The Con ibu ion o G ow h and In e es Ra e Diffe en ials o he
Pe sis ence o Real Exchange Ra es
Dimi ios Mallia opulos∗Eka e ini Panopoulou†
Theologos Pan elidis‡Niki as Pi is§
Feb ua y 2006
Abs ac
This pape employs a new me hodology o measu ing he con ibu ion o g ow h and in e es
a e diffe en ials o he hal -li e o de ia ions om Pu chasing Powe Pa i y (PPP). Ou me hod
is based on di ec ly compa ing he impulse esponse unc ion o a VAR model, whe e he eal
exchange a e is G ange caused by hese a iables wi h he impulse esponse unc ion o a
uni a ia e ARMA model o he eal exchange a e. We show ha he impulse esponse unc ion
o he VAR model is no , in gene al, he same wi h he impulse esponse unc ion ob ained om
he equi alen ARMA ep esen a ion, i he eal exchange a e is G ange caused by o he
a iables in he sys em. The diffe ence be ween he wo unc ions cap u es he effec s o he
G ange -causing a iables on he hal -li e o de ia ions om PPP. Ou empi ical esul s o a
se o ou cu encies sugges ha eal and nominal long e m in e es a e diffe en ials and eal
GDP g ow h diffe en ials accoun o 22% o 50% o he hal -li e o de ia ions om PPP.
Keywo ds: eal exchange a e; pe sis ence measu es; VAR; impulse esponse unc ion; PPP.
JEL Classi ica ion: F31, C32.
Acknowledgmen s: Financial suppo om he G eek Minis y o Educa ion and he
Eu opean Union unde “H aklei os” g an is g ea ly app ecia ed. The usual disclaime applies.
∗Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus and EFG-Eu obank.
†Na ional Uni e si y o I eland, Maynoo h and Uni e si y o Pi aeus. 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], phone: 00353 1 7083793, ax: 00353 1 7083934.
‡Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus.
§Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus.
1In oduc ion
Long- un Pu chasing Powe Pa i y (PPP) s a es ha eal exchange a es, de ined as he ela i e
p ice o a baske o goods exp essed in a common cu ency, should be s a iona y, implying ha
changes in he eal exchange a e should be a bi aged away in he long un. Ye , one cha ac e is ic
o eal exchange a es is ha hey a e highly pe sis en p ocesses. In o he wo ds, he speed a
which a gi en shock o he eal exchange a e dissipa es is e y slow. One measu e o pe sis ence
is hal -li e, de ined as he numbe o pe iods equi ed o a gi en shock o educe o hal i s ini ial
alue. A la ge numbe o empi ical s udies has ound ha eal exchange a es a e s a iona y, bu
highly pe sis en p ocesses wi h hal -li es o de ia ions om PPP be ween h ee and i e yea s.1
The empi ical e idence o an ex emely slow speed o con e gence owa ds PPP canno be
easily econciled wi h he s ylized ac ha sho - e m de ia ions om PPP a e bo h la ge and
ola ile. Indeed, he sho - e m ola ili y o eal exchange a es is o he same o de o magni ude
as he ola ili y o nominal exchange a es. Combined wi h his s ylized ac , he inding o high
pe sis ence o he eal exchange a e cons i u es a puzzle as o he na u e o he shocks d i ing eal
exchange a es.2
The majo i y o empi ical s udies compu e hal -li es o PPP de ia ions wi hin a uni a ia e
