PPP o e a cen u y: Coin eg a ion and s uc u al change
Eka e ini Panopoulou
Na ional Uni e si y o I eland, Maynoo h∗
Feb ua y 2006
Abs ac
The pu pose o his pape is o in es iga e he abili y o pa ame e ins abili y es s in eg essions
wi h I(1) p ocesses o disc imina e be ween changes in he coin eg a ing ela ionship and changes in he
ma ginal dis ibu ion o he eg esso s. Using annual da a o he G-7 coun ies and he Pu chasing Powe
Pa i y, we conclude ha he eg ession coefficien be ween he p ice le el diffe en ial and he exchange
a e has indeed emained s able du ing he 20 h cen u y and ind ample e idence suppo ing he PPP.
JEL classi ica ion: F31, C13, C22
Keywo ds: coin eg a ion es ima o s; PPP; s uc u al change; small-sample p ope ies; s uc u al s a-
bili y es s;
Acknowledgemen s: I am g a e ul o N. Pi is and M. Roche o help ul sugges ions and commen s.
The usual disclaime applies.
∗Co espondence o: Eka e ini Panopoulou Depa men o Economics, Na ional Uni e si y o I eland, Maynoo h,
Co. Kilda e, Republic o I eland. E-mail: [email protected]. Tel: 00353 1 7083793. Fax: 00353 1 7083934.
1. In oduc ion
Tes ing o eg ession pa ame e cons ancy has a long his o y in economics. The main body
o he li e a u e, howe e , is de o ed o eg essions in ol ing s a iona y se ies (see Pe on, 2005,
o a li e a u e e iew and he e e ences he ein). Since Engle and G ange ’s seminal pape
in 1987, he issue o es ima ing coin eg a ion pa ame e s has ecei ed conside able a en ion in
bo h he heo e ical and applied econome ics li e a u e. Consequen ly, pa ame e cons ancy
es s alid o coin eg a ing ela ionships we e de eloped. Hansen (1992) was he i s o ex end
s uc u al b eak es s o eg essions in ol ing in eg a ed a iables. These es s a e de i ed as
Lang ange- Mul iplie (LM) es s in co ec ly speci ied likelihood p oblems. The au ho , using
he ully modi ied es ima o o Phillips and Hansen (1990), de i es he limi ing dis ibu ion o
he espec i e es s a is ics unde he null o coefficien cons ancy, which ollow asymp o ically
non-s anda d, bu nuisance ee dis ibu ions.
The pu pose o his pape is o in es iga e he abili y o hese pa ame e ins abili y es s in
eg essions wi h I(1) p ocesses o disc imina e be ween changes in he coin eg a ing ela ionship
and changes in he ma ginal dis ibu ion o he eg esso s.1The e alua ion akes place wi hin a
bi a ia e coin eg a ing amewo k o he mos popula pa i y ela ionship in In e na ional Eco-
nomics, he Pu chasing Powe Pa i y (PPP). Using annual da a o he G-7 coun ies, we ind
ha he eg ession coefficien be ween he p ice le el diffe en ial and he exchange a e has indeed
emained s able du ing he 20 h cen u y. Ou esul s sugges ha when a single b eak occu s
in he p ocess ha d i es he eg esso , while he pa ame e o in e es emains s able, he es s
display se e e size dis o ions, leading o an o e ejec ion o he null. By means o Mon e Ca lo
simula ions, we ob ain small sample c i ical alues o he es s along wi h he empi ical dis ib-
u ion o some commonly used coin eg a ion es ima o s. On he whole, we es ablish he s abili y
o he coin eg a ing coefficien and ind ample e idence suppo ing PPP in he G-7 coun ies o e
he las cen u y.
The es o he pape is o ganised as ollows: Sec ion 2 b ie ly e iews he s abili y es s.
Sec ion 3 epo s he empi ical and Mon e Ca lo esul s and sec ion 4 concludes he pape .
