The Contribution of Growth and Interest Rates Differentials to the Persistence of Real Exchange Rates

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 Re e ences [1] Abua , N. and P. Jo ion (1990). Pu chasing powe pa i y in he long- un. Jou nal o Finance, 45, 157-174. 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Wo king Pape Se ies no. 5032, Na ional Bu eau o Economic Resea ch. [40] Wu, Y. (1996). A e eal exchange a es non s a iona y? E idence om a panel-da a es . Jou nal o Money, C edi , and Banking, 28, 54-63. 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