said and Dickey (1984), Phillips (1987), Phillips and Pe on (1988), and Ellio e al. (1996), impose ei he
a cons an o a linea end model. I , howe e , he de e minis ic end componen is nonlinea , highly pe sis en
end-s a iona y p ocesses can be ha dly dis inguishable om uni oo p ocesses (see, e.g., Bie ens, 1997; Becke
e al., 2006).
I is no only a misspeci ied end model ha may lead o high powe losses, as an o e pa ame e ized model can
also educe he powe o uni oo es s. The e o e, many au ho s ha e sugges ed applying end models ha seem
mo e sui able o mac o da a. B oken end models wi h one- ime changes in mean o slope wi h known b eakpoin
we e i s s udied by Pe on (1989) and Rappopo and Reichlin (1989). Ch is iano (1992) demons a ed ha a
b oken end model wi h an unknown b eakpoin is mo e adequa e, and Zi o and And ews (1992), as well as
Bane jee e al. (1992), p oposed uni oo es s o his amewo k. S uc u al changes in inno a ion a iances we e
s udied by Hamo i and Tokihisa (1997), Kim e al. (2002), and Ca alie e (2005), while Ca alie e e al. (2011)
conside ed uni oo es ing unde b oken ends oge he wi h non-s a iona y ola ili y. Leybou ne e al. (1998),
Kape anios e al. (2003), and Kílíç (2011) allowed o exponen ial smoo h ansi ions om one end egime o
ano he . Bie ens (1997) app oxima ed a nonlinea mean unc ion wi h Chebyshe polynomials, and Ende s and
Lee (2012) p oposed a Fou ie se ies app oxima ion o he end, which a e app oaches ha can be used when he
exac o m and da e o s uc u al changes a e unknown. Fo a comp ehensi e e iew on he esea ch on uni oo
es ing see Choi (2015).
*Co espondence o: S en O o, Uni e si y o Bonn, Ins i u e o Finance S a is ics, Adenaue allee 24, 53113 Bonn, Ge many. E-mail:
[email p o ec ed]
© 2020 The Au ho s. Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and
ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
86 S. OTTO
Dickey–Fulle - ype es s a e based on he -s a is ic o he i s -o de au o eg essi e pa ame e . In case o a
cons an end, he es ima o is de i ed om a eg ession o Δy on (y −1−y),whe eyis he sample mean. Schmid
and Phillips (1992) es ima ed he cons an by he ini ial obse a ion, which esul s in a eg ession o Δy on (y −1−
y1). Whe eas a cons an is o en no a good global app oxima ion, in a small block, a smoo hly a ying end can
be app oxima ed qui e closely by a cons an . To exploi his ac , we p opose a block p ocedu e o il e ou he
unknown end componen . Blocking was also used in Rooch e al. (2019) o es ima e he ac ional in eg a ion
pa ame e in a simila si ua ion. We di ide he se ies in o T−Bo e lapping blocks o leng h B. As he blocks can
be conside ed as uni s o a panel, we ollow he panel uni oo es s p oposed by B ei ung (2000) and Le in e al.
(2002) and conside a pooled eg ession o Δyj+ on (yj+ −1−yj) o 2≤ ≤Tand 1≤j≤T−B. The de e minis ic
unc ion is app oxima ed locally by a cons an . One could also use highe o de local app oxima ions o he end
unc ion, bu un epo ed simula ions indica e ha hese app oxima ions do no wo k well in samples o usual size.
Fo his eason, we ocus on cons an local app oxima ions. Unde a gene al class o piecewise con inuous end
unc ions, he esul ing pooled es ima o is consis en as B,T→∞. The limi ing null dis ibu ion o he -s a is ic
is a unc ional o a B ownian mo ion unde ixed-basymp o ics. Unde small-basymp o ics, a no mal dis ibu ion
is ob ained.
The a icle is o ganized as ollows: in Sec ion 2 he au o eg essi e model wi h independen and he e oskedas ic
e o s is analyzed oge he wi h he asymp o ic beha io o he pooled leas squa es es ima o in he p esence o
a gene al nonlinea end componen . Fo bo h ixed-band small-bblock asymp o ics, he limi ing dis ibu ions
a e de i ed unde bo h he uni oo hypo hesis and unde local al e na i es. In he p esence o he e oskedas ic
e o s, nuisance pa ame e s appea in he limi ing dis ibu ions, and he es ima ion o hese pa ame e s is discussed.
Sec ion 3 conside s pseudo - es s o he uni oo hypo hesis, and he e oskedas ici y- obus es s a is ics a e
p o ided. In Sec ion 4, a p e-whi ening p ocedu e is p oposed o accoun o sho - un dynamics, while Sec ion 5
epo s on Mon e Ca lo simula ions. The es s a e ound o ha e only mino size dis o ions in small samples and
a e sized co ec ly in la ge samples. I is shown ha in he p esence o slowly a ying ends, pooled es s end o
yield highe powe han con en ional uni oo es s. Finally, Sec ion 6 p esen s he conclusion.
While some p oo s including hose o he main heo ems a e p esen ed in he Appendix, he mo e echnical
p oo s a e a ailable as Suppo ing in o ma ion. In he ollowing, W( )deno es a s anda d B ownian mo ion and
‘⇒’ s ands o weak con e gence on he càdlàg space D[0,1] oge he wi h a sui able no m. Θ(⋅)deno es he exac
o de Landau symbol, ha is, aT=Θ(bT)i and only i aT=O(bT)and bT=O(aT),asT→∞. Mo eo e , ⌊⋅⌋
is he in ege pa o i s a gumen , and Δy s ands o he di e enced se ies y −y −1. Finally, 
−→and p
−→deno e
con e gence in dis ibu ion and con e gence in p obabili y.
2. THE POOLED ESTIMATOR
We a e in e es ed in in e ence conce ning he au o eg essi e pa ame e 𝜌in he model
y =d +x ,x =𝜌x −1+u , =1,…,T,(1)
whe e 𝜌is close o equal o one. The de e minis ic end componen d is ea ed as non-s ochas ic and ixed in
epea ed samples, whe e i s unc ional o m is non-pa ame ic and unknown.
Assump ion 1 ( end componen ).The end componen is gi en by d =d( ∕T),whe ed( )is a piecewise
Lipschi z con inuous unc ion.
No e ha any con inuously di e en iable unc ion is Lipschi z con inuous. Lipschi z unc ions a e locally close
o a cons an alue in he sense ha he e exis s some C<∞such ha |d( )−d(s)|≤C| −s| o all ,s∈ℝ.The
piecewise Lipschi z condi ion allows o a pa i ion wi h a ini e numbe o in e als, such ha d( )is Lipschi z
con inuous on each in e al. This includes bo h smoo h changes and ab up b eaks in he end unc ion. Fo he
wileyonlinelib a y.com/jou nal/j sa © 2020 The Au ho s. J. Time Se . Anal. 42: 85–106 (2021)
Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12557
UNIT ROOT TESTING WITH SLOWLY VARYING TRENDS 87
ini ial alue, i is assumed ha E[x2
0]<∞. We in oduce he pooled es ima o and he uni oo es s a is ics unde
he ollowing assump ions on he e o e m:
Assump ion 2 (he e oskedas ic e o s).The p ocess {u } ∈ℕis independen ly dis ibu ed wi h E[u ]=0,E[u2
]=
𝜎2
and E[u4
]<∞,whe e𝜎 =𝜎( ∕T). The unc ion 𝜎( )is càdlàg, non-s ochas ic, s ic ly posi i e, and bounded.
The p incipal app oach o dealing wi h a gene al, slowly a ying end is o app oxima e he unknown end
locally by a cons an . Le Bbe some blockleng h ha sa is ies 2≤B<T. We di ide he ime se ies in o T−B
o e lapping blocks o leng h Band hen block-wise es ima e 𝜌 ia OLS unde a cons an end speci ica ion. In
he ashion o Schmid and Phillips (1992), as well as B ei ung and Meye (1994), he cons an end is es ima ed
by he i s obse a ion in each block, which co esponds o he maximum likelihood es ima o unde he uni oo
hypo hesis 𝜌=1. The ea e , by pooling he T−Bindi idual block eg essions, we ob ain he eg ession equa ion
Δy +j=𝜙(y +j−1−yj)+u +j, =2,…,B,j=1,…,T−B,
whe e 𝜙=𝜌−1. The pooled OLS es ima o is o mula ed as