2. S uc u al s abili y es s
The model employed by Hansen (1992) is a s anda d mul iple eg ession coin eg a ing model:
y =Ax +u1 , =1,2,...n, whe e x =(x0
1 ,x
0
2 )0gi enby heequa ions: x0
1 =k1 ,x
0
2 =
Q
1
k1 +Q
2
k2 +x0
2 and x0
2 =x0
2 −1+u2 .The elemen s o u and k a e a sequence o mean ze o
andom ec o s and nonega i e in ege powe s o ime, espec i ely. Associa ed wi h he p e ious
sys em a e he long- un co a iance ma ix, Ω, and he one-sided co a iance ma ix, Λ,de ined
as Ω=lim
n→∞
1
n
n
P
=1
n
P
j=1
E(uju0
)and Λ=lim
n→∞
1
n
n
P
=1
P
j=1
E(uju0
).We pa i ion Ωand Λcom o mably
wi h uand ha e Ω=ÃΩ11 Ω12
Ω21 Ω22 !and Λ=ÃΛ11 Λ12
Λ21 Λ22 !.ThenΩ1.2=Ω11 −Ω12Ω−1
22 Ω21
ep esen s he long- un a iance o u1 condi ional on u2 and Λ+
21 =Λ21 −Λ22Ω−1
22 Ω21 he bias
due o he endogenei y o he eg esso s.
1Hansen (2000) shows ha in a s a iona y en i onmen he asymp o ic dis ibu ion o hese es s is affec ed by
changes in he ma ginal dis ibu ion o he eg esso s.
1
Hansen (1992) p oposed h ee es s o pa ame e cons ancy in he con ex o coin eg a ion,
namely he SupF,MeanF,andLc.They all equi e an asymp o ic mixed no mal es ima e o A
and ha e he same null hypo hesis bu diffe in hei choice o al e na i e hypo hesis. The null
hypo hesis o he h ee es s is ha he coefficien Ais cons an and he iming o he b eak is
unknown. The es s a is ic employed by he i s wo es s is
Fn = ec(Sn )0(b
Ω1·2⊗Vn )−1 ec(Sn )= {S0
n V−1
n Sn b
Ω−1
1·2}
whe e Sn =
P
i=1
ˆsi,ˆs =³x ˆu+
1 −¡0
ˆ
Λ+
21¢´,V
n =Mn −Mn M−1
nn Mn ,and Mn =
P
i=1
xix0
i.The main
diffe ence is he ea men o al e na i e hypo heses.
The i s es modelsAas unde going a single s uc u al b eak a unknown ime and he
al e na i e hypo hesis is H1:A16=A2.This es s a is ic is simply supF =supFn
[ /n]∈T
,[ /n]∈T,
whe e Tis some compac subse o (0,1) and [·]deno es in ege pa . The second and hi d
es model he pa ame e A as a ma ingale p ocess: A =A −1+ε ;E(ε /T −1)=0and
E(ε ε0
)=δ2G ,whe e G =(
b
Ω1·2⊗Vn )−1.Then he null hypo hesis can be o mula ed as
a cons ain ha he a iance o he ma ingale diffe ences is 0: H0:δ2=0.The al e na i e
hypo hesis is H2:δ2>0,wi h es s a is ic MeanF =1
n∗PFn
[ /n]∈T
,whe e n∗=P1
[ /n]∈T
and [ /n]∈T.
The hi d al e na i e is H3:δ2>0,wi h es s a is ic Lc= {M−1
nn
n
P
=1
S b
Ω1·2S0
}.
