Journal of International Money and Finance 19 (2000) 917–941 www.elsevier.nl/locate/econbase The monetary model in the presence of I(2) components: long-run relationships, short-run dynamics and forecasting of the Greek drachma Panayiotis F. Diamandis a, Dimitris A. Georgoutsos b, Georgios P. Kouretas c,* a b Department of Business Administration, Athens University of Economics and Business, Athens GR-10434, Greece Department of International and European Economic Studies, Athens University of Economics and Business, Athens GR-10434, Greece c Department of Economics, University of Crete, Rethymno GR-74100, Greece Abstract This paper re-examines the long-run properties of the monetary exchange rate model using data for the drachma–dollar and drachma–mark exchange rates under the hypothesis that the system contains variables that are I(2). Using the recent I(2) test by Paruolo (On the determination of integration indices in I(2) systems. J. Economet. 72 (1996) 313–356) to examine the presence of I(2) and I(1) components in a multivariate context we find that the system contains two I(2) variables in both cases and this finding is reconfirmed by the estimated roots of the companion matrix (Do purchasing power parity and uncovered interest rate parity hold in the long-run? An example of likelihood inference in a multivariate time-series model. Juselius, J. Economet. 69 (1995) 211–240). The I(2) component led to the transformation of the estimated model by imposing long-run but not short-run proportionality between domestic and foreign money. Two statistically significant cointegrating vectors were found and, by imposing linear restrictions on each vector as suggested by Johansen and Juselius (Identification of the longrun and the short-run structure: an applicaion to the ISLM model. J. Economet. 63 (1994) 7– 36) and Johansen (Identifying restrictions of linear equations with applications to simultaneous equations and cointegration. J. Economet. 69 (1995b) 111–132), the order and rank conditions for identification are satisfied, but the test for overidentifying restrictions was not significant only for the case of the drachma/mark rate. The main findings suggest that we reject the forward-looking version of the monetary model for the drachma/dollar case but not when the drachma/mark rate is used, a result that is attributed to the monetary and exchange rate policy * Corresponding author. Fax: +30 831 77406. E-mail address: kouretas@econ.soc.uoc.gr (G.P. Kouretas). 0261-5606/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 1 - 5 6 0 6 ( 0 0 ) 0 0 0 4 0 - 1 918 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 followed by the Greek authorities since Greece’s joining of the European Union. Furthermore, we test for parameter stability using the tests developed by Hansen and Johansen (Recursive estimation in cointegrated VAR-models. Working paper (1993) University of Copenhagen) and it is shown that the dimension of the cointegration rank is sample independent while the estimated coefficients do not exhibit instabilities in recursive estimations. Finally, it is shown that the monetary model outperforms the random walk model in an out-of-sample forecasting contest.  2000 Elsevier Science Ltd. All rights reserved. JEL classification: F31; F33; C32; C51; C52 Keywords: I(2) cointegration; Exchange rates; Monetary model; Identification; Stability; Forecasting 1. Introduction Modeling and forecasting the exchange rate has been an important issue in international finance since the inception of the flexible exchange rate regime in foreign exchange markets more than two decades ago. During the 1970s a wide range of theoretical models arose to explain our experience with flexible exchange rates. These models were mainly developed within the monetary theory of the exchange rate, whose basic rationale is that since an exchange rate is the price of one country’s money in terms of that of another, it is important to analyze the determinants of that price in terms of the outstanding stocks of and demand for the two monies. Although this model is theoretically very appealing, its empirical validity is surrounded by controversy. Hodrick (1978) and Bilson (1978) provided some early tests for the recent floating exchange rate experience. Using data up to the end of 1978 their results tend to be supportive of the theory. However, later studies by Rasulo and Wilford (1980) and Driskill and Sheffrin (1981) employing data after 1978 have led to results which provide little support for the model. Several explanations have been offered for the apparent breakdown of the model in the extended samples, ranging from inconsistent estimates of the coefficients to the instability in the underlying money demand functions. Moreover, it has been argued that this class of models has inferior out-of-sample forecasting ability compared to naive models such as a random walk (Meese and Rogoff, 1983). In contrast, studies that have examined the monetary model from a long-run perspective, such as MacDonald and Taylor (1991, 1993, 1994a,b) and Kouretas (1997) and Diamandis et al. (1998) among others, provide evidence for the long-run validity of the model as well as its out-of-sample forecasting performance for a number of key currencies. In this paper we re-examine the long-run properties of the monetary exchange rate model for the drachma–dollar and drachma–mark exchange rates during the recent float by applying several developments in the econometrics of nonstationarities and cointegration. Our approach is novel in a number of ways. First, we provide a new analysis for