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

The Contribution of Growth and Interest Rate Differentials to the Persistence of Real Exchange Rates Dimitrios Malliaropulos∗Ekaterini Panopoulou† Theologos Pantelidis‡Nikitas Pittis§ February 2006 Abstract This paper employs a new methodology for measuring the contribution of growth and interest rate differentials to the half-life of deviations from Purchasing Power Parity (PPP). Our method is based on directly comparing the impulse response function of a VAR model, where the real exchange rate is Granger caused by these variables with the impulse response function of a univatiate ARMA model for the real exchange rate. We show that the impulse response function of the VAR model is not, in general, the same with the impulse response function obtained from the equivalent ARMA representation, if the real exchange rate is Granger caused by other variables in the system. The difference between the two functions captures the effects of the Granger-causing variables on the half-life of deviations from PPP. Our empirical results for a set of four currencies suggest that real and nominal long term interest rate differentials and real GDP growth differentials account for 22% to 50% of the half-life of deviations from PPP. Keywords: real exchange rate; persistence measures; VAR; impulse response function; PPP. JEL Classification: F31, C32. Acknowledgments: Financial support from the Greek Ministry of Education and the European Union under “Hrakleitos” grant is greatly appreciated. The usual disclaimer applies. ∗Department of Banking and Financial Management, University of Piraeus and EFG-Eurobank. †National University of Ireland, Maynooth and University of Piraeus. Correspondence to: Ekaterini Panopoulou, Department of Economics, National University of Ireland Maynooth, Co.Kildare, Republic of Ireland. E-mail: [email protected], phone: 00353 1 7083793, fax: 00353 1 7083934. ‡Department of Banking and Financial Management, University of Piraeus. §Department of Banking and Financial Management, University of Piraeus. 1Introduction Long-run Purchasing Power Parity (PPP) states that real exchange rates, defined as the relative price of a basket of goods expressed in a common currency, should be stationary, implying that changes in the real exchange rate should be arbitraged away in the long run. Yet, one characteristic of real exchange rates is that they are highly persistent processes. In other words, the speed at which a given shock to the real exchange rate dissipates is very slow. One measure of persistence is half-life, defined as the number of periods required for a given shock to reduce to half its initial value. A large number of empirical studies has found that real exchange rates are stationary, but highly persistent processes with half-lifes of deviations from PPP between three and five years.1 The empirical evidence of an extremely slow speed of convergence towards PPP cannot be easily reconciled with the stylized fact that short-term deviations from PPP are both large and volatile. Indeed, the short-term volatility of real exchange rates is of the same order of magnitude as the volatility of nominal exchange rates. Combined with this stylized fact, the finding of high persistence of the real exchange rate constitutes a puzzle as to the nature of the shocks driving real exchange rates.2 The majority of empirical studies compute half-lives of PPP deviations within a univariate framework, typically by estimating a first-order autoregressive, AR(1), model of the real exchange rate. In such a specification, the error term, which accounts for the variation of the real exchange rate, can be thought of as a ‘composite shock’ that incorporates various individual shocks, such as monetary shocks or shocks to tastes and technology. As a result, impulse response analysis (IRA) within the univariate framework cannot identify the effect of each individual shock, but simply tells us how fast the real exchange rate adjusts to a disturbance of unknown origins. This paper aims to shed some light on the causes of persistence of real exchange rates. In particular, we are interested in quantifying the relative importance of a set of macroeconomic variables which are considered to be fundamental determinants of real exchange rates on the persistence 1See, e.g. Frankel (1986, 1990), Abuaf and Jorion (1990), Glen (1992), Froot and Rogoff(1995), Lothian and Taylor (1996) and Rogoff(1996), among others. Studies using panel data, find only slightly shorter half-lifes, see, e.g. Frankel and Rose (1996), Oh (1996), Wu (1996), Lothian (1997)and Papell (1997), among others. Recent work with panel data, however, casts doubt on the stationarity of real exchange rates, see e.g. O’Connel (1998) and Breuer et al. (2001, 2002). 