𝜙=𝜌 −1=∑T−B
j=1∑B
=2Δy +j(y +j−1−yj)
∑T−B
j=1∑B
=2(y +j−1−yj)2
.
In he ollowing, we de i e he asymp o ic p ope ies o he nume a o and he denomina o sepa a ely. The
nume a o and denomina o s a is ics a e de ined as
1,T=1
B3∕2T1∕2
T−B
∑
j=1
B
∑
=2
Δy +j(y +j−1−yj),2,T=1
B2T
T−B
∑
j=1
B
∑
=2
(y +j−1−yj)2,
such ha √BT(𝜌 −1)=1,T∕2,T. Thei coun e pa s wi hou de e minis ics a e gi en by
1,T=1
B3∕2T1∕2
T−B
∑
j=1
B
∑
=2
Δx +j(x +j−1−xj),2,T=1
B2T
T−B
∑
j=1
B
∑
=2
(x +j−1−xj)2.
In wha ollows, we show ha , unde he block p ocedu e, he de e minis ic componen can be igno ed asymp o -
ically. All asymp o ic esul s a e join ly de i ed o B,T→∞. While he s a is ics 1,Tand 2,Ta e in easible i
d is unknown, hey can be well app oxima ed by 1,Tand 2,Tin he ollowing sense:
Lemma 1. Le 𝜌=1−c∕√BT wi h c≥0,le d sa is y Assump ion 1, and le u sa is y Assump ion 2. Then,
as B,T→∞,1,T−1,T=OP(B−1∕2),and2,T−2,T=OP(T−1∕2).
Acco dingly, we ob ain (1,T−1,T,2,T−2,T)p
−→(0,0)join ly, and he block p ocedu e il e s ou he end
componen in he nume a o and he denomina o asymp o ically. Hence, applying Slu sky’s heo em, we can w i e
√BT(𝜌 −1)=1,T
2,T
=1,T
2,T
+oP(1).
This esul is alid wi hou any a e es ic ions o B. To ob ain he limi ing dis ibu ion, we o mula e some
p ope ies o he nume a o and denomina o s a is ics.
J. Time Se . Anal. 42: 85–106 (2021) © 2020 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12557 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
88 S. OTTO
Lemma 2. Le 𝜌=1−c∕√BT wi h c≥0,andle u sa is y Assump ion 2. Then, as B,T→∞, he ollowing
s a emen s hold ue:
(a) 1,T=∑T
j=1qj,T−c⋅T,whe eqj,T,j≤T,T∈ℕis a ma ingale di e ence a ay wi h
qj,T=B−3∕2T−1∕2∑ ∈j∑ −1
k=1ujuj−k,j={ ∈ℕ∶1≤ ≤B,j+B−T≤ ≤j−1},and
T=0.5∫1
0𝜎2( )d +OP(B1∕2T−1∕2).
(b) Va [1,T]=Θ(1)and Va [2,T]=Θ(BT−1).
(c) I c=0and 𝜎2
=𝜎2 o all ∈ℕ,
2
T∶= 𝜎2Va [1,T]
E[2,T]=(T−B)(2B−1)−2(B−2)
3B(T−B).
The p e ious esul s sugges dis inguishing be ween di e en a es o B, which leads o wo undamen ally
di e en ypes o blockleng h asymp o ics. The ixed-bapp oach deno es he case whe e he ela i e blockleng h
B∕Tcon e ges o some alue bwi h 0<b<1, such ha Band Tg ow a he same a e. In he small-bapp oach,
we conside a ela i e blockleng h ha con e ges o ze o, while B,T→∞.1As he blocks a e o e lapping, he
e o e ms in he pooled eg ession equa ion a e co ela ed, bu , o una ely, he co ela ion s uc u e is known by
cons uc ion. Toge he wi h he cen al limi heo em o ma ingale di e ence a ays, he ollowing asymp o ic
esul can be es ablished o he small-bcase:
Theo em 1. Le 𝜌=1−c∕√BT wi h c≥0,le d sa is y Assump ion 1, and le u sa is y Assump ion 2. Le
B∕T→0as B,T→∞. Then,
1,T