The supF da es back o Quand (1960) and en ails he choice o he egion T. Following
And ews (1993) and Hansen (1992), we selec T=[.15,.85].Some imming is also equi ed
o he meanF, which is simply he a e age Fn es . The easie o all o compu e is he Lc
es , which equi es no imming. The asymp o ic dis ibu ions o he es s a is ics a e non
s anda d and depend on he s ochas ic p ocess o he eg esso s. In his espec , asymp o ic
c i ical alues we e calcula ed by Mon e Ca lo simula ions (Tables 1-3, Hansen, 1992). Mo eo e ,
he es s a e dependen on he es ima ion p ocedu e o he coin eg a ing ela ionship. In his
s udy, we calcula e he es s ia he Fully Modi ied (FM-S) es ima ion echnique o Phillips and
Hansen (1990), which in u n equi es he selec ion o a ke nel and he de e mina ion o he
bandwid h. We employ he Quad a ic Spec al ke nel and selec he bandwid h pa ame e by
applying he And ews (1991) da a-dependen p ocedu e. Mo eo e , he “p ewhi ened” e sion
o FM-S (FM-PW) which il e s he e o ec o b
u p io o es ima ing Ωand ∆is employed.
3. Empi ical esul s and Mon e Ca lo simula ions
3.1. Da a
Ou da a se consis s o annual da a on he bila e al exchange a e o e USD and he WPI o
he G-7 coun ies (excluding Ge many).2The sample co e s a cen u y o da a, namely he pe iod
1800-1999.3In pa icula , we ocus on he PPP ela ionship, linking nominal exchange a es o
p ice diffe en ials and equi ing one- o-one adjus men o he o me o he la e . Fo mally, PPP
2Da a o Ge many a e no eliable due o he uni ica ion p ocess.
3The da a a e ob ained om Global Financial Da a, Annual Wo kshee s. The coun ies conside ed a e he
ollowing (wi h he i s yea o he sample in b acke s): Canada (1900), F ance (1900), I aly (1910), Japan
(1900), UK (1900), US (1900).
2
can be exp essed by he ollowing eg ession s =a+b(p∗
−p )+ ,whe e s and p∗
−p is he
nominal exchange a e and he p ice gap be ween wo ma ke s.4The null hypo hesis o be es ed
can ake he o m: PPP holds ⇔b=1.Usually, he li e a u e ag ees on he ac ha s and
(p∗
−p )a e coin eg a ed p ocesses. On he o he hand, es ima es o bappea o be signi ican ly
diffe en om uni y, hus cas ing doub s on he pa i y. As a i s s ep, we ca ied ou ypical
uni oo es s and coin eg a ion es s. On he whole, he esul s (no epo ed) b oadly con i m
hose o o he s udies, i.e. nominal exchange a es and p ice diffe en ials seem o be I(1) and
coin eg a ed p ocesses.
3.2. S uc u al s abili y es s
A e es ablishing ha he in ol ed a iables a e in eg a ed and coin eg a ed, we p oceed
wi h es ing o he s abili y o he coin eg a ion pa ame e since ejec ing he null o s abili y o
he coin eg a ing coefficien cons i u es an empi ical ailu e o he PPP and p obably es ima ion
has o be done in diffe en subsamples.
Table 1 epo s he esul s o he s abili y es s. On he basis o he supF es , we ejec he
null o s abili y in all coun ies wi h he b eak es ima ed a a ound he end o Wo ld wa II. The
o he wo es s p o ide mixed e idence on he issue wi h esul s being mo e obus o Canada,
I aly and Japan.5Employing he s anda d o he p ewhi ened e sion o he FMLS es ima o
does no in gene al lead o con lic ing esul s, wi h he excep ion o he iming o he b eak in
I aly and UK, which is shi ed by 5 and 8 pe iods, espec i ely.
[INSERT TABLE 1]
3.3. Mon e Ca lo simula ions
He e, we examine whe he he p esence o a b eak in he ma ginal dis ibu ion o he eg esso s
affec s he size o he es s which p obably e oneously de ec i as a coefficien b eak. To do
his, we assume ha s and (p∗
−p )a e gene a ed by he ollowing bi a ia e iangula DGP:
s =a+b(p∗
−p )+u1 (1)
∆(p∗
−p )=u2 (2)
We u he assume ha u =[u1 ,u
2 ]0is an I(0) p ocess and ollows a VAR(1) p ocess, i.e
u =A u −1+e :and e ∼NIID(0,Σ ).The model pa ame e s a e calib a ed om he exchange
a e and ela i e p ice da a om he coun ies a hand. In pa icula , he pa ame e o in e es ,
b, is se equal o one o he whole sample, while he co a iance ma ix Σ and he ansi ion
ma ix A is allowed o unde go a single b eak a he ime indica ed by he sup F es (see Table
1, FM-PW).6We conside a sample size o 150 obse a ions and eplica e simula ions 5000 imes.