2Rogoff(1996) termed this the “PPP puzzle”. 1 of deviations from PPP. This set of variables includes output growth differentials and long-term interest rate differentials (both nominal and real) between the domestic and the foreign economy. In order to measure the relative contribution of these variables to the persistence of deviations from PPP, we compare the half-life estimates obtained from a VAR model which includes these variables along with the real exchange rate with the half-life estimates obtained from univariate models of the real exchange rate. The difference between the two half-life estimates is a measure of the contribution of these variables to the persistence of the real exchange rate. Our choise of macroeconomic determinants of real exchange rates has two motivations: First, sticky-price theories of exchange rates suggest that deviations from PPP are closely related to this set of macroeconomic variables.3Second, given the trend to globalization of both financial markets and economies, policymakers and practitioners are interested to know how much faster real exchange rates would revert towards PPP if business cycles and monetary policy were fully synchronized across major economies. In order to motivate our method, let us first define the real exchange rate, y1t,astherelative price of foreign goods in terms of domestic goods. In log form: y1t≡st−(pt−p∗ t) where stis the nominal exchange rate, measured in units of domestic currency per unit of foreign currency, and pt(p∗ t) is the domestic (foreign) price index. Furthermore, let Yt=[y1t,y2t]0be an (n×1)−vector of variables where y2tis an (n−1)-vector of macroeconomic variables, which affect the dynamic adjustment of the real exchange rate towards the PPP level. Let us further assume that Ytfollows a n−variate VAR(1) model.4It is well known that each variable in the VAR(1) model (including y1t) has an equivalent univariate ARMA(n, n −1) representation, where nand n−1are the maximum orders of the autoregressive and moving average parts, respectively (see Lutkepohl, 1993). In view of this ‘equivalence’, there is no specification error involved in one’s decision to employ the ARMA model for estimating the response of the real exchange rate to a unit shock in the error term, say et.The latter, however, is a combination of the 3See Dornbusch (1976, 1989), Frankel (1979) and Meese and Rogoff(1988). 4The VAR(1) model is assumed at this stage for expositional purposes only. 2 errors in the VAR model, which in turn implies that the origins of this shock cannot be identified. Assume for simplicity that there is no contemporaneous correlation among the elements of Yt,and consider the first equation of the VAR model, that is the one for the real exchange rate. The error term in this equation, say ε1t,describes the shocks in the real exchange rate not accounted for by y2t,that is it describes the effects of any other random factors that affect the exchange rate. The VAR-response, IRV,of y1tto a unit shock in ε1tshould now be faster than its equivalent ARMAresponse, IRA,to a unit shoch in etif the variables y2thave actually a role to play. Indeed, the difference, D=IRA−IRV,describes the dynamic adjustment path of the real exchange rate which is solely due to the observed variables y2t.Obviously, the effects of other factors that influence the real exchange rate not taken into account in the VAR specification are captured by IRAitself. The bigger Dis, the more (less) important the role of y2t(other factors) for the persistence of the real exchange rate will be. To further clarify our point, assume that the half-life of PPP deviations, estimated within the ARMA model for the real exchange rate is 20 quarters. On the other hand, assume that the half-life estimate obtained from the VAR model, which includes y1tand y2tis only 12 quarters. This means that the contribution of y2tto the half-life of y1tis 20-12=8 quarters. The remaining 12 quarters is the number of periods required for y1tto adjust (by half) to shocks in other factors. In such a scenario, y2taccounts for 40% (=8/20) of the persistence of the real exchange rate. The remainder of the paper is structured as follows. Section 2 focuses on the econometric methodology. In the context of a first-order bivariate VAR model, it compares the impulse response function (IRF) of the first variable of the VAR model with the IRF obtained from the univariate ARMA representation of this variable. It also derives conditions under which these two IRFs are identical. Section 3 motivates our choice of the macroeconomic variables in our empirical application. Section 4 reports the empirical results and section 5 concludes. 