amewo k, ypically by es ima ing a i s -o de au o eg essi e, AR(1), model o he eal exchange
a e. In such a speci ica ion, he e o e m, which accoun s o he a ia ion o he eal exchange
a e, can be hough o as a ‘composi e shock’ ha inco po a es a ious indi idual shocks, such as
mone a y shocks o shocks o as es and echnology. As a esul , impulse esponse analysis (IRA)
wi hin he uni a ia e amewo k canno iden i y he effec o each indi idual shock, bu simply ells
us how as he eal exchange a e adjus s o a dis u bance o unknown o igins.
This pape aims o shed some ligh on he causes o pe sis ence o eal exchange a es. In pa -
icula , we a e in e es ed in quan i ying he ela i e impo ance o a se o mac oeconomic a iables
which a e conside ed o be undamen al de e minan s o eal exchange a es on he pe sis ence
1See, e.g. F ankel (1986, 1990), Abua and Jo ion (1990), Glen (1992), F oo and Rogoff(1995), Lo hian and
Taylo (1996) and Rogoff(1996), among o he s. S udies using panel da a, ind only sligh ly sho e hal -li es, see, e.g.
F ankel and Rose (1996), Oh (1996), Wu (1996), Lo hian (1997)and Papell (1997), among o he s. Recen wo k wi h
panel da a, howe e , cas s doub on he s a iona i y o eal exchange a es, see e.g. O’Connel (1998) and B eue e
al. (2001, 2002).
2Rogoff(1996) e med his he “PPP puzzle”.
1
o de ia ions om PPP. This se o a iables includes ou pu g ow h diffe en ials and long- e m
in e es a e diffe en ials (bo h nominal and eal) be ween he domes ic and he o eign economy.
In o de o measu e he ela i e con ibu ion o hese a iables o he pe sis ence o de ia ions
om PPP, we compa e he hal -li e es ima es ob ained om a VAR model which includes hese
a iables along wi h he eal exchange a e wi h he hal -li e es ima es ob ained om uni a ia e
models o he eal exchange a e. The diffe ence be ween he wo hal -li e es ima es is a measu e
o he con ibu ion o hese a iables o he pe sis ence o he eal exchange a e.
Ou choise o mac oeconomic de e minan s o eal exchange a es has wo mo i a ions: Fi s ,
s icky-p ice heo ies o exchange a es sugges ha de ia ions om PPP a e closely ela ed o
his se o mac oeconomic a iables.3Second, gi en he end o globaliza ion o bo h inancial
ma ke s and economies, policymake s and p ac i ione s a e in e es ed o know how much as e
eal exchange a es would e e owa ds PPP i business cycles and mone a y policy we e ully
synch onized ac oss majo economies.
In o de o mo i a e ou me hod, le us i s de ine he eal exchange a e, y1 ,as he ela i e
p ice o o eign goods in e ms o domes ic goods. In log o m:
y1 ≡s −(p −p∗
)
whe e s is he nominal exchange a e, measu ed in uni s o domes ic cu ency pe uni o o eign
cu ency, and p (p∗
) is he domes ic ( o eign) p ice index. Fu he mo e, le Y =[y1 ,y2 ]0be an
(n×1)− ec o o a iables whe e y2 is an (n−1)- ec o o mac oeconomic a iables, which affec
he dynamic adjus men o he eal exchange a e owa ds he PPP le el.
Le us u he assume ha Y ollows a n− a ia e VAR(1) model.4I is well known ha
each a iable in he VAR(1) model (including y1 ) has an equi alen uni a ia e ARMA(n, n −1)
ep esen a ion, whe e nand n−1a e he maximum o de s o he au o eg essi e and mo ing a e age