−→(−c
2∫1
0
𝜎2( )d ,1
3∫1
0
𝜎4( )d ),and 2,T
p
−→1
2∫1
0
𝜎2( )d .
Since 2,Tcon e ges in p obabili y o a cons an , we ha e join con e gence o (1,T,2,T), and he pooled es i-
ma o is asymp o ically no mally dis ibu ed unde small-basymp o ics. Unde he uni oo hypo hesis 𝜌=1,o ,
equi alen ly, i c=0, i ollows ha
√BT(𝜌 −1)
−→(0,4
3
∫1
0𝜎4( )d
(∫1
0𝜎2( )d )2).
The asymp o ic a iance o 𝜌 in ol es in eg als o he second- and ou h-o de powe s o he unc ion 𝜎( ),
whe e he ac o ∫1
0𝜎4( )d ∕(∫1
0𝜎2( )d )2is equal o uni y in case o homoskedas ici y. This ac o also appea s
in he asymp o ic a iance ma ix o he OLS es ima o o he au o eg essi e coe icien unde uncondi ional
he e oskedas ici y (see Phillips and Xu, 2006).
Ca alie e (2005) showed ha pe manen changes in ola ili y induce a ime-shi in he igh -hand side p ocess o
he unc ional cen al limi heo em. A a iance- ans o med B ownian p ocess W𝜂( )appea s in he limi ing dis i-
bu ions o Dickey–Fulle - ype uni oo es s. Gi en he a iance p o ile 𝜂,whe e𝜂(s)=(
∫1
0𝜎2( )d )−1∫s
0𝜎2( )d ,
he ans o med p ocess is de ined as W𝜂( )=W(𝜂( )),whe eW( )is a s anda d B ownian mo ion. When impos-
ing ixed-basymp o ics, he nume a o and denomina o s a is ics can be ep esen ed as a pa ial sum p ocess o
he inno a ions, which leads o he ollowing limi ing esul :
1No e ha he e minology ‘ ixed-band small-basymp o ics’ was also used in he con ex o long- un a iance es ima ion. Whe eas Kie e
and Vogelsang (2005) used his wo ding o he asymp o ics o he a io o he unca ion poin o he sample size, we conside he a io o he
blockleng h o he sample size.
wileyonlinelib a y.com/jou nal/j sa © 2020 The Au ho s. J. Time Se . Anal. 42: 85–106 (2021)
Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12557

UNIT ROOT TESTING WITH SLOWLY VARYING TRENDS 89
Theo em 2. Le 𝜌=1−c∕√BT wi h c≥0,le d sa is y Assump ion 1, and le u sa is y Assump ion 2. Le
0<b<1,andle B∕T→bas B,T→∞. Then,
(1,T
2,T)
−→(0.5b−3∕2∫1
0𝜎2( )d (∫1−b
0(Jc,b,𝜂(b+ )−Jc,b,𝜂( ))2−b(1−b))
b−2∫1
0𝜎2( )d ∫1−b
0∫b+
(Jc,b,𝜂(s)−Jc,b,𝜂( ))2dsd ),
whe e Jc,b,𝜂( )=∫
0e−( −s)c∕bdW𝜂(s).
The limi ing dis ibu ions a e ep esen ed as unc ionals o he p ocess Jc,b,𝜂, which is an O ns ein–Uhlenbeck
ype p ocess ha is d i en by a a iance- ans o med Wiene p ocess. Consequen ly, he pooled es ima o is asymp-
o ically ep esen ed as a unc ional o a s anda d B ownian mo ion. I 𝜌=1, he con inuous mapping heo em
and Theo em 2 imply ha
√BT(𝜌 −1)
−→
b1∕2∫1−b
0(W𝜂(b+ )−W𝜂( ))2d +b3∕2(1−b)
2∫1−b
0∫b+
(W𝜂(s)−W𝜂( ))2dsd
unde ixed-basymp o ics. In compa ison o he limi ing dis ibu ion o he 𝜌-s a is ic in he Dickey–Fulle
amewo k, he unc ional includes an addi ional in eg al, which esul s om pooling he block eg essions.
To es ima e he unknown pa ame e s in he limi ing dis ibu ions, we conside he esiduals u =y −𝜌y −1 o
=2,…,Tand hei sample mean u=(T−1)−1∑T
j=2uj. Le , o no a ional con enience, u1=0,andle
𝜎2=1
T−2
T
∑
j=2
(uj−u)2,𝜅2=∑T−B
j=1∑B
=1(uj+1−u)2(uj+ −1
B∑B
k=1uj+k)2
∑T−B
j=1∑B
=1(uj+ −1
B∑B
k=1uj+k)2,
𝜂(s)=∑⌊sT⌋
j=2(uj−1
⌊sT⌋−1∑⌊sT⌋
k=2uk)2
+(sT −⌊sT⌋)(u⌊sT⌋+1−1
⌊sT⌋∑⌊sT⌋+1
k=2uk)2
∑T
j=2(uj−u)2
,
whe e s∈[0,1]. We ob ain he ollowing consis ency esul s:
Lemma 3. Le 𝜌=1−c∕√BT wi h c≥0,le d sa is y Assump ion 1, and le u sa is y Assump ion 2.
(a) 𝜎2p
−→∫1
0𝜎2( )d ,asB,T→∞.
(b) sups∈[0,1]|𝜂(s)−𝜂(s)|p
−→0,asB,T→∞.
(c) 𝜅2p
−→∫1
0𝜎4( )d ∕∫1
0𝜎2( )d ,asB,T→∞and B∕T→0.
3. PSEUDO -STATISTICS FOR UNIT ROOT TESTING
The p incipal concep o Dickey–Fulle - ype uni oo es s is o conside a pseudo - es o he null hypo h-
esis H0∶𝜌=1. Following his app oach in he pooled eg ession amewo k, he usual s anda d e o is
gi en by s𝜌 =𝜎(∑T−B
j=1∑B
=2(y +j−1−yj)2)−1∕2=𝜎(2,TB2T)−1∕2and he con en ional -s a is ic is ep esen ed as
(𝜌−1)∕s𝜌 =√B1,T∕√𝜎22,T, which di e ges in p obabili y unde H0. Acco dingly, we conside a scaled pseudo
-s a is ic o he o m
𝜏=𝜌 −1
s𝜌√B
=1,T
𝜎√2,T
,(2)
which is OP(1),asB,T→∞.
J. Time Se . Anal. 42: 85–106 (2021) © 2020 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12557 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
90 S. OTTO
In wha ollows, pseudo - es s a e de ined o bo h small-band ixed-bblock asymp o ics. To ge a
nuisance-pa ame e - ee limi ing dis ibu ion unde small-basymp o ics, we eplace 𝜎 by 𝜅 in (2). The small-b
pseudo -s a is ic is gi en as
𝜏-SB =1,T
𝜅 T√2,T
=∑T−B
j=1∑B
=2Δy +j(y +j−1−yj)
𝜅 T√B∑T−B
j=1∑B
=2(y +j−1−yj)2
.
The ac o Tis de ined in Lemma 2. Since T→2∕3, his e m p o ides a ini e-sample co ec ion and scales
he asymp o ic a iance o he -s a is ic o uni y. Unde ixed-basymp o ics, a nuisance e m appea s in he Gaus-
sian p ocess i sel . By means o ans o ming he da a wi h i s in e se a iance p o ile, Ca alie e and Taylo
(2007) showed ha he ime- ans o ma ion in he Gaussian limi ing p ocesses can be in e ed. The a iance p o-
ile es ima o 𝜂(s)is s ic ly inc easing and admi s he unique in e se unc ion 𝜂−1(s). Acco dingly, we conside
he ime- ans o med se ies y =y⌊𝜂−1( ∕T)T⌋ o =1,…,T. We eplace he o iginal se ies in he es s a is ic by y
and de ine