3 2 Impulse Response Analysis: Multivariate Models and their Equivalent Univariate Representations This section highlights our main methodological point, namely that the impulse response analysis within a VAR model differs in general from that conducted within the equivalent univariate ARMA models. For illustrative purposes and in order to avoid unnecessary complications, we focus on the simplest possible case, namely that of a zero-mean bivariate VAR(1) model. The results extend to the case of a k−variate VAR(p) model in a straightforward way. Let Yt=(y1t,y 2t)0follow a stable VAR(1) process: Yt=AYt−1+Ut(1) where A=   a11 a12 a21 a22   ,aij ∈R. The error vector Ut=(u1t,u 2t)0is a white noise process, that is, E(Ut)=0,E(UtU0 t)=Σu=   σ11 σ12 σ12 σ22   and E(UtU0 s)=0for t6=s. Thecovariancematrix Σuis assumed to be non-singular. Following Lutkepohl (1993), each component series yit,i=1,2of Ythas an equivalent univariate ARMA(p, q) representation where p≤2and q≤1.5To be specific, the ARMA(2,1) representation of y1tis as follows: y1t−(a11 +a22)y1t−1+(a11a22 −a21a12)y1t−2=e1t+γ1e1t−1(2) where Var(e1t)=σ2 1,γ1=S±√Q+R Fand σ2 1=G1 γ1.6 Furthermore, 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), 5For a proof, see Corollary 6.1.1. in Lutkepohl (1993), page 232. 6Note that we have to choose the invertible solution for γ1,i.e. the value of γ1that satisfies |γ1|<1. 4 G1=a12σ12 −a22σ11. It is interesting to note that the MA error term, w1t≡e1t+γ1e1t−1,is related to the original VAR errors as follows: w1t=u1t−a22u1t−1+a12u2t−1(3) This relationship shows that the error in the univariate representation of y1tcan be thought of as an aggregation of the original errors in the VAR model. As a result, the variation of w1tis due to the variation of either u1tor u2tor both. Furthermore the above relationships show that the variance, σ2 1,of the error term, e1t,is a complicated function of the VAR parameters. This means that the shock e1tof y1tin the context of the ARMA model is determined by the structure of the intertemporal interactions between y1tand y2tand the second moments of u1tand u2t.As a consequence, its ‘origins’ are far from clear. Let us now examine the response of y1tto a unit shock in its innovations, in the context of both the VAR(1) and the ARMA(2,1) models. Before we proceed any further, it is important to emphasize the role of σ12 6=0on the interpretation of the errors in the VAR model.If σ12 6=0,then the error, u1t,in the first equation of the VAR model, cannot be interpreted as the innovations driving y1t.On the other hand, if σ12 =0,thenu1tregains its status as ‘the innovations’ of y1tin the VAR model and can be thought of as summarizing the factors that contribute to the variability of y1t, other than y1t−1and y2t−1.We are interested in comparing the impulse response function, IRFu,of y1t,from the univariate model with the impulse response function, IRFm,of y1tfrom the multivariate model. Note that IRFmrefers to the response of y1tto a unit shock in u1t.7The cases σ12 =0and σ12 6=0are analyzed in subsections 2.1 and 2.2 respectively.8 7In the case of the VAR model, a response in y1tmay be caused by an impulse in u2t,evenifσ12 =0. 8The diagonality restrictions on the covariance matrix are tested in the empirical part of the paper for all the countries under consideration. 5 2.1 The Case of a Diagonal Covariance Matrix, σ12 =0 Throughout this subsection we assume σ12 =0.The impulse response functions under consideration, IRFuand IRFm,aredefined as follows: IRFu(k)=γk+ k X j=1 ajIRFu(k−j) where k=1,2,3,....,IRFu(0) = 1,γk=0for k>1,a1=(a11 +a22),a2=(a21a12 −a11a22)and ak=0for k>2. On the other hand, IRFmis usually defined in the context of the infinite moving average representation of Yt,thatisYt=∞ X i=0 ΦiUt−iwhere Φi=Ai. Then, it is easy to show that IRFm(k)=φ11,k where φ11,k is the upper left element of Φk. We are interested in comparing IRFu(k)with IRFm(k).We present our results in the form of the following propositions. Proposition 1: IRFu(k)is in general not equivalent to IRFm(k)for some k<∞.9 Proof: See Appendix. Due to the presence of γ1in IRFu(k),it is analytically impossible to identify all the cases where IRFu(k)>IRF m(k).If, however, we impose some additional parameter restrictions, then the following result can be established: Proposition 2: If a11 >0,a22 >0and a12a21 >0,IRFu(k)>IRF m(k)for every k∈N. Proof: See Appendix. However, there is one case where IRFu(k)=IRFm(k)for every k.Specifically, this case arises when y2tdoes not Granger cause y1t.Hence: Lemma 1 When a12 =0,IRFu(k)=IRFm(k)for every k≥0. Proof: See Appendix. It is important to note that only when a12 =0, the AR(1) model is the correct univariate specification for y1t.In the opposite case, the AR(1) is a misspecified model, thus producing misleading 9Given the stability of (1), both IRFuand IRFmtend to zero as k−→ ∞ . 