pa s, espec i ely (see Lu kepohl, 1993). In iew o his ‘equi alence’, he e is no speci ica ion
e o in ol ed in one’s decision o employ he ARMA model o es ima ing he esponse o he eal
exchange a e o a uni shock in he e o e m, say e .The la e , howe e , is a combina ion o he
3See Do nbusch (1976, 1989), F ankel (1979) and Meese and Rogoff(1988).
4The VAR(1) model is assumed a his s age o exposi ional pu poses only.
2
e o s in he VAR model, which in u n implies ha he o igins o his shock canno be iden i ied.
Assume o simplici y ha he e is no con empo aneous co ela ion among he elemen s o Y ,and
conside he i s equa ion o he VAR model, ha is he one o he eal exchange a e. The e o
e m in his equa ion, say ε1 ,desc ibes he shocks in he eal exchange a e no accoun ed o by
y2 , ha is i desc ibes he effec s o any o he andom ac o s ha affec he exchange a e. The
VAR- esponse, IRV,o y1 o a uni shock in ε1 should now be as e han i s equi alen ARMA-
esponse, IRA, o a uni shoch in e i he a iables y2 ha e ac ually a ole o play. Indeed, he
diffe ence, D=IRA−IRV,desc ibes he dynamic adjus men pa h o he eal exchange a e which
is solely due o he obse ed a iables y2 .Ob iously, he effec s o o he ac o s ha in luence he
eal exchange a e no aken in o accoun in he VAR speci ica ion a e cap u ed by IRAi sel . The
bigge Dis, he mo e (less) impo an he ole o y2 (o he ac o s) o he pe sis ence o he eal
exchange a e will be.
To u he cla i y ou poin , assume ha he hal -li e o PPP de ia ions, es ima ed wi hin he
ARMA model o he eal exchange a e is 20 qua e s. On he o he hand, assume ha he hal -li e
es ima e ob ained om he VAR model, which includes y1 and y2 is only 12 qua e s. This means
ha he con ibu ion o y2 o he hal -li e o y1 is 20-12=8 qua e s. The emaining 12 qua e s
is he numbe o pe iods equi ed o y1 o adjus (by hal ) o shocks in o he ac o s. In such a
scena io, y2 accoun s o 40% (=8/20) o he pe sis ence o he eal exchange a e.
The emainde o he pape is s uc u ed as ollows. Sec ion 2 ocuses on he econome ic
me hodology. In he con ex o a i s -o de bi a ia e VAR model, i compa es he impulse esponse
unc ion (IRF) o he i s a iable o he VAR model wi h he IRF ob ained om he uni a ia e
ARMA ep esen a ion o his a iable. I also de i es condi ions unde which hese wo IRFs
a e iden ical. Sec ion 3 mo i a es ou choice o he mac oeconomic a iables in ou empi ical
applica ion. Sec ion 4 epo s he empi ical esul s and sec ion 5 concludes.
3
2 Impulse Response Analysis: Mul i a ia e Models and hei Equi -
alen Uni a ia e Rep esen a ions
This sec ion highligh s ou main me hodological poin , namely ha he impulse esponse analysis
wi hin a VAR model diffe s in gene al om ha conduc ed wi hin he equi alen uni a ia e ARMA
models. Fo illus a i e pu poses and in o de o a oid unnecessa y complica ions, we ocus on he
simples possible case, namely ha o a ze o-mean bi a ia e VAR(1) model. The esul s ex end o
he case o a k− a ia e VAR(p) model in a s aigh o wa d way.
Le Y =(y1 ,y
2 )0 ollow a s able VAR(1) p ocess:
Y =AY −1+U (1)
whe e A=