1,T=1
B3∕2T1∕2
T−B
∑
j=1
B
∑
=2
Δy +j(y +j−1−yj),
2,T=1
B2T
T−B
∑
j=1
B
∑
=2
(y +j−1−yj)2,
which yields he ixed-bs a is ic
𝜏-FB =

1,T
𝜎√
2,T
=∑T−B
j=1∑B
=2Δy +j(y +j−1−yj)
𝜎√B∑T−B
j=1∑B
=2(y +j−1−yj)2
.
In p ac ice, he ime- ans o med se ies y can ha e duplica e en ies in low ola ili y pe iods and he e o e may
no include all in o ma ion o he o iginal se ies in high ola ili y pe iods. Howe e , we do no need o disca d any
obse a ions when ans o ming he da a. We may a i icially ex end he se ies. An auxilia y sample size 
T≥T
can be chosen in such a way ha 𝜂−1( ∕
T)−𝜂−1(( −1)∕ 
T)≥
T−1 o all =1,…,
T. Then, he g id o wid h 1∕
T
is dense enough such ha y =y⌊𝜂−1( ∕
T)
T⌋, =1,…,
T, includes all sample poin s o he o iginal se ies, and he
ixed-bs a is ic may be applied o his auxilia y se ies. No e ha he auxilia y ime se ies is no necessa y om a
heo e ical poin o iew, bu i leads o be e es esul s in small samples.
Theo em 3. Le 𝜌=1−c∕√BT wi h c≥0,le d sa is y Assump ion 1, and le u sa is y Assump ion 2.
(a) Le B∕T→0as B,T→∞. Then,
𝜏-SB 
−→(−c√3
2
∫1
0𝜎2( )d
√∫1
0𝜎4( )d
,1).
(b) Le 0<b<1,andle B∕T→bas B,T→∞. Then,
𝜏-FB 
−→∫1−b
0(Jc,b(b+ )−Jc,b( ))2d −b(1−b)
2√b∫1−b
0∫b+
(Jc,b(s)−Jc,b( ))2dsd
,
whe e Jc,b( )=∫
0e−( −s)c∕bdW(s)is a s anda d O ns ein–Uhlenbeck p ocess.
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Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12557
UNIT ROOT TESTING WITH SLOWLY VARYING TRENDS 91
Table I. Asymp o ic c i ical alues o he ixed-b es
B∕T
𝛼−0.1−0.2−0.3−0.4−0.5−0.6−0.7−0.8−0.9
0.2 −0.788 −0.812 −0.815 −0.799 −0.761 −0.701 −0.623 −0.520 −0.377
0.1 −1.126 −1.128 −1.104 −1.055 −0.987 −0.903 −0.798 −0.664 −0.486
0.05 −1.403 −1.375 −1.327 −1.257 −1.169 −1.067 −0.939 −0.781 −0.573
0.04 −1.486 −1.446 −1.391 −1.318 −1.222 −1.113 −0.978 −0.814 −0.600
0.03 −1.582 −1.534 −1.471 −1.394 −1.291 −1.169 −1.025 −0.855 −0.630
0.02 −1.709 −1.650 −1.579 −1.489 −1.374 −1.246 −1.094 −0.909 −0.669
0.01 −1.904 −1.830 −1.745 −1.639 −1.511 −1.361 −1.191 −0.995 −0.729
0.001 −2.431 −2.320 −2.203 −2.042 −1.882 −1.692 −1.480 −1.226 −0.905
No e: The sample pa hs o he s anda d B ownian mo ions con ained in he asymp o ic null dis ibu ion o 𝜏-FB a e simula ed by a disc e ized
e sion o W( )on a g id o 50,000 equidis an poin s. The empi ical quan iles a e ob ained om 100,000 Mon e Ca lo epe i ions.
The uni oo hypo hesis is ejec ed in a o o s a iona i y i he es s a is ic is smalle han he 𝛼-quan ile o
he limi ing dis ibu ion o he case c=0,whe e𝛼is he signi icance le el. Fo 𝜏-SB we can ely on s anda d
no mal quan iles as c i ical alues. The limi ing dis ibu ion o 𝜏-FB is non-s anda d. No e ha Jc( )=W( )i
c=0. Table I p esen s simula ed le - ailed quan iles o he null dis ibu ion o a ious ela i e blockleng hs B∕T
and signi icance le els.
F om he poin o iew o a p ac i ione , he 𝜏-SB es has a numbe o ad an ages: he dis ibu ion is s anda d
no mal; hus, he e is no need o eso o new ables, and p- alues a e easy o implemen . In ac , he simula ions in
Sec ion 5 indica e ha he s anda d no mal app oxima ion is qui e accu a e in small samples i B=Θ(T𝛾),whe e
0.5≤𝛾≤0.8. Fu he mo e, he uni oo es is obus o he e oskedas ici y wi hou using any da a modi ica ion
me hod such as hose in Ca alie e and Taylo (2007) and Bea e (2018) o wild boo s ap implemen a ions (see
Ca alie e and Taylo , 2008a).
4. TESTING UNDER SHORT-RUN DYNAMICS
A mo e ealis ic scena io o mac oeconomic a iables is ha e o e ms a e se ially co ela ed. We impose
Assump ion 3 on he e o p ocess:
Assump ion 3 (se ially co ela ed e o s).The p ocess {u } ∈ℤpossesses he mo ing a e age ep esen a ion
u =𝜓(L)𝜖 =∑∞
i=0𝜓i𝜖 −iwi h ∑∞
i=0|𝜓i|<∞,whe eLis he usual lag ope a o . Mo eo e , all solu ions zo he
equa ion 𝜓(z)=0sa is y |z|>1. The p ocess {𝜖 } ∈ℤis independen ly dis ibu ed wi h E[𝜖 ]=0,E[𝜖2
]=𝜎2
and
E[𝜖4
]<∞,whe e𝜎 =𝜎( ∕T). The unc ion 𝜎( )is càdlàg, non-s ochas ic, s ic ly posi i e, and bounded.
Assump ion 3 implies ha he mo ing a e age ep esen a ion o u is in e ible, and we may w i e 𝜃(L)u =
u −∑∞
i=1𝜃iu −i=𝜖 ,whe e𝜃(z)=1−∑∞
i=1𝜃izi,and∑∞
i=1|𝜃i|<∞. To co ec o he e ec o sho - un dynamics,
we ollow B ei ung and Das (2005), among o he s, and conside he p e-whi ened se ies x∗
=𝜃(L)x .By(1),i
ollows ha
x∗
=𝜃(L)𝜌x −1+𝜃(L)u =𝜌x∗
−1+𝜖 ,
whe e 𝜖 sa is ies he same condi ions as u unde Assump ion 2. Consequen ly, i he uni oo s a is ics a e de ined
in e ms o
∗
1,T=1
B3∕2T1∕2
T−B
∑
j=1
B
∑
=2
Δx∗
+j(x∗
+j−1−x∗
j),∗
2,T=1
B2T
T−B
∑
j=1
B
∑
=2
(x∗
+j−1−x∗
j)2
ins ead o 1,Tand 2,T, hei limi ing dis ibu ions coincide wi h hose p esen ed in he p e ious sec ions.
J. Time Se . Anal. 42: 85–106 (2021) © 2020 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12557 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
92 S. OTTO
Since he au o eg essi e pa ame e s o he e o p ocess a e unknown, hey need o be es ima ed. In he ashion
o Said and Dickey (1984) and Chang and Pa k (2002), we ix some lag o de pTand conside he AR(pT) e o
ep esen a ion u =∑pT
i=1𝜃iu −i+𝜖pT, wi h 𝜖pT, =∑∞
i=pT+1𝜃iu −i+𝜖 . Then,
Δx =𝜙x −1+
pT
∑
i=1
𝜃iu −i+𝜖pT,T,(3)
which is equal o ∑pT
i=1𝜃iΔx −i+𝜖pT,Tunde he uni oo hypo hesis. The lag o de pTis allowed o g ow wi h he
sample size T. In wha ollows, we show ha he di e enced de e minis ic e ms a e asymp o ically negligible, as
pT→∞wi h pT=o(B1∕2), and we may eplace Δx −iby Δy −i o all i≥0in he augmen ed eg ession equa ion.
Le (𝜑,
𝜃1,…,
𝜃pT)′be he leas squa es coe icien ec o om he eg ession o Δy on y −1,Δy −1,…,Δy −pT, o
=pT+1.…,T.
Lemma 4. Le 𝜌=1−c∕√BT wi h c≥0,le d sa is y Assump ion 1, and le u sa is y Assump ion 3. Then,
∑pT
i=1(
𝜃i−𝜃i)=OP(pTB−1∕2),aspT,B,T→∞.
The es ima ed p e-whi ened se ies is de ined as y∗
=y −∑pT
i=1
𝜃iy −i, and he co esponding nume a o and
denomina o s a is ics a e gi en by