6 results in every aspect of statistical inference. This has direct implications on the wide application of the AR(1) model as the univariate representation of the real exchange rate. In the presence of even a single Granger-causing variable for the real exchange rate, the AR(1) model is clearly inappropriate. 2.2 The Case of a Non-Diagonal Covariance Matrix, σ12 6=0 In this case, the error term, u1t,in the first equation of the VAR(1) does not coincide with the innovations driving y1t.Following standard practice, we restore the orthogonality of the errors by utilizing the Cholesky decomposition of Σu,thatisΣu=PP0,wherePis a lower triangular matrix. After some algebra, we obtain the following representation for Yt: y1t=a11y1t−1+a12y2t−1+v1t(4) y2t=σ12 σ11 y1t+(a21 −σ12 σ11 a11)y1t−1+(a22 −σ12 σ11 a12)y2t−1+v2t where Vt=   v1t v2t   =   u1t u2t−σ12 σ11 u1t   with covariance matrix ΣV=   σ11 0 0σ22 −σ2 12 σ11   .This particular representation was obtained by assuming that y1tis causally prior to y2t.This means that the current values of y1tdo not react contemporaneously to changes in y2t.The error term, v1t,in the first equation of (4) is orthogonal to y1t−1and y2t−1,that is it can be thought of as summarizing all the other factors that contribute to the variability of y1t,apart from y1t−1and y2t−1.Based on (4), we obtain the following infinite MA representation of Yt: Yt=∞ X i=0 ΘiWt−i where Θi=ΦiPand Wt=(w1tw2t)0=P−1Ut10. We now define the Impulse Response Function, IRFmo, of y1tto be: IRFmo(k)= θ11,k √σ11 10By construction, the variance-covariance matrix of Wtis ΣW=I2. 7 where θ11,k is the upper left element of Θk.Bydefinition, IRFmo(k)is the response of y1tto a unit shock in its innovations, v1t,after kperiods. Therefore, IRFmo(k)is directly comparable to IRFu(k). The following proposition holds: Proposition 3 In general, IRFmo(k)6=IRFu(k)for some finite k. Proof: See Appendix. The following lemma provides the sufficient condition to obtain equivalence of IRFmo(k)and IRFu(k).11 Lemma 2 When a12 =0,IRFu(k)=IRFmo(k)for every k≥0. Proof: See Appendix. 3 Choice of Economic Variables Economic theory has identified two main sets of determinants of real exchange rates: (a) real variables which describe the evolution of tastes and technology and determine the long-run equilibrium real exchange rate,12 and (b) monetary/aggregate demand variables which describe the deviations of real exchange rates from PPP.13 While real disturbances, such as changes in tastes and technology, are likely to explain longterm changes in the real exchange rate, mediumand short-term changes are more likely to reflect monetary or aggregate demand shocks. Such shocks can have substantial effects on the real economy in the presence of short-term nominal price rigidities. This is a central feature of the Dornbusch (1976) sticky-price monetary model. In this model, monetary disturbances lead to overshooting of the real exchange rate due to short-term price stickiness. During the adjustment to long-term equilibrium, deviations from PPP are related to output and interest rate differentials between the domestic and the foreign economy. Frankel (1979) derives an alternative representation of the real exchange rate in terms of real interest rate differentials.14 11Despite our best efforts, we have not yet succeeded in proving that IRFu(k)=IRFmo(k)for some sensible parameter configurations. Nevertheless, extentive simulation results seem to support such a conjecture. 12See, e.g. Balassa (1964) and Samuelson (1964). According to the so-called “Balassa-Samuelson hypothesis”, the long-run equilibrium real exchange rate is determined by the share of nontradable goods in the consumer basket (i.e. by consumer preferences) and relative total factor productivity in the tradables and non-tradables sector. 13See, e.g. Dornbusch (1976, 1989) and Meese and Rogoff(1988). 14In an empirical paper, Baxter (1994) finds a strong correlation between real exchange rates and real interest rate differentials. 