a11 a12
a21 a22


,aij ∈R. The e o ec o U =(u1 ,u
2 )0is a whi e noise p ocess, ha
is, E(U )=0,E(U U0
)=Σu=


σ11 σ12
σ12 σ22


and E(U U0
s)=0 o 6=s. Theco a iancema ix
Σuis assumed o be non-singula .
Following Lu kepohl (1993), each componen se ies yi ,i=1,2o Y has an equi alen uni a ia e
ARMA(p, q) ep esen a ion whe e p≤2and q≤1.5To be speci ic, he ARMA(2,1) ep esen a ion
o y1 is as ollows:
y1 −(a11 +a22)y1 −1+(a11a22 −a21a12)y1 −2=e1 +γ1e1 −1(2)
whe e Va (e1 )=σ2
1,γ1=S±√Q+R
Fand σ2
1=G1
γ1.6
Fu he mo e,
S=(1+a2
22)σ11 −2a12a22σ12 +a2
12σ22,
Q=(1+a4
22 −2a2
22)σ2
11 +a4
12σ2
22 +(4a2
12a2
22 −4a2
12)σ2
12 −4(a12a3
22 −a22a12)σ11σ12,
R=(2a2
12 +2a2
22a2
12)σ11σ22 −4a3
12a22σ12σ22,
F=2(a12σ12 −a22σ11),
5Fo a p oo , see Co olla y 6.1.1. in Lu kepohl (1993), page 232.
6No e ha we ha e o choose he in e ible solu ion o γ1,i.e. he alue o γ1 ha sa is ies |γ1|<1.
4