∗
1,T=1
B3∕2T1∕2
T−B
∑
j=1
B
∑
=2
Δy∗
+j(y∗
+j−1−y∗
j),
∗
2,T=1
B2T
T−B
∑
j=1
B
∑
=2
(y∗
+j−1−y∗
j)2.
Lemma 5. Le 𝜌=1−c∕√BT wi h c≥0,le d sa is y Assump ion 1, and le u sa is y Assump ion 3. Then,

∗
1,T−∗
1,T=OP(pTB−1∕2),and 
∗
2,T−∗
2,T=OP(pTT−1∕2),aspT,B,T→∞.
As a di ec consequence, (
∗
1,T−∗
1,T,
∗
2,T−∗
2,T)p
−→(0,0)i pT=o(B1∕2).Le 𝜌∗be gi en by √BT(𝜌∗−1)=

∗
1,T∕
∗
2,Tand le he p e-whi ened esiduals be de ined as u∗
=y∗
−𝜌∗y∗
−1, o =pT+1,…,T. Fo no a ional
con enience, le u∗
1=…=u∗
pT=0. The p e-whi ened coun e pa s o he es ima o s om Lemma 3 a e de ined as
𝜎∗2=1
T−2
T
∑
j=2
(u∗
j−u∗)2,𝜅∗2=∑T−B
j=1∑B
=1(u∗
j+1−u∗)2(u∗
j+ −1
B∑B
k=1u∗
j+k)2
∑T−B
j=1∑B
=1(u∗
j+ −1
B∑B
k=1u∗
j+k)2,
𝜂∗(s)=∑⌊sT⌋
j=2(u∗
j−1
⌊sT⌋−1∑⌊sT⌋
k=2u∗
k)2
+(sT −⌊sT⌋)(u∗
⌊sT⌋+1−1
⌊sT⌋∑⌊sT⌋+1
k=2u∗
k)2
∑T
j=2(u∗
j−u∗)2
.
Analogously, we conside he ime- ans o med p e-whi ened se ies y∗
=y∗
⌊𝜂∗−1( ∕T)T⌋ o all =1,…,T,whe e
𝜂∗−1(s)is he unique in e se o 𝜂∗(s), and we de ine

∗
1,T=1
B3∕2T1∕2
T−B
∑
j=1
B
∑
=2
Δy∗
+j(y∗
+j−1−y∗
j),
∗
2,T=1
B2T
T−B
∑
j=1
B
∑
=2
(y∗
+j−1−y∗
j)2.
Fo any lag o de pT≥0, he p e-whi ened e sions o he es s a is ics a e gi en by
𝜏-SBpT=