8 quarters from the ARMA models. This suggests that real and nominal long term interest rate differentials and real GDP growth differentials account for a substantial fraction of the half-life of PPP deviations. The difference between the ARMA estimate of half-life, HLu(as reported in Table 2), and the VAR estimate of half-life, HLm,is 3.75 quarters, in line with estimates of persistence of real exchange rates from calibrated international business cycle models with nominal price rigidities such as Chari et al. (2002). The remaining seven quarters of the half-life of deviations from PPP can be attributed to other (unspecified) sources of persistence. By comparing the half-life estimates of the multivariate models with the half-life estimates of their equivalent univariate representations, we can compute the fraction of half-life attributable to the set of macroeconomic variables included in the VAR model as (HLu-HLm)/HLu. As reported in the last column of Table 8, the fraction of halflife due to real and nominal long term interest rate differentials and real GDP growth differential ranges from 22% in the UK to 50% in Germany, with an average across the four country-pairs of 34%. The 95% confidence intervals of half-lifes are considerably tighter than in the univariate context, suggesting that our estimates of half-lifes are more precise. The lower bound of the asymptotic confidence intervals is estimated at four quarters for all country pairs, compared with 5-7 quarters in the univariate models. The upper bounds range from 13 to 30 quarters, compared to 16-23 in the univariate models. Interestingly, the Monte Carlo confidence intervals are tighter than those based on the asymptotic distribution of the impulse response function (lower bound: 3-4 quarters, upper bound: 12-22 quarters). It is important to note that our estimates break the consensus view at the lower end of its range without accounting for a series of potential econometric pitfalls, such as temporal aggregation bias,22 nonlinear adjustment23 or cross-sectional aggregation bias.24 Correcting for these econometric issues would certainly reduce estimated half-lifes even further. 22For an extensive analysis of temporal aggregation bias in half-life estimates see Taylor (2001). 23See, for instance, Michael et al. (1997), Taylor and Peel (2000) and Taylor (2001). 24See, for instance, Imbs et al. (2005). 15 5Conclusions In this paper, we estimated the half-life of PPP deviations in the context of a Vector Autoregressive model, where the real exchange rate is allowed to interact with a set of macroeconomic variables, suggested by theories of exchange rate determination. By doing this, we were able to discern the relative effect of these variables on the speed of adjustment of the real exchange rate towards long-run PPP. We first showed that the impulse response function of a variable participating in the VAR model is not, in general, the same with the impulse response function obtained from the equivalent ARMA representation of this variable, if the latter is Granger caused by other variables in the system. The difference between the two impulse response functions captures the effect of the Granger-causing variables on the dynamic adjustment process of the variable of interest. We investigate the implications of our analytical results for the speed of adjustment of four real exchange rates vis-a-vis the US dollar (French franc, German mark, Italian lira and UK pound) during the post-Bretton Woods period. Our empirical results suggest that real exchange rates are in fact Granger caused by these variables. As a result, the adjustment horizons of deviations from PPP decrease substantially. The average half-life estimate across the four pairs of real exchange rates is below two years, suggesting that real or nominal interest rate differentials and GDP growth differentials account for a significant fraction of deviations from PPP. Comparing the half-life estimates of the univariate models with the half-life estimates of the VAR model, we conclude that between 22% and 50% of the half-life of deviations from PPP is due to these variables. Of course, although real or nominal interest rate differentials and GDP growth differentials explain a significant fraction of deviations from PPP, our results leave a good bit of variation in real exchange rates to unknown sources. These sources still account on average for a half-life of just below two years, hence, a puzzle remains as to whether real sources are volatile enough to explain the observed movements of real exchange rates. However, recent work on the PPP puzzle suggests that standard methods of estimation used in the literature largely overestimate the size of real exchange rates half-lifes because they fail to correct for a number of biases stemming from parameter heterogeneity, temporal aggregation and nonlinear adjustment. Our method is not able to identify whether