G1=a12σ12 −a22σ11.
I is in e es ing o no e ha he MA e o e m, w1 ≡e1 +γ1e1 −1,is ela ed o he o iginal
VAR e o s as ollows:
w1 =u1 −a22u1 −1+a12u2 −1(3)
This ela ionship shows ha he e o in he uni a ia e ep esen a ion o y1 can be hough
o as an agg ega ion o he o iginal e o s in he VAR model. As a esul , he a ia ion o w1 is
due o he a ia ion o ei he u1 o u2 o bo h. Fu he mo e he abo e ela ionships show ha
he a iance, σ2
1,o he e o e m, e1 ,is a complica ed unc ion o he VAR pa ame e s. This
means ha he shock e1 o y1 in he con ex o he ARMA model is de e mined by he s uc u e
o he in e empo al in e ac ions be ween y1 and y2 and he second momen s o u1 and u2 .As a
consequence, i s ‘o igins’ a e a om clea .
Le us now examine he esponse o y1 o a uni shock in i s inno a ions, in he con ex o
bo h he VAR(1) and he ARMA(2,1) models. Be o e we p oceed any u he , i is impo an o
emphasize he ole o σ12 6=0on he in e p e a ion o he e o s in he VAR model.I σ12 6=0, hen
he e o , u1 ,in he i s equa ion o he VAR model, canno be in e p e ed as he inno a ions
d i ing y1 .On he o he hand, i σ12 =0, henu1 egains i s s a us as ‘ he inno a ions’ o y1 in
he VAR model and can be hough o as summa izing he ac o s ha con ibu e o he a iabili y
o y1 , o he han y1 −1and y2 −1.We a e in e es ed in compa ing he impulse esponse unc ion,
IRFu,o y1 , om he uni a ia e model wi h he impulse esponse unc ion, IRFm,o y1 om he
mul i a ia e model. No e ha IRFm e e s o he esponse o y1 o a uni shock in u1 .7The cases
σ12 =0and σ12 6=0a e analyzed in subsec ions 2.1 and 2.2 espec i ely.8
7In he case o he VAR model, a esponse in y1 may be caused by an impulse in u2 ,e eni σ12 =0.
8The diagonali y es ic ions on he co a iance ma ix a e es ed in he empi ical pa o he pape o all he
coun ies unde conside a ion.
5
2.1 The Case o a Diagonal Co a iance Ma ix, σ12 =0
Th oughou his subsec ion we assume σ12 =0.The impulse esponse unc ions unde conside a-
ion, IRFuand IRFm,a ede ined as ollows:
IRFu(k)=γk+
k
X
j=1
ajIRFu(k−j)
whe e k=1,2,3,....,IRFu(0) = 1,γk=0 o k>1,a1=(a11 +a22),a2=(a21a12 −a11a22)and
ak=0 o k>2. On he o he hand, IRFmis usually de ined in he con ex o he in ini e mo ing
a e age ep esen a ion o Y , ha isY =∞
X
i=0
ΦiU −iwhe e Φi=Ai. Then, i is easy o show ha
IRFm(k)=φ11,k
whe e φ11,k is he uppe le elemen o Φk.
We a e in e es ed in compa ing IRFu(k)wi h IRFm(k).We p esen ou esul s in he o m o
he ollowing p oposi ions.
P oposi ion 1: IRFu(k)is in gene al no equi alen o IRFm(k) o some k<∞.9
P oo : See Appendix.
Due o he p esence o γ1in IRFu(k),i is analy ically impossible o iden i y all he cases
whe e IRFu(k)>IRF
m(k).I , howe e , we impose some addi ional pa ame e es ic ions, hen
he ollowing esul can be es ablished:
P oposi ion 2: I a11 >0,a22 >0and a12a21 >0,IRFu(k)>IRF
m(k) o e e y k∈N.
P oo : See Appendix.
Howe e , he e is one case whe e IRFu(k)=IRFm(k) o e e y k.Speci ically, his case a ises
when y2 does no G ange cause y1 .Hence:
Lemma 1 When a12 =0,IRFu(k)=IRFm(k) o e e y k≥0.
P oo : See Appendix.
I is impo an o no e ha only when a12 =0, he AR(1) model is he co ec uni a ia e spec-
i ica ion o y1 .In he opposi e case, he AR(1) is a misspeci ied model, hus p oducing misleading
9Gi en he s abili y o (1), bo h IRFuand IRFm end o ze o as k−→ ∞ .
6
esul s in e e y aspec o s a is ical in e ence. This has di ec implica ions on he wide applica ion
o he AR(1) model as he uni a ia e ep esen a ion o he eal exchange a e. In he p esence
o e en a single G ange -causing a iable o he eal exchange a e, he AR(1) model is clea ly
inapp op ia e.
2.2 The Case o a Non-Diagonal Co a iance Ma ix, σ12 6=0
In his case, he e o e m, u1 ,in he i s equa ion o he VAR(1) does no coincide wi h he
inno a ions d i ing y1 .Following s anda d p ac ice, we es o e he o hogonali y o he e o s by
u ilizing he Cholesky decomposi ion o Σu, ha isΣu=PP0,whe ePis a lowe iangula ma ix.
A e some algeb a, we ob ain he ollowing ep esen a ion o Y :
y1 =a11y1 −1+a12y2 −1+ 1 (4)
y2 =σ12
σ11
y1 +(a21 −σ12
σ11
a11)y1 −1+(a22 −σ12
σ11
a12)y2 −1+ 2
whe e V =