the determination of the order of integration of the variables. Although testing for unit roots has become a standard procedure it has been made clear that if the data are being determined in a multivariate framework, a univariate model is P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 919 at best a bad approximation of the multivariate counterpart, while at worst, it is completely misspecified leading to arbitrary conclusions. Therefore, we employ the recently developed testing methodology suggested by Paruolo (1996) which allows us to reveal the existence of I(2) and I(1) components in a multivariate context. Testing successively less and less restricted hypotheses according to the Pantula (1989) principle does this analysis. Additionally, we make use of the information provided by the roots of the companion matrix which allows us to make firmer conclusions about the rank of the cointegration space (Juselius, 1995). To test for the existence of cointegration and to estimate the coefficients of the cointegration vectors we apply the Johansen (1988, 1991) multivariate cointegration technique. Second, since in a multivariate framework, such as the one given by the monetary model, a vector error correction model may contain multiple cointegrating vectors, a question arises as to whether one can associate all of them with the monetary model and if not, which vector is identified with it and what is the interpretation given to the others. Thus, following Johansen and Juselius (1994) and Johansen (1995b) we impose independent linear restrictions on the coefficients of the accepted cointegrating vectors. Third, given that at least one statistically significant cointegrating vector has been found we examine the stability of the long-run relationships through time. Hansen and Johansen (1993) propose three tests for parameter stability in cointegrated-VAR systems, which allow us to provide evidence of the sample independency of the cointegration rank as well as of parameter stability. Fourth, our finding of cointegration facilitates an examination of the short-run monetary model using a dynamic error correction model. There are several important findings that stem from our estimation approach. First, we find evidence of cointegration between the drachma–dollar, and the drachma– mark exchange rates and the corresponding monetary variables while simultaneously we have established the presence of a common I(2) component between the domestic and foreign money series. Second, given the presence of an I(2) component we adopt a data transformation that allows us to move to the I(1) model, which can simplify the empirical analysis considerably, since the statistical inference of the I(2) model is not yet as developed as that of the I(1). Therefore, we tested whether long-run proportionality between domestic and foreign money could be imposed on the data. Third, given that two cointegrating vectors were found to be statistically significant, for both the drachma–dollar and the drachma–mark exchange rates, we imposed independent linear restrictions so that we associated one vector with the monetary model and the other with a proposed variant of the uncovered interest parity (UIP) condition which is consistent with the monetary and exchange rate policy that the Greek monetary authorities adopted in the late 1980s. This joint structure is shown to be overidentified and the joint restrictions are rejected for the drachma–dollar case but we are unable to reject them for the drachma–mark case. Fourth, the application of the recursive tests of Hansen and Johansen (1993) show that the dimension of the cointegration space is sample independent and the estimated coefficients do not exhibit instabilities in recursive estimations. Finally, the chosen error correction model satisfies a number of in-sample diagnostics and it outperforms a random walk model, over a 2-year post-estimation sample period, using dynamic forecasts for the 920 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 identified drachma–mark exchange rate model. This latter finding may suggest that by modeling the data dynamics adequately and imposing appropriate coefficient restrictions, the monetary model of the exchange rate can still be used for explaining exchange rate movements. The plan of the remainder of the paper is as follows. Section 2 presents the monetary model. In Section 3 we discuss the econometric methodology for modeling and testing cointegration in the presence of I(2) components. The data used and the multivariate cointegration results are presented in Section 4. Section 5 reports the short-run dynamics of the monetary model and the out-of-sample forecasting results. The final section presents our concluding remarks. 2. The monetary model The monetary model of exchange rate determination is an extension of the quantity theory of money to the case of an open economy. It assumes that: (i) real income and money supply are determined exogenously; (ii) capital and goods are perfectly mobile; (iii) foreign and domestic assets are perfect substitutes; (iv) goods’ prices are perfectly flexible; and (v) domestic (foreign) money is demanded only by domestic (foreign) residents. The early, flexible-price monetary model relies on the twin assumptions of continuous purchasing power parity (PPP) and the existence of stable money demand functions for the domestic and foreign economies. Recent experience with flexible exchange rates has shown, however, that real exchange rates have fluctuated substantially over the years causing shifts in international competitiveness. Stickiness