the persistence of real exchange rates is due to real 16 or monetary shocks and, hence, does not address the so-called “PPP puzzle”. However, it opens the way to assess the role of fundamental determinants of real exchange rates identified by different theories on the persistence of deviations from PPP. Further work is needed to address the issue of identification. Finally, our method is general enough to assess the importance of fundamental determinants on the observed persistence of a wide range of economic and financial variables, such as inflation, real wages, dividend-price ratios etc. 17 References [1] Abuaf, N. and P. Jorion (1990). Purchasing power parity in the long-run. Journal of Finance, 45, 157-174. [2] Balassa, B. (1964). The purchasing power parity doctrine: a reappraisal, Journal of Political Economy, 72, 584-596. [3] Baxter, M. (1994). Real exchange rates and real interest differentials. Have we missed the business-cycle relationship? Journal of Monetary Economics, 33, 5-37. [4] Breuer, J.B., R. McNown and M. Walace (2001). Misleading inferences from panel unit-root tests with an illustration from purchasing power parity. Review of International Economics,9, 482-493. [5] Breuer, J.B., R. McNown and M. Walace (2002). Series-specific unit root tests with panel data. Oxford Bulletin of Economics and Statistics, 64, 527-546. [6] Chari, V.V., P. Kehoe and E. McGrattan (2002). Can sticky prices generate volatile and persistent real exchange rates? Review of Economic Studies, 69, 533-563. [7] Cheung, Y.W. and K. S. Lai (1998). Parity revision in real exchange rates during the postBretton Woods period. Journal of International Money and Finance, 17, 597-614. [8] Cheung, Y.W and K.S. Lai (2000).On the purchasing power parity puzzle, Journal of International Economics, 52, 321-330. [9] Dickey, D.A. and W.A. Fuller (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427—431. [10] Dornbusch, R. (1976). Expectations and exchange rate dynamics. Journal of Political Economy, 84, 1161-1176. [11] Dornbusch, R. (1989). Real exchange rates and macroeconomics: A selective survey. Scandinavian Journal of Economics, 91, 401-432. 18 [12] Elliott, G., T.J. Rothenberg and J.H. Stock (1996). Efficient tests for an autoregressive unit root. Econometrica 64, 813-836. [13] Frankel, J. (1979). On the mark: A theory of floating exchange rates based on real interest differentials. American Economic Review, 69(4), 610-622. [14] Frankel, J. (1986). International capital mobility and crowding-out in the U.S. economy: Imperfect integration of financial markets or goods markets? In: Hafer R.W. (ed.): How Open is the U.S. Economy? Lexington: Lexington Books. [15] Frankel, J. (1990). Zen and the art of modern macroeconomics: A commentary. In: Haraf, W.S. and T.A. Willett (eds): Monetary policy for a volatile global economy, 117-123. AEI Press, Washington, DC. [16] Frankel, J. and A. Rose (1996). A panel project on purchasing power parity: Mean reversion within and between countries. Journal of International Economics, 40, 209-224. [17] Froot, K. and K. Rogoff(1995). Perspectives on PPP and long-run real exchange rates. In: Grossman G. M. and K. Rogoff(eds): The Handbook of International Economics. Amsterdam: North-Holland. [18] Glen, J. D. (1992). Real exchange rates in the short, medium, and long run. Journal of International Economics, 33, 147-66. [19] Higgins, M., and E. Zakrajsek (1999). Purchasing power parity: Three stakes through the heart of the unit root null. Federal Reserve Bank of New York. [20] Huizinga, J. (1987). An empirical investigation of the long-run behavior of real exchange rates. Carnegie-Rochester Conference Series on Public Policy, 27, 149-215. [21] Imbs, J., H. Mumtaz, M.O. Ravn and H. Rey (2005). PPP strikes back: Aggregation and the real exchange rate. The Quarterly Journal of Economics, 120(1), 1-43. [22] Kwiatkowski, D., P.C.B. Phillips, P. Schmidt and Y. Shin (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics, 54, 159-178. 