1
2


=


u1
u2 −σ12
σ11 u1


wi h co a iance ma ix ΣV=


σ11 0
0σ22 −σ2
12
σ11


.This
pa icula ep esen a ion was ob ained by assuming ha y1 is causally p io o y2 .This means
ha he cu en alues o y1 do no eac con empo aneously o changes in y2 .The e o e m,
1 ,in he i s equa ion o (4) is o hogonal o y1 −1and y2 −1, ha is i can be hough o as
summa izing all he o he ac o s ha con ibu e o he a iabili y o y1 ,apa om y1 −1and
y2 −1.Based on (4), we ob ain he ollowing in ini e MA ep esen a ion o Y :
Y =∞
X
i=0
ΘiW −i
whe e Θi=ΦiPand W =(w1 w2 )0=P−1U 10.
We now de ine he Impulse Response Func ion, IRFmo, o y1 o be:
IRFmo(k)= θ11,k
√σ11
10By cons uc ion, he a iance-co a iance ma ix o W is ΣW=I2.
7
whe e θ11,k is he uppe le elemen o Θk.Byde ini ion, IRFmo(k)is he esponse o y1 o a
uni shock in i s inno a ions, 1 ,a e kpe iods. The e o e, IRFmo(k)is di ec ly compa able o
IRFu(k). The ollowing p oposi ion holds:
P oposi ion 3 In gene al, IRFmo(k)6=IRFu(k) o some ini e k.
P oo : See Appendix.
The ollowing lemma p o ides he sufficien condi ion o ob ain equi alence o IRFmo(k)and
IRFu(k).11
Lemma 2 When a12 =0,IRFu(k)=IRFmo(k) o e e y k≥0.
P oo : See Appendix.
3 Choice o Economic Va iables
Economic heo y has iden i ied wo main se s o de e minan s o eal exchange a es: (a) eal a i-
ables which desc ibe he e olu ion o as es and echnology and de e mine he long- un equilib ium
eal exchange a e,12 and (b) mone a y/agg ega e demand a iables which desc ibe he de ia ions
o eal exchange a es om PPP.13
While eal dis u bances, such as changes in as es and echnology, a e likely o explain long-
e m changes in he eal exchange a e, medium- and sho - e m changes a e mo e likely o e lec
mone a y o agg ega e demand shocks. Such shocks can ha e subs an ial effec s on he eal economy
in he p esence o sho - e m nominal p ice igidi ies. This is a cen al ea u e o he Do nbusch
(1976) s icky-p ice mone a y model. In his model, mone a y dis u bances lead o o e shoo ing
o he eal exchange a e due o sho - e m p ice s ickiness. Du ing he adjus men o long- e m
equilib ium, de ia ions om PPP a e ela ed o ou pu and in e es a e diffe en ials be ween he
domes ic and he o eign economy. F ankel (1979) de i es an al e na i e ep esen a ion o he eal
exchange a e in e ms o eal in e es a e diffe en ials.14
11Despi e ou bes effo s, we ha e no ye succeeded in p o ing ha IRFu(k)=IRFmo(k) o some sensible
pa ame e con igu a ions. Ne e heless, ex en i e simula ion esul s seem o suppo such a conjec u e.
12See, e.g. Balassa (1964) and Samuelson (1964). Acco ding o he so-called “Balassa-Samuelson hypo hesis”, he
long- un equilib ium eal exchange a e is de e mined by he sha e o non adable goods in he consume baske (i.e.
by consume p e e ences) and ela i e o al ac o p oduc i i y in he adables and non- adables sec o .
13See, e.g. Do nbusch (1976, 1989) and Meese and Rogoff(1988).
14In an empi ical pape , Bax e (1994) inds a s ong co ela ion be ween eal exchange a es and eal in e es a e
diffe en ials.
8
qua e s om he ARMA models. This sugges s ha eal and nominal long e m in e es a e
diffe en ials and eal GDP g ow h diffe en ials accoun o a subs an ial ac ion o he hal -li e o
PPP de ia ions.
The diffe ence be ween he ARMA es ima e o hal -li e, HLu(as epo ed in Table 2), and
he VAR es ima e o hal -li e, HLm,is 3.75 qua e s, in line wi h es ima es o pe sis ence o eal
exchange a es om calib a ed in e na ional business cycle models wi h nominal p ice igidi ies such
as Cha i e al. (2002). The emaining se en qua e s o he hal -li e o de ia ions om PPP can be
a ibu ed o o he (unspeci ied) sou ces o pe sis ence. By compa ing he hal -li e es ima es o he
mul i a ia e models wi h he hal -li e es ima es o hei equi alen uni a ia e ep esen a ions, we
can compu e he ac ion o hal -li e a ibu able o he se o mac oeconomic a iables included in
he VAR model as (HLu-HLm)/HLu. As epo ed in he las column o Table 8, he ac ion o hal -
li e due o eal and nominal long e m in e es a e diffe en ials and eal GDP g ow h diffe en ial
anges om 22% in he UK o 50% in Ge many, wi h an a e age ac oss he ou coun y-pai s o
34%.
The 95% con idence in e als o hal -li es a e conside ably igh e han in he uni a ia e con ex ,
sugges ing ha ou es ima es o hal -li es a e mo e p ecise. The lowe bound o he asymp o ic
con idence in e als is es ima ed a ou qua e s o all coun y pai s, compa ed wi h 5-7 qua e s
in he uni a ia e models. The uppe bounds ange om 13 o 30 qua e s, compa ed o 16-23 in
he uni a ia e models. In e es ingly, he Mon e Ca lo con idence in e als a e igh e han hose
based on he asymp o ic dis ibu ion o he impulse esponse unc ion (lowe bound: 3-4 qua e s,
uppe bound: 12-22 qua e s). I is impo an o no e ha ou es ima es b eak he consensus
iew a he lowe end o i s ange wi hou accoun ing o a se ies o po en ial econome ic pi alls,
such as empo al agg ega ion bias,22 nonlinea adjus men 23 o c oss-sec ional agg ega ion bias.24
Co ec ing o hese econome ic issues would ce ainly educe es ima ed hal -li es e en u he .
22Fo an ex ensi e analysis o empo al agg ega ion bias in hal -li e es ima es see Taylo (2001).
23See, o ins ance, Michael e al. (1997), Taylo and Peel (2000) and Taylo (2001).
24See, o ins ance, Imbs e al. (2005).
15