∗
1,T
𝜅∗ T√
∗
2,T
,𝜏-FBpT=

∗
1,T
𝜎∗√
∗
2,T
.
No e ha 𝜏-SB0=𝜏-SB and 𝜏-FB0=𝜏-FB. To summa ize, we ob ain he ollowing limi ing dis ibu ions:
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Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12557
UNIT ROOT TESTING WITH SLOWLY VARYING TRENDS 99
Table VII. Size and powe esul s o obus es s unde b eaks in end and a iance
Sample size T=100 T=300
𝜌𝜌=1𝜌=0.9𝜌=1𝜌=0.9
𝜆234234234234
Sha p b eak in a iance
𝜏-SB, B=T0.50.067 0.069 0.069 0.344 0.337 0.329 0.057 0.056 0.056 0.847 0.806 0.767
𝜏-SB, B=T0.60.071 0.075 0.077 0.420 0.421 0.416 0.062 0.061 0.061 0.954 0.933 0.909
𝜏-SB, B=T0.70.081 0.095 0.107 0.526 0.565 0.585 0.068 0.072 0.074 0.992 0.987 0.981
𝜏-SB, B=T0.80.085 0.124 0.162 0.569 0.683 0.756 0.082 0.116 0.147 0.999 1.000 1.000
𝜏-FB, B=0.2T0.040 0.039 0.040 0.283 0.261 0.238 0.045 0.044 0.042 0.947 0.882 0.812
𝜏-FB, B=0.4T0.043 0.042 0.041 0.346 0.308 0.276 0.047 0.046 0.045 0.982 0.935 0.876
𝜏-FB, B=0.6T0.042 0.040 0.042 0.349 0.327 0.307 0.045 0.045 0.045 0.989 0.974 0.947
Sha p b eak in end and a iance
𝜏-SB, B=T0.50.066 0.068 0.070 0.324 0.305 0.283 0.057 0.056 0.056 0.836 0.786 0.737
𝜏-SB, B=T0.60.071 0.074 0.076 0.391 0.370 0.344 0.062 0.061 0.061 0.946 0.916 0.881
𝜏-SB, B=T0.70.080 0.091 0.099 0.474 0.470 0.444 0.067 0.071 0.073 0.988 0.976 0.959
𝜏-SB, B=T0.80.095 0.145 0.194 0.526 0.595 0.627 0.081 0.111 0.136 0.997 0.996 0.993
𝜏-FB, B=0.2T0.040 0.039 0.038 0.260 0.228 0.197 0.046 0.044 0.043 0.935 0.854 0.765
𝜏-FB, B=0.4T0.044 0.042 0.043 0.295 0.240 0.200 0.047 0.046 0.046 0.960 0.872 0.772
𝜏-FB, B=0.6T0.042 0.043 0.047 0.292 0.240 0.205 0.046 0.045 0.046 0.954 0.866 0.766
No e: Simula ion esul s a e epo ed o 100,000 eplica ions. The e o s u a e simula ed independen ly as s anda d no mal andom a iables,
and he se ies a e no p e-whi ened (p=0). The sha p b eak speci ica ion is de ined by a b eak in he a iance a 2∕3o he sample. The
ejec ion equencies a e based on he asymp o ic c i ical alues o a signi icance le el o 5%.
The powe o he pooled es s depends on he blockleng h. In case o no b eak, a la ge blockleng h implies highe
powe esul s, which is in line wi h he heo e ical indings ha hose es s ha e powe in a 1∕√BT neighbo hood
o he uni oo hypo hesis. Fo blockleng hs o B=T0.8in he small-bcase and B=0.6Tin he ixed-bcase, he
powe esul s a e simila o hose om he ADF es and he Dickey–Fulle GLS es , whe e he o de ing depends
on he ini ial condi ion (c . Figu e 2). Hence, none o he es s domina es he pooled es s uni o mly ac oss hese
small-sample speci ica ions (al hough, asymp o ically, hose es s ha e powe in a 1∕Tneighbo hood o he uni
oo hypo hesis). Fu he mo e, smalle blockleng hs, such as T0.6in he small-bcon ex and 0.2Tin he ixed-b
con ex , s ill yield easonably high powe . In pa icula , he EL es pe o ms much wo se in all cases. The size
and powe esul s ob ained unde he AR(1) e o speci ica ion wi h bo h ixed and lexible lag augmen a ion o
he p e-whi ening scheme a e simila o hose p oduced by i.i.d. e o s.
As he es s a e designed o yield highe powe in he p esence o slowly a ying ends and b eaks, we compa e
he size-adjus ed powe s o he es s unde he end speci ica ions p esen ed in Table II and Figu e 1. Fo la ge
b eak sizes 𝜆, i is shown ha he smalle he blockleng h, he g ea e he powe esul s. In mos cases, he pooled
es s ha e g ea e powe han he ADF, he DF-GLS, he DF-GLS- end, and he EL es . Fu he mo e, he powe
esul s o he pooled es s a e qui e uni o m ac oss di e en end speci ica ions when compa ed o hose o he
con en ional es s.
Table VI shows ha he pooled es s ha e easonable size and powe p ope ies unde he p esence o AR(1)
e o s and di e en end speci ica ions. Fu he mo e, om Table VII, we can conclude ha he es s a e sized
co ec ly and ha e good powe p ope ies in he p esence o a b eak in he a iance and in he end unc ion.
The blockleng h Bis a uning pa ame e ha needs o be chosen ca e ully, and any op imali y esul would
depend on he ac ual end model. In p ac ice, howe e , he end model is unknown, which makes i ha d o
de i e an op imal blockleng h. Al hough heo e ical ecommenda ions canno be o mula ed based on he cu en
analysis, he small-b es s wi h B=T0.7and he ixed-b es s wi h T=0.2Byield e y p omising esul s o all
end unc ions s udied in his a icle and a e he e o e ecommended as he de aul se ings.
6. CONCLUSION
We ha e p esen ed wo a ian s o a uni oo es unde an unknown end speci ica ion ha a e obus unde bo h
he e oskedas ici y and au oco ela ion. When applied o ini e samples, he es s show good size p ope ies. The
J. Time Se . Anal. 42: 85–106 (2021) © 2020 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12557 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.