in prices (Dornbusch, 1976) in conjunction with the uncovered interest parity (UIP) condition are usually invoked in order to allow for short-term deviations of both the nominal and the real exchange rates from their long-run levels as determined by the PPP. Moreover, the UIP condition is necessary for the derivation of the forward-looking version of the monetary model, under which the exchange rate depends on all the expected realizations of the forcing variables, that is, the monetary aggregates and the output variables. Under these assumptions a typical monetary reduced form equation is obtained (see Baillie and McMahon, 1989; MacDonald and Taylor, 1992): et5b01b1mt1b2m∗t 1b3yt1b4y∗t 1b5it1b6i∗t 1ut (1) where et is the spot exchange rate (home currency price of foreign currency); mt denotes the domestic money supply; yt denotes domestic income; it denotes the shortterm domestic interest rate; corresponding foreign magnitudes are denoted by an asterisk; ut is a disturbance error; and all variables apart from the interest rate terms, are expressed in natural logarithms. The expected signs of the coefficients in (1) are: b0.0, b1.0, b2,0, b3,0, b4.0, b5.0, b6,0. The Keynesian (sticky-price) model assumes opposite signs for the interest rates. Different signs of the interest rate coefficients in Eq. (1) will also be produced under imperfect substitutability between the assets of the two countries. Associated with Eq. (1) is a set of coefficient restrictions that are regularly imposed P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 921 and tested. The most important restriction is whether proportionality exists between the exchange rate and relative monies (b1=2b2=1). Moreover, we test the assumptions that the income and interest rate elasticities for money demand are equal in both countries (b3=2b4) and (b5=2b6). 3. Econometric methodology Our cointegration analysis is based on the multivariate cointegration technique developed by Johansen (1988, 1991) and extended by Johansen and Juselius (1990, 1992) which is a Full Information Maximum Likelihood (FIML) estimation method. It makes use of the information incorporated in the dynamic structure of the model and it also estimates the entire space of the long-run relationships among a set of variables, without imposing a normalization on the dependent variable a priori. Although the Johansen procedure is well known we briefly discuss it in light of some recent extensions of the methodology that we apply in this paper. Consider a p-dimensional vector autoregressive model which in error correction form is given by k21 Dzt5 O GiDzt−i1Pzt−11gDt1m1et, t51,…,T, (2) i51 where zt=[e, m, m*, y, y*, i, i*]t as defined in Section 2, zk+1, …, z0 are fixed and et|Niidp(0,S). The adjustment of the variables to the values implied by the steady state relationship is not immediate due to a number of reasons like imperfect information or costly arbitrage. Therefore, the correct specification of the dynamic structure of the model, as expressed by the parameters (G1, …, Gk21, g), is important in order that the equilibrium relationship be revealed. The matrix P=ab9 defines the cointegrating relationships, b, and the rate of adjustment, a, of the endogenous variables to their steady state values. Dt is a vector of non-stochastic variables, such as centered seasonal dummies which sum to zero over a full year by construction and are necessary to account for short-run effects which could otherwise violate the Gaussian assumption, and/or intervention dummies. Here m is a drift and T is the sample size. If we allow the parameters of the model q=(G1, …, Gk21, P, g, m, S) to vary unrestrictedly then model (2) corresponds to the I(0) model. The I(1) and I(2) models are obtained if certain restrictions are satisfied. Thus, the higher-order models are nested within the more general I(0) model. It has been shown (Johansen, 1991) that if zt|I(1), then that matrix P has reduced rank r,p, and there exist p×r matrices a and b such that P=ab9. Furthermore, k−1 C=a9>(G)b> has full rank, where G=I2 S Gi and a> and b> are p×(p2r) matrices i=1 orthogonal to a and b, respectively. Following this parameterization, there are r linearly-independent stationary relations given by the cointegrating vectors b and p2r linearly-independent non- 922 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 stationary relations. These last relations define the common stochastic trends of the system and the MA representation shows how they contribute to the various variables. By contrast the AR representation of model (2) is useful for the analysis of the long-run relations of the data. The I(2) model is defined by the first reduced rank condition of the I(1) model and that C=a>9Gb>=fh9 is of reduced rank s1, where f and h are (p2r)×s1 matrices and s1,(p2r). Under these conditions we may re-write (2) as k22 D2zt5Pzt−12GDzt−11 O CiD2zt−11gDt1m1et, (3) i51 where k21 Ci52 O Gi, i51,…,k22. j5i11 Johansen (1997) shows that the space spanned by the vector zt can be decomposed into r stationary directions, b, and p2r nonstationary directions, b>, and the latter into the directions (b1>, b2>), where b1>=b>h is of dimension p×s1 and b2>=b>(b>9b>)−1h> is of dimension p×s2 and s1+s2=p2r. The properties of the process are described by: I(2):{b2>9zt}, I(1):{b9zt}, {b1>9zt}, I(0):{b1>9Dzt}, {b2>9D2zt}, {b9zt1w9Dzt} where w is a p×r matrix of weights, designed to pick out the I(2) components of zt (Johansen, 1992a, 1995a). Thus, we have that the cointegrating vectors b9zt are actually I(1) and require a linear combination of the differenced process