5Conclusions
In his pape , we es ima ed he hal -li e o PPP de ia ions in he con ex o a Vec o Au o eg essi e
model, whe e he eal exchange a e is allowed o in e ac wi h a se o mac oeconomic a iables,
sugges ed by heo ies o exchange a e de e mina ion. By doing his, we we e able o disce n he
ela i e effec o hese a iables on he speed o adjus men o he eal exchange a e owa ds
long- un PPP. We i s showed ha he impulse esponse unc ion o a a iable pa icipa ing in
he VAR model is no , in gene al, he same wi h he impulse esponse unc ion ob ained om he
equi alen ARMA ep esen a ion o his a iable, i he la e is G ange caused by o he a iables
in he sys em. The diffe ence be ween he wo impulse esponse unc ions cap u es he effec o he
G ange -causing a iables on he dynamic adjus men p ocess o he a iable o in e es .
We in es iga e he implica ions o ou analy ical esul s o he speed o adjus men o ou eal
exchange a es is-a- is he US dolla (F ench anc, Ge man ma k, I alian li a and UK pound)
du ing he pos -B e on Woods pe iod. Ou empi ical esul s sugges ha eal exchange a es a e in
ac G ange caused by hese a iables. As a esul , he adjus men ho izons o de ia ions om PPP
dec ease subs an ially. The a e age hal -li e es ima e ac oss he ou pai s o eal exchange a es
is below wo yea s, sugges ing ha eal o nominal in e es a e diffe en ials and GDP g ow h
diffe en ials accoun o a signi ican ac ion o de ia ions om PPP. Compa ing he hal -li e
es ima es o he uni a ia e models wi h he hal -li e es ima es o he VAR model, we conclude
ha be ween 22% and 50% o he hal -li e o de ia ions om PPP is due o hese a iables.
O cou se, al hough eal o nominal in e es a e diffe en ials and GDP g ow h diffe en ials
explain a signi ican ac ion o de ia ions om PPP, ou esul s lea e a good bi o a ia ion in
eal exchange a es o unknown sou ces. These sou ces s ill accoun on a e age o a hal -li e o
jus below wo yea s, hence, a puzzle emains as o whe he eal sou ces a e ola ile enough o
explain he obse ed mo emen s o eal exchange a es. Howe e , ecen wo k on he PPP puzzle
sugges s ha s anda d me hods o es ima ion used in he li e a u e la gely o e es ima e he size
o eal exchange a es hal -li es because hey ail o co ec o a numbe o biases s emming om
pa ame e he e ogenei y, empo al agg ega ion and nonlinea adjus men .
Ou me hod is no able o iden i y whe he he pe sis ence o eal exchange a es is due o eal
16
o mone a y shocks and, hence, does no add ess he so-called “PPP puzzle”. Howe e , i opens
he way o assess he ole o undamen al de e minan s o eal exchange a es iden i ied by diffe en
heo ies on he pe sis ence o de ia ions om PPP. Fu he wo k is needed o add ess he issue
o iden i ica ion. Finally, ou me hod is gene al enough o assess he impo ance o undamen al
de e minan s on he obse ed pe sis ence o a wide ange o economic and inancial a iables, such
as in la ion, eal wages, di idend-p ice a ios e c.
17
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21
Appendix
P oo o P oposi ion1
I is easy o show ha in he con ex o (1), IRFm(1) = a11. On he o he hand, IRFu(1) =
a11 +a22 +γ1. Simila ly, IRFm(2) = a2
11 +a12a21,whe easIRFu(2) = (a11 +a22)(γ1+a11 +a22)−
a11a22 +a21a12. Simila esul s a e ob ained o k>2. The e o e, in gene al, IRFu(k)6=IRFm(k).
P oo o P oposi ion2
A e some algeb a we ha e ha
IRFm(k)−IRFu(k)=(2
−1−k((a11 +a22 −x)k
−(a11 +a22 +x)k)((−1+a2
22)σ11 −a2
12σ22 +
+q(σ11 +a2
22σ11 +a2
12σ22)2−4a2
22σ2
11))/(xa22σ11)
o al e na i ely:
IRFm(k)−IRFu(k)=(
1
2(λk
2−λk
1)((−1+a2
22)σ11 −a2
12σ22 +
+q(σ11 +a2
22σ11 +a2
12σ22)2−4a2
22σ2
11))/(xa22σ11)
whe e
x=p(a11 −a22)2+4a12a21
and λ1and λ2a e he eigen alues o A.25 I is easy o show ha λ1>|λ2|o (λk
2−λk
1)<0 o
e e y ini e k. Then, wha emains o be p o ed is ha
((−1+a2
22)σ11 −a2
12σ22 +q(σ11 +a2
22σ11 +a2
12σ22)2−4a2
22σ2
11)>0.
Indeed,
25λ1=1
2(a11 +a22 +s(a11 −a22 )2+4a12a21)and λ2=1
2(a11 +a22 −s(a11 −a22 )2+4a12a21).
22
q(σ11 +a2
22σ11 +a2
12σ22)2−4a2
22σ2
11 =q(σ11 −a2
22σ11 +a2
12σ22)2+4a2
22a2
12σ11σ22 >
>q(σ11 −a2
22σ11 +a2
12σ22)2=σ11 −a2
22σ11 +a2
12σ22.
Thus,
((−1+a2
22)σ11 −a2
12σ22 +q(σ11 +a2
22σ11 +a2
12σ22)2−4a2
22σ2
11)>0
whichin u nimplies ha IRFu(k)≥IRFm(k) o e e y k∈N.
P oo o Lemma1
Be o e we p o e his Lemma, we need o ake an in e media e s ep, as desc ibed in he ollowing
ema k:
Rema k 1 Le A=