100 S. OTTO
ixed-bpooled es s a is ic con e ges o a unc ional o a B ownian mo ion unde he uni oo hypo hesis, while
he small-b a ian shows a s anda d no mal dis ibu ion in he limi . Au oco ela ion- obus e sions o he es s
we e in oduced using a p e-whi ening scheme. Mon e Ca lo simula ions indica e ha , while unde he ze o- end
speci ica ion, he ixed-band small-b es s pe o m simila o he con en ional es s in e ms o size and powe ,
unde sha p b eaks as well as smoo h changes in he end, hei powe is much highe . Fu he mo e, he powe s
o he es s a e less sensi i e o he ini ial alue when compa ed o he augmen ed Dickey–Fulle es and he
Dickey–Fulle GLS es .
ACKNOWLEDGEMENTS
I hank Jö g B ei ung and Ma ei Deme escu o hei ex ensi e ad ice and suppo . My hanks also go o Hans
Manne , Ma kus Kösle , Robinson K use-Beche , Dominik Wied, Naza ii Salish, Uwe Hassle , Ma in Wagne ,
he Co-Edi o , and wo anonymous e e ees o hei help ul commen s. The sugges ions made by pa icipan s
a ending he 2015 RMSE mee ing in Cologne, he SMYE con e ence 2017 in Halle (Saale), he SNDE con e ence
2017 in Pa is, and he IAAE con e ence 2017 in Sappo o a e also highly app ecia ed. Fu he mo e, he usage o he
CHEOPS HPC clus e o pa allel compu ing and a con e ence g an o he In e na ional Associa ion o Applied
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APPENDIX A. PROOFS
A.1. Auxilia y Resul s
Lemma A.1. Le 𝜌=1−c∕√BT wi h c≥0,le d sa is y Assump ion 1, and le u sa is y Assump ion 2.
Fu he mo e, le 1≤s≤B. Then,
(a) ∑B
=1||∑T−B
j=1Δd +jΔds+j||=O(1).
(b) ∑B
=1||∑T−B
j=1Δd +jΔxs+j||=OP(T1∕2).
The p oo is a ailable in he suppo ing in o ma ion in he online e sion o his a icle.
J. Time Se . Anal. 42: 85–106 (2021) © 2020 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12557 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
102 S. OTTO
A.2. P oo o Lemma 1
Fi s , we e o mula e he nume a o and denomina o s a is ics. No e ha
Δy +j(y +j−1−yj)−Δx +j(x +j−1−xj)=Δd +j(d +j−1−dj)+Δd +j(x +j−1−xj)+Δx +j(d +j−1−dj),
and
(y +j−1−yj)2−(x +j−1−xj)2=(d +j−1−dj)2+2(x +j−1−xj)(d +j−1−dj).
We decompose 1,T−1,T=S1+S2+S3and 2,T−2,T=S4+S5,whe e
S1=∑T−B
j=1∑B
=2Δd +j(d +j−1−dj)
B3∕2T1∕2,S2=∑T−B
j=1∑B
=2Δd +j(x +j−1−xj)
B3∕2T1∕2,
S3=∑T−B
j=1∑B
=2Δx +j(d +j−1−dj)
B3∕2T1∕2,S4=∑T−B
j=1∑B
=2(d +j−1−dj)2
B2T,
S5=∑T−B
j=1∑B
=22(x +j−1−xj)(d +j−1−dj)
B2T.
Lemma A.1 yields S1+S2+S3=OP(B−1∕2),andS4+S5=OP(T−1∕2), and he asse ion ollows by Slu sky’s
heo em.
A.3. P oo o Lemma 2
The p oo is a ailable in he suppo ing in o ma ion in he online e sion o his a icle.
A.4. P oo o Theo em 1
F om Lemma 2(a), i ollows ha E[q2
j,T]=O(T−1) o any j≤T, which implies ha Va [∑T
j=1qj,T]=
∑T−B
j=B+1E[q2
j,T]+o(1). The iden i y ∑n
=2∑ −1
k=1ak=∑n−1
k=1(n−k)akholds ue o any sequence (a ) ∈ℕ, which ollows
by induc ion on n. Then, o B+1≤j≤T−B,
B3∕2T1∕2qj,T=
B
∑
=2
−1
∑
k=1
ujuj−k=
B
∑
k=1
(B−k)uj−k=
B−1
∑
k=1
kujuj−B+k,
which yields
Va [T
∑
j=1
qj,T]=∑T−B
j=B+1∑B−1
k=1k2E[u2
j]E[u2
j−B+k]
B3T+o(1)
=∫T−B
T
B
T∫1
0
s2𝜎2( )𝜎2(j−⌊(1−s)B⌋
T)dsd +o(1)=∫1
0∫1
0
s2𝜎4( )dsd +o(1)
=1
3∫1
0
𝜎4( )d +o(1).
Mo eo e , we ha e max1≤j≤TE[q2
j,T]=o(1), and Jensen’s and Ma ko ’s inequali ies yield max1≤j≤T|qj,T|=oP(1).
Since {qj,T}is a ma ingale di e ence a ay, we can apply he cen al limi heo em om Theo em 24.3 in Da idson
(1994), which implies ha ∑T
j=1qj,T∕√Va [∑T
j=1qj,T]
−→(0,1),asT→∞. Fu he mo e, om Lemma 2,
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Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12557
UNIT ROOT TESTING WITH SLOWLY VARYING TRENDS 103
E[1,T]=−c∕2∫1
0𝜎2( )d +o(1), and he i s s a emen ollows om Lemma 1. Fo he second s a emen , no e
ha ,
E[2,T]= 1
B2T
T−B
∑
j=1
B
∑
=2
E[( −1
∑
k=1
Δxj+k)2]=1
B2T
T−B
∑
j=1
B
∑
=2
E[( −1
∑
k=1
uj+k+𝜙xj+k−1)2]
=1
B2T
T−B
∑
j=1
B
∑
=2
E[( −1
∑
k=1
uj+k)2]+o(1)= 1
B2T
T−B
∑
j=1
B
∑
=2
−1
∑
k=1
𝜎2
j+k+o(1)
=1
B2T
T−B
∑
j=1
B−1
∑
k=1
(B−k)𝜎2(j+k
T)+o(1)=∫T−B
T
0∫1
0
(1−s)𝜎2( +sB
T)dsd +o(1)
=∫1
0∫1
0
(1−s)𝜎2( )dsd +o(1)=1
2∫1
0
𝜎2( )d +o(1).
Fu he mo e, om Lemma 2, Va [2,T]=o(1), and he asse ion ollows by Chebyshe ’s inequali y oge he wi h
Lemma 1.
A.5. P oo o Theo em 2
Le XT( )=T−1∕2∑⌊ T⌋
k=1ukand YT( )=T−1∕2x⌊ T⌋ o ≥0. F om Lemmas 1 and 2 in Ca alie e (2005), i
ollows ha XT⇒𝜎W𝜂,whe e𝜎2=∫1
0𝜎2( )d deno es he a e age a iance. Fo no a ional con enience, we se
u0=x0. No e ha a Taylo expansion a ound 0 yields e−x=1−x+o(x), which implies ha 𝜌=1−c∕√BT =
exp(−c∕√BT)+o(1∕√BT). Then, wi h he con inuous mapping heo em, we ob ain
1
𝜎√T
x⌊ T⌋=⌊ T⌋
∑
k=0
𝜌⌊ T⌋−kuk
𝜎√T
=⌊ T⌋
∑
k=0
e−(⌊ T⌋−k)c∕√BT uk
𝜎√T
+oP(1)
=∫
0
e−( −s)c∕bdXT(s)+oP(1)⇒∫
0
e−( −s)c∕bdW𝜂(s)=Jc,b,𝜂( ),(A1)
which yields YT⇒𝜎Jc,b,𝜂. We ew i e
Δx +jx +j−1=Δx +j(x +j−1+x +j−Δx +j)
2
=(x +j−x +j−1)(x +j+x +j−1)−(Δx +j)2
2=
x2
+j−x2
+j−1−(Δx +j)2
2
such ha
B
∑
=2
Δx +j(x +j−1−xj)=
B
∑
=1
x2
+j−x2
+j−1−(Δx +j)2
2−Δx +jxj
=1
2(x2
j+B−x2
j)−(xj+Bxj−x2
j)−1
2
B
∑
=1
(Δx +j)2=(xj+B−xj)2
2−1
2
B
∑
=1
(Δx +j)2.
J. Time Se . Anal. 42: 85–106 (2021) © 2020 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12557 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
104 S. OTTO
Then, wi h Lemma 1,
1,T=1,T+oP(1)=∑T−B
j=1(xB+j−xj)2−∑T−B
j=1∑B
=1(Δx +j)2
2B3∕2T1∕2
=∫1−b
0(YT(b+ )−YT( ))2d −1
T2∑T−B
j=1∑B
=1(Δx +j)2
2b3∕2+oP(1).
F om Δx =u , i ollows ha
E[1
T2
T−B
∑
j=1
B
∑
=1
(Δx +j)2]=1
T2
T−B
∑
j=1
B
∑
=1
E[u2
+j]=b(1−b)∫1
0
𝜎2( )d +o(1),
which implies ha
1,T=∫1−b
0(YT(b+ )−YT( ))2d −b(1−b)∫1
0𝜎2( )d
2b3∕2+oP(1).(A2)
Fu he mo e, Lemma 1 yields
2,T=2,T+oP(1)= 1
b2∫1−b
0∫b+
(YT(s)−YT( ))2dsd +oP(1).(A3)
The asse ion ollows om (A1), oge he wi h he con inuous mapping heo em.
A.6. P oo o Lemma 3
Since (1−𝜌)=OP(B−1∕2T−1∕2)and x =OP(T1∕2), he esiduals sa is y
u =y −𝜌y −1=Δy +(1−𝜌)y −1
=Δd +u +(𝜌−1)x −1+(1−𝜌)y −1=u +OP(B−1∕2)
and u=OP(T−1∕2). Then, o any s∈[0,1],
1
T⌊sT⌋
∑
j=1
(uj−u)2=1
T⌊sT⌋
∑
j=1
u2
j+OP(B−1∕2)=∫s
0
𝜎2( )d +oP(1),(A4)
and (a) ollows wi h s=1. Fu he mo e, by Slu sky’s heo em, 𝜂(s)=𝜂(s)+oP(1)holds poin wise o all s∈[0,1].
Then, (b) ollows by Dini’s heo em since bo h 𝜂(s)and 𝜂(s)a e con inuous, mono one, and bounded. Fo (c),
no e ha
1
T−B
T−B
∑
j=1(uj+ −1
B
B
∑
k=1
uj+k)2
=1
T−B
T−B
∑
j=1
u2
j+ +OP(B−1∕2),(A5)
o any =1,…,B. Equa ions (A4) and (A5) yield
1
(T−B)B
T−B
∑
j=1
B
∑
=1(uj+ −1
B
B
∑
k=1
uj+k)2
=∫1
0
𝜎2( )d +oP(1),
wileyonlinelib a y.com/jou nal/j sa © 2020 The Au ho s. J. Time Se . Anal. 42: 85–106 (2021)
Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12557