To accoun o he effec o he ini ial condi ions, he i s 50 obse a ions a e disca ded. The
pe cen ejec ions o he null o a nominal 5% size along wi h he a e age es ima ed iming o he
4All a iables in logs.
5Following Hansen (1992) he cu -offpoin p- alue is se a 0.20.
6The pa ame e alues o he co a iance ma ix Σ and he ansi ion ma ix A o he wo subsamples and
he i e coun ies unde sc u iny a e no epo ed o b e i y bu a e a ailable upon eques om he au ho .
3
b eak and i s a endan s anda d e o o he h ee es s and he coun ies a hand a e epo ed
in Table 2. We also calcula e new c i ical alues co esponding o an empi ical size o 5% and
employ hem o assess whe he s abili y is ejec ed o no . These esul s along wi h he desicion
o he es s a e epo ed in Table 3.
[INSERT TABLE 2 & 3]
The in o ma ion con en in hese Tables may be summa ised as ollows:
(i) All he es s a e o e sized o all coun ies wi h he excep ion o he Lc es (FM-PW
e sion) o Canada. Size dis o ions in some cases each 100%. Fo example, he size o bo h
he meanF and sup F o I aly is 99%.
(ii) The Lc es u ns ou o be he leas dis o ed. Wi h he excep ion o I aly, he espec i e
alues o he emaining coun ies ange om 3.6% o 13%.
(iii) Employing he p ewhi ened e sion o FMLS does no necessa ily lead o size gains. On
he con a y, he size o he es s inc eases conside ably o F ance, I aly and Japan.
(i ) The a e age iming o he b eak is es ima ed wi h a la ge s anda d de ia ion, which can
each 23 yea s in some cases.
( ) On he basis o he small sample c i ical alues o he es s (Table 3), we o e all canno
ejec he null o s abili y o he coin eg a ion coefficien . The only cases we ejec he null o
s abili y is on he g ounds o he Lc es o Canada and he SupF es o F ance.
O e all ou esul s sugges ha du ing he las cen u y he ma ginal dis ibu ions o exchange
a es and p ice diffe en ials ha e wi nessed a s uc u al b eak, while he pa ame e o in e es
has indeed emained cons an . Nex , we es ima e he coin eg a ing pa ame e and es whe he
i is equal o one o he coun ies a hand.
3.4. Coin eg a ion es ima o s
The coin eg a ion es ima o s we employ a e he O dina y Leas Squa es (OLS), he Au-
o eg essi e Dis ibu ed Lag (ADL) (Pesa an and Shin, 1999), he Johansen’s (JOH) maximum
likelihood (1988, 1991) and he Fully Modi ied leas squa es es ima o s (FM-PW) (Phillips and
Hansen, 1990) es ima o s. The es ima es o ba e epo ed in Table 4 along wi h he associa ed
s anda d e o s and he - es s o he null hypo hesis o in e es b=1 o all he coun ies unde
conside a ion. We also es he hypo hesis o in e es based on he 2.5% ( 0.025)and he 97.5%
( 0.975)poin s in he empi ical dis ibu ions o he ele an -s a is ics h ough Mon e Ca lo sim-
ula ions.7These simula ed c i ical alues along wi h he empi ical sizes o he - es s o nominal
sizes o 5% a e epo ed in Table 5.