Dzt to achieve stationarity. Johansen (1991) shows how the model can be written in moving average form, while Johansen (1997) derives the FIML solution to the estimation problem for the I(2) model. Furthermore, Johansen (1995a) provides an asymptotically equivalent two-step procedure which computationally is simpler. It applies the standard eigenvalue procedure derived for the I(1) model twice, first to estimate the reduced rank of the P matrix, and then for given estimates of a and b, to estimate the reduced rank of â>9Gb̂>, (Juselius 1994, 1995, 1998). An equally important issue, along with the existence of at least one cointegration vector, is the issue of the stability of such a relationship through time as well as the stability of the estimated coefficients of such a relationship. Thus, Sephton and Larsen (1991) have shown that Johansen’s test may be characterized by sample dependency. Hansen and Johansen (1993) have suggested methods for the evaluation of parameter constancy in cointegrated VAR models, formally using estimates obtained from the Johansen FIML technique. Three tests have been constructed under P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 923 the two VAR representations. In the “Z-representation” all the parameters of model (2) are re-estimated during the recursions while under the “R-representation” the short-run parameters Gi=1, …, k21 are fixed to their full sample values and only the long-run parameters a and b are re-estimated. The first test is called the Rank test and we examine the null hypothesis of sample independency of the cointegration rank of the system. This is accomplished by first estimating the model over the full sample, and the residuals corresponding to each recursive subsample are used to form the standard sample moments associated with Johansen’s reduced rank. The eigenvalue problem is then solved directly from these subsample moment matrices. The obtained sequence of trace statistics is scaled by the corresponding critical values, and we accept the null hypothesis that the chosen rank is maintained regardless of the subperiod for which it has been estimated if it takes values greater than one. A second test deals with the null hypothesis of constancy of the cointegration space for a given cointegration rank. Hansen and Johansen (1993) propose a likelihood ratio test that is constructed by comparing the likelihood function from each recursive subsample to the likelihood function computed under the restriction that the cointegrating vector estimated from the full sample falls within the space spanned by the estimated vectors of each individual sample. The test statistic is a c2 distributed with (p2r)r degrees of freedom. The third test examines the constancy of the individual elements of the cointegrating vectors b. However, when the cointegration rank is greater than one, the elements of those vectors can not be identified, except under restrictions. Fortunately, one can exploit the fact that there is a unique relationship between the eigenvalues and the cointegrating vectors. Therefore, when the cointegrating vectors have undergone a structural change this will be reflected in the estimated eigenvalues. Hansen and Johansen (1993) have derived the asymptotic distribution as well as the asymptotic variance of the estimated eigenvalues. In a multivariate context, such as the one given by the monetary model, a vector error correction model may contain multiple cointegrating vectors, and in such a case the individual cointegrating vectors are underidentified in the absence of sufficient linear restrictions on each of the vectors. The issue of identification in cointegrated systems has recently been addressed by Johansen and Juselius (1994) and Johansen (1995b). Consider again the long run matrix P=ab9 and let F be any r×r matrix of full rank. Then P=aF−1Fb9=a∗b∗9, where a∗=aF−1 and b∗=Fb9 and without imposing restrictions on a and b so that to limit the admissible matrices, F, the cointegrating vectors are not unique. In fact given the normalization under which both a and b are calculated, only the space spanned by the b vectors is uniquely determined. Thus, we need to impose restrictions implied by economic theory, for example homogeneity and zero restrictions, so that we are able to discriminate between them. The necessary and sufficient conditions for identification in a cointegrated system in terms of linear restrictions on the columns of b are analogous to the classical identification problem that we face in the simultaneous equations problem. Thus, the 924 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 order condition for identification of each of the r cointegrating vectors is that we can impose at least r21, just identifying restrictions and one normalization on each vector without changing the likelihood function. This is a necessary condition. The necessary and sufficient condition for identification of the ith cointegration vector, the Rank condition, is that the rank (Ri9H1,…Ri9Hk)$k, where i and k=1, …, r21 and kÞi (Johansen and Juselius, 1994). The linear restrictions of the model are of the form Ri9bi=0, where Ri is a (p×ki) matrix, or equivalently by Ri9Hi=0, i=1,…,r, where Hi is a known (p×si) design matrix which satisfies bi=Hi9ti and ti is a (si×1) vector of freely varying parameters (ki+si)=p. For example, if there are two accepted cointegrating vectors among the seven variables of our model, the exact identification, according to the order condition requires one linear restriction on each cointegrating vector and the Rank condition is satisfied if rank (Ri9Hj )$1, iÞj. Furthermore, Johansen and Juselius (1994) provide a likelihood ratio statistic to test for overidentifying restrictions that is distributed as a c2 with n=Si(p2r+12si) degrees of freedom, where p and r are given by the dimension p×r of b, and si is the number of freely estimated parameters t, in vector i, which comply with bi=Hiti. 4. Empirical results The data for this study, relating to the drachma–dollar and drachma–mark exchange rates and Greek, US and German macroeconomic variables, are all taken from the International Monetary’s Fund International Financial Statistics CD-ROM, except for the Greek monetary aggregate, and run from January 1976 through October 1997. In particular, the exchange rates are expressed in units of home currency per foreign currency and they are end-of-month quotations (line ae); the money supply aggregate is M1 (line 34 for Germany, line 59 for the US while the Greek M1 is taken from the monthly bulletin of the Bank of Greece); the income variable is industrial output for Germany and the US (line 66c) and manufacturing production for Greece (line 66ey); and the short term interest rates are the three months treasury bill rate for Germany and the US (line 60c) and for Greece the 3–12 months deposit rate (line 60l). To account for a number of important institutional events we also include three shift dummy variables. D83 and D85 account for the effects of the two devaluations of a magnitude of 15% and they take the value of 1 on 1983.01 and 1985.10, respectively. In addition, for the drachma–dollar exchange rate case, a shift dummy variable, D82, is included to allow for a possible break in the deterministic trends in velocities and interest rates that occurred around the end of 1981 (Hoffman et al., 1995). It takes the value of 1 for t=1982.01 … 1997.10, 0 otherwise.1 1 We have chosen M1 as the monetary aggregate for reasons of comparison with previous studies. Additionally, Hoffman et al. (1995) have used M1 in estimating long-run money demand functions for the USA and Germany which are found to be stable by applying the same methodology as in this paper. Psaradakis (1993) has found similar results for the case of the Greek money demand when M1 is used. Industrial output (and manufacturing output for Greece) is considered as a proxy for income since GDP measures are not available on a monthly basis. Finally, treasury bill interest rates are the standard shortterm interest rates used in most of the relevant literature. However since treasury bill rates are available for Greece since early 90s’, data on the 3–12 months deposit rate have been employed. P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 925 4.1. Determination of the cointegration rank and the order of integration The first step in the analysis is the determination of the cointegration rank index, r and the order of integration of the variables. In order for the procedure to be valid we have to check that the assumptions underlying the model are satisfied. In particular we investigate that the estimated residuals do not deviate from being Gaussian white noise errors. A structure of three lags for both the drachma–dollar and the drachma–mark exchange rates was chosen based on several univariate and multivariate residual misspecification tests reported in Table 1. We note that our conditional VAR model is well specified, except for the presence of non-normality especially in the case of the Greek deposit rate which was administratively determined till mid 80s. However, the asymptotic properties of Johansen’s method only depends on the i.i.d. assumption of the errors and thus the normality assumption is not such a serious problem. Table 1 also reports the estimates of the short-run effects of the dummy variables. The Johansen–Juselius multivariate cointegration technique, as explained in Section 3, is applicable only in the presence of variables that are realizations of I(1) processes or a mixture of I(1) and I(0) processes, in systems used for testing for the order of cointegration rank. Until recently the order of integration of each series was determined via the standard unit root tests. However, it has been made clear by now that if the data are being determined in a multivariate framework, a univariate model is at best a bad approximation of the multivariate counterpart, while at worst, it is completely misspecified leading to arbitrary conclusions. Thus, in the presence of I(1) series, Johansen and Juselius (1990) developed a multivariate stationarity test which has become the standard tool for determining the order of integration of the series within the multivariate context. Additionally, when the data are I(2), one also has to determine the number of I(2) trends, s2 among the p2r common trends. The two-step procedure discussed in Section 3 is used to determine the order of integration and the rank of the two matrices. The hypothesis that the number of I(1) trends=s1 and the rank=r is tested against the unrestricted H0 model based on a likelihood ratio test procedure discussed in Paruolo (1996). Table 2 reports the trace test statistics for all possible values of r and s1=p2r2s2, under the assumption that the data contain linear but no quadratic trends. The 95% critical test values reported in italics below the calculated test values are taken from the asymptotic distributions reported in Paruolo (1996, Table 5). Starting from the most restricted hypothesis {r=0, s1=0, s2=7} and testing successively less and less restricted hypotheses according to the Pantula (1989) principle, it is shown that the case in favor of the