UNIT ROOT TESTING WITH SLOWLY VARYING TRENDS 105
1
(T−B)B
T−B
∑
j=1
B
∑
=1
(uj+1−u)2(uj+ −1
B
B
∑
k=1
uj+k)2
=∫1
0
𝜎4( )d +oP(1),
as B,T→∞and B∕T→0, and he esul ollows by Slu sky’s heo em.
A.7. P oo o Theo em 3
No e ha T→√2∕3,and 𝜅 T√2,T
p
−→√∫1
0𝜎4( )d ∕3, which ollows om Theo em 1 and Lemma 3. Then,
(a) ollows oge he wi h Slu sky’s heo em. Fo (b), le x⌊ T⌋=x⌊𝜂−1( )T⌋and u⌊ T⌋=u⌊𝜂−1( )T⌋.Fu he mo e,le

XT( )=T−1∕2∑⌊ T⌋
k=1ukand 
YT( )=T−1∕2x⌊ T⌋. Theo em 1 in Ca alie e and Taylo (2008b) s a es ha 
XT⇒𝜎W,
whe e 𝜎2=∫1
0𝜎2( )d , and, analogously o (A1), i ollows ha 
YT⇒Jc,b. Following (A2) and (A3), we ob ain
𝜏-FB =(∫1−b
0(
Y(b+ )−
Y( ))2d −b(1−b)𝜎2)∕(2b3∕2)
√𝜎∫1−b
0∫b+
(
Y(s)−
Y( ))2dsd ∕b2
+oP(1),
and he asse ion ollows wi h he con inuous mapping heo em and Slu sky’s heo em.
A.8. P oo o Lemma 4
The p oo is a ailable in he suppo ing in o ma ion in he online e sion o his a icle.
A.9. P oo o Lemma 5
The p oo is a ailable in he suppo ing in o ma ion in he online e sion o his a icle.
A.10. P oo o Theo em 4
Le , o no a ional con enience, 𝜃0=
𝜃0=−1,andle 
𝜃(z)=1−∑pT
i=1
𝜃izi, which yields y∗
=
𝜃(L)y .Le

d∗
=
𝜃(L)d and x∗
=
𝜃(L)x . Analogously o he p oo o Lemma 3, we ha e
u∗
=y∗
−𝜌∗y −1=Δy∗
+(1−𝜌∗)y∗
−1=Δx∗
+O(B−1∕2)=𝜖 +oP(1).
The consis encies o 𝜎∗2,𝜅∗2,and 𝜂∗(s) ollow om he ac ha
1
T⌊sT⌋
∑
j=1
(u∗
j−u∗)2=1
T⌊sT⌋
∑
j=1
𝜖2
j+oP(1)=∫s
0
𝜎2( )d +oP(1),s∈[0,1],
and
1
T−B
T−B
∑
j=1(u∗
j+ −1
B
B
∑
k=1
u∗
j+k)2
=1
T−B
T−B
∑
j=1
𝜖2
j+ +oP(1),
1
(T−B)B
T−B
∑
j=1
B
∑
=1(u∗
j+ −1
B
B
∑
k=1
u∗
j+k)2
=∫1
0
𝜎2( )d +oP(1),
J. Time Se . Anal. 42: 85–106 (2021) © 2020 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12557 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
106 S. OTTO
1
(T−B)B
T−B
∑
j=1
B
∑
=1
(u∗
j+1−u∗)2(u∗
j+ −1
B
B
∑
k=1
u∗
j+k)2
=∫1
0
𝜎4( )d +oP(1),
whe e he las wo equa ions hold ue as B∕T→0, analogously o Lemma 3.
Finally, since he p e-whi ened nume a o and denomina o s a is ics (∗
1,T,∗
2,T)unde Assump ion 3 ha e
he same p ope ies as (1,T,2,T)unde Assump ion 2, he asse ion ollows wi h Lemma 5 and he p oo o
Theo em 3.
wileyonlinelib a y.com/jou nal/j sa © 2020 The Au ho s. J. Time Se . Anal. 42: 85–106 (2021)
Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12557