19 [23] Lothian, J. R. (1997). Multi-country evidence on the behavior of purchasing power parity under the current float. Journal of International Money and Finance, 16,19-35. [24] Lothian, J.R. and M.P. Taylor (1996). Real exchange rate behavior: The recent float from the perspective of the past two centuries. Journal of Political Economy, 104, 488-509. [25] Lutkepohl, H. (1993). Introduction to multiple time series analysis, Second Edition. New York: Springer-Verlag. [26] Meese, R. and K. Rogoff(1988). Was it real? The exchange rate - interest differential relation over the modern floating-rate period. Journal of Finance, 43, 933-948. [27] Michael, P., A.R. Nobay and D.A. Peel (1997). Transaction costs and nonlinear adjustment in real exchange rates: An empirical investigation. Journal of Political Economy, 105, 862-879. [28] Murray, C.J. and D.H. Papell (2002). The purchasing power parity persistence paradigm. Journal of International Economics, 56, 1-19. [29] Ng, S. and P. Perron (2001). Lag length selection and the construction of unit root tests with good size and power. Econometrica, 69, 1519-1554. [30] O’Connell, P. (1998). The overvaluation of purchasing ower parity. Journal of International Economics, 44, 1-19. [31] Oh, K.Y. (1996). Purchasing power parity and unit root tests using panel data. Journal of International Money and Finance, 15, 405-418. [32] Papell, O. (1997). Searching for stationarity: Purchasing power parity under the current float. Journal of International Economics, 43, 313-332. [33] Phillips, P.C.B. and P. Perron (1988). Testing for a unit root in time series regression. Biometrika, 75, 335—346. [34] Rogoff, K. (1996). The purchasing power parity puzzle. Journal of Economic Literature, 34, 647-668. 20 [35] Rossi, B. (2005). Confidence intervals for half-life deviations from purchasing power parity. Journal of Business and Economic Statistics, 23(4), 432-442. [36] Samuelson, P.A. (1964). Theoretical notes on trade problems. Review of Economics and Statistics, 51, 239-246. [37] Taylor, A.M. (2001). Potential pitfalls for the purchasing-power parity puzzle? Sampling and specification biases in mean-reversion tests of the law of one price. Econometrica, 69, 473-498. [38] Taylor, M.P. and D.A. Peel (2000). Nonlinear adjustment, long-run equilibrium and exchange rate fundamentals. Journal of International Money and Finance, 19, 33-53. [39] Wei, S.-J. and D.C. Parsley (1995). Purchasing power dis-parity during the floating rate period: Exchange rate volatility, trade barriers, and other culprits. Working Paper Series no. 5032, National Bureau of Economic Research. [40] Wu, Y. (1996). Are real exchange rates non stationary? Evidence from a panel-data test. Journal of Money, Credit, and Banking, 28, 54-63. 21 Appendix ProofofProposition1 It is easy to show that in the context of (1), IRFm(1) = a11. On the other hand, IRFu(1) = a11 +a22 +γ1. Similarly, IRFm(2) = a2 11 +a12a21,whereasIRFu(2) = (a11 +a22)(γ1+a11 +a22)− a11a22 +a21a12. Similar results are obtained for k>2. Therefore, in general, IRFu(k)6=IRFm(k). ProofofProposition2 After some algebra we have that 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) or alternatively: 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) where x=p(a11 −a22)2+4a12a21 and λ1and λ2are the eigenvalues of A.25 It is easy to show that λ1>|λ2|or (λk 2−λk 1)<0for every finite k. Then, what remains to be proved is that ((−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 whichinturnimpliesthatIRFu(k)≥IRFm(k)for every k∈N. ProofofLemma1 Before we prove this Lemma, we need to take an intermediate step, as described in the following remark: Remark 1 Let A=   a11 0 a21 a22   where aij ∈R. Then, for every integer d>0,Ad=    ad 11 0 q1ad 22   where q1is a function of aij . Proof: We prove the remark by induction. For d=1,Ad=A=   a11 0 a21 a22   ,whichisoftheform:   ad 11 0 q1ad 22   with q1=a21. Assume that Ad=   ad 11 0 q1ad 22   where q1is a function of aij. Then, we must show that 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 of the form:    ad+1 11 0 q0 1ad+1 22   . Now, we proceed with the proof of the lemma. We have defined IRFm(k)to be equal to the upper left element, φ11,k,of Φk=Ak. By means of the previous remark, we have that Φkis of the form:    ak 11 0 q1ak 22   where q1is a function of aij. Therefore, IRFm(k)=ak 11. Next, it is easy to show that when a12 =0,i.e.y2tdoes not Granger cause y1t, the univariate representation of y1t is the following AR(1) model: y1t=a11 ∗y1t−1+e1t, which in turn implies that IRFu(k)=ak 11. Thus, IRFu(k)=IRFm(k)for every k. ProofofProposition3 It is straightforward to show that IRFmo(1) = a11 +a12 σ12 σ11 , which is in general different than IRFu(1) = a11 +a22 +γ1. Similarly, IRFmo(2) = a2 11 +a11a12 σ12 σ11 +a12a21 +a12a22 σ12 σ11 whereas IRFu(2) = (a11 +a22)(γ1+a11 +a22)−a11a22 +a21a12 Similar results are obtained for k>2. Therefore, in general, IRFu(k)6=IRFmo(k). ProofofLemma2 We have already shown that when a12 =0,IRFu(k)=ak 11,k≥0. In addition, Φkis of the form:    ak 11 0 q1ak 22   (see lemma 1) where q1is a function of aij. Given that Pis lower triangular, it is easy to show that Θk=ΦkPhas the following form: Θk=   ak 11√σ11 0 q1q2   ,whereq1and q2 are functions of aij and σij,i, j =1,2.Thus,IRFmo(k)= θ11,k √σ11 =ak 11 =IRFu(k). 24