a11 0
a21 a22


whe e aij ∈R. Then, o e e y in ege d>0,Ad=



ad
11 0
q1ad
22


whe e q1is a unc ion o aij .
P oo : We p o e he ema k by induc ion.
Fo d=1,Ad=A=


a11 0
a21 a22


,whichiso he o m:


ad
11 0
q1ad
22


wi h q1=a21.
Assume ha Ad=


ad
11 0
q1ad
22


whe e q1is a unc ion o aij. Then, we mus show ha
Ad+1 =


ad+1
11 0
q0
1ad+1
22


.Now,
Ad+1 =AdA=


ad
11 0
q1ad
22





a11 0
a21 a22


=


ad+1
11 0
a11q1+a21ad
22 ad+1
22



23
which is o he o m: 


ad+1
11 0
q0
1ad+1
22


.
Now, we p oceed wi h he p oo o he lemma. We ha e de ined IRFm(k) o be equal o he
uppe le elemen , φ11,k,o Φk=Ak. By means o he p e ious ema k, we ha e ha Φkis o he
o m: 


ak
11 0
q1ak
22


whe e q1is a unc ion o aij. The e o e, IRFm(k)=ak
11. Nex , i is easy o
show ha when a12 =0,i.e.y2 does no G ange cause y1 , he uni a ia e ep esen a ion o y1
is he ollowing AR(1) model: y1 =a11 ∗y1 −1+e1 , which in u n implies ha IRFu(k)=ak
11.
Thus, IRFu(k)=IRFm(k) o e e y k.
P oo o P oposi ion3
I is s aigh o wa d o show ha IRFmo(1) = a11 +a12 σ12
σ11 , which is in gene al diffe en han
IRFu(1) = a11 +a22 +γ1. Simila ly,
IRFmo(2) = a2
11 +a11a12
σ12
σ11
+a12a21 +a12a22
σ12
σ11
whe eas
IRFu(2) = (a11 +a22)(γ1+a11 +a22)−a11a22 +a21a12
Simila esul s a e ob ained o k>2. The e o e, in gene al, IRFu(k)6=IRFmo(k).
P oo o Lemma2
We ha e al eady shown ha when a12 =0,IRFu(k)=ak
11,k≥0. In addi ion, Φkis o he
o m: 


ak
11 0
q1ak
22


(see lemma 1) whe e q1is a unc ion o aij. Gi en ha Pis lowe iangula ,
i is easy o show ha Θk=ΦkPhas he ollowing o m: Θk=


ak
11√σ11 0
q1q2


,whe eq1and q2
a e unc ions o aij and σij,i, j =1,2.Thus,IRFmo(k)= θ11,k
√σ11 =ak
11 =IRFu(k).
24