[INSERT TABLE 4&5]
On he basis o asymp o ic c i ical alues, PPP su i es he empi ical e idence only o
Canada and I aly i espec i e o he es ima o employed. Fo he emaining coun ies e idence
is mixed as we canno ejec he null o a uni coefficien o F ance based on JOH and o he
UK based on he ADL and FM-PW es ima o s. Na u ally, he pe o mance o he es ima o s is
affec ed by he p esence o a b eak. Ou simula ions sugges ha he dis ibu ions o he ele an
-s a is ics become less lep oku ic and in mos cases shi o he le (Table 5). This leads o
signi ican size inc eases, which o he asymp o ically efficien es ima o s ange om 9.6% o
7The design o he Mon e ca lo expe imen is simila o he one in Sec ion 3.3.
4

44.4% o a nominal 5% size. Consequen ly, employing c i ical alues leads o ewe ejec ions o
a uni coefficien hypo hesis. Speci ically, we ind e idence in a o o PPP o all he coun ies a
hand when small sample c i ical alues and he asymp o ically efficien es ima o s a e employed.
4. Conclusions
In hispape weha eexamined heeffec s o a b eak in he ma ginal dis ibu ion o in eg a ed
eg esso s on he size o some commonly used s abili y es s o he coin eg a ion coefficien ,
namely he SupF,MeanF,andLc(Hansen, 1992). By means o Mon e Ca lo simula ions we
showed ha he es s e oneously de ec a b eak in he coin eg a ing coefficien when he b eak
ac ually occu s in he ma ginal dis ibu ion o he eg esso s. These issues we e add essed in an
empi ical amewo k, speci ically he Pu chasing Powe Pa i y o he G-7 coun ies o which we
ound ample e idence.
Re e ences
[1]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.
[2]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.
[3]Engle, R.F. and C.W.J. G ange . 1987, Coin eg a ion and e o co ec ion ep esen a ion,
es ima ion and es ing, Econome ica 55, 251-276.
[4]Hansen, B.E., 1992, Tes s o pa ame e ins abili y in eg essions wi h I(1) p ocesses, Jou nal
o Business and Economic S a is ics 20, 45-59.
[5]Hansen, B.E., 2000, Tes ing o s uc u al change in condi ional models, Jou nal o Econo-
me ics 97, 93-115.
[6]Johansen, S., 1988, S a is ical analysis o coin eg a ing ec o s, Jou nal o Economic Dynam-
ics and Con ol, 12, 231-254.
[7]Johansen, S., 1991, Es ima ion and hypo hesis es ing o coin eg a ion ec o s in Gaussian
ec o au o eg essi e models, Econome ica, 59, 1551-1580.
[8]Pe on, P., 2005, Dealing wi h s uc u al b eaks, Palg a e Handbook o Econome ics, Vol.1:
Econome ics Theo y.
[9]Pesa an H.M. and Y. Shin (1999), An Au o eg essi e Dis ibu ed Lag Modelling App oach
o Coin eg a ion Analysis, Econome ics and Economic Theo y in he 20 h Cen u y: The
Ragna F isch Cen ennial Symposium, chap e 11. Camb idge Uni e si y P ess, Camb idge.
[10]Phillips, P.C B and Hansen, B.E, 1990, S a is ical in e ence in ins umen al a iables eg es-
sion wi h I(1) p ocesses, Re iew o Economic S udies 57, 99-125.
[11]Quand , R. , 1960, Tes s o he hypo hesis ha a linea eg ession sys em obeys wo sepa a e
egimes, Jou nal o he Ame ican s a is ical Associa ion 55, 324-330.
5
Table 1: S uc u al S abili y Tes s
Coun y Canada F ance I aly Japan UK
FM-S FM-PW FM-S FM-PW FM-S FM-PW FM-S FM-PW FM-S FM-PW
Lc1.027 0.665 0.177 0.150 0.278 0.324 0.308 0.275 0.327 0.116
(0.01) (0.01) (0.20) (0.20) (0.17) (0.13) (0.14) (0.18) (0.13) (0.20)
MeanF 8.204 5.302 2.647 2.199 4.810 11.922 9.677 11.698 2.904 2.732
(0.01) (0.03) (0.20) (0.20) (0.04) (0.01) (0.01) (0.01) (0.20) (0.20)
SupF 19.429 14.918 19.753 18.130 33.754 88.843 8.892 34.42 4.376 7.448
(0.01) (0.02) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01)
B eak 1946 1946 1945 1945 1934 1939 1948 1948 1954 1946
No e: p- alues in pa en heses below es s.
Table 2: Size o S uc u al S abili y Tes s
Coun y Canada F ance I aly Japan UK
FM-S FM-PW FM-S FM-PW FM-S FM-PW FM-S FM-PW FM-S FM-PW
Lc9.10 3.60 5.85 6.45 55.00 81.75 6.95 12.50 12.25 5.35
MeanF 16.40 7.50 8.70 9.50 99.15 100 29.15 42.45 39.55 20.80
SupF 18.00 9.10 6.70 8.55 98.80 100 21.35 36.55 37.95 18.10
B eak 35 36 27 26 46 47 45 45 45 39
(s.e.)(23) (23) (20) (20) (14) (14) (19) (19) (21) (23)
No e: Empi ical size o a nominal 5% size.
Table 3: Mon e Ca lo C i ical Values
Coun y Canada F ance I aly Japan UK
FM-S FM-PW FM-S FM-PW FM-S FM-PW FM-S FM-PW FM-S FM-PW
Lc0.729 0.507 0.611 0.648 1.813 2.845 0.622 0.765 0.745 0.55
No No Yes Yes Yes Yes Yes Yes Yes Yes
MeanF 8.619 5.328 5.431 5.656 83.35 150.88 10.29 16.79 7.424 4.888
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
SupF 28.47 16.27 13.23 14.376 277.75 492.53 26.49 44.21 19.04 10.39
Yes Yes No No Yes Yes Yes Yes Yes Yes
No e: “Yes” indica es non- ejec ion o he null o coefficien cons ancy and “No” indica es
ejec ion o he null.
6
Table 4: Es ima ion esul s
Coun y Canada F ance I aly Japan UK
Es ima o b
s a b
s a b
s a b
s a b
s a
OLS 0.914 -0.694 1.040 2.938 0.963 -1.663 0.942 -4.425 0.783 -4.227
(0.124) (0.014) (0.023) (0.013) (0.051)
ADL 1.056 0.241 1.043 2.243 0.978 -0.731 0.927 -2.114 0.839 -1.508
(0.234) (0.019) (0.030) (0.034) (0.107)
JOH 1.174 0.994 1.122 0.705 0.995 -0.208 0.952 -2.400 0.754 -2.625
(0.175) (0.173) (0.024) (0.020) (0.094)
FM −PW 1.064 0.405 1.044 3.143 0.997 -0.143 1.049 2.631 1.295 1.815
(0.158) (0.014) (0.021) (0.019) (0.163)
No e: S anda d e o s in pa en heses below coefficien s.
Table 5: Small-sample pe o mance o coin eg a ion es ima o s
Canada F ance I aly Japan UK
Es ima o 0.025 0.975 0.025 0.975 0.025 0.975 0.025 0.975 0.025 0.975
OLS -13.221 7.231 -2.780 2.266 -6.672 8.538 -4.051 3.839 -4.781 4.908
(67.80) (14.05) (59.20) (30.30) (40.30)
ADL -4.448 3.044 -2.633 2.255 -3.636 2.950 -2.491 2.549 -2.158 2.812
(24.10) (10.60) (20.05) (9.90) (11.40)
JOH -5.221 4.143 -2.499 2.378 -2.820 2.055 -3.250 3.445 -2.816 2.967
(44.35) (10.15) (9.60) (22.10) (16.15)
FM −PW -2.010 2.665 -2.576 2.623 -4.433 1.874 -3.024 3.211 -2.393 2.756
(10.85) (13.90) (25.10) (17.90) (12.50)
No e: Empi ical size o he - es s o a 5% empi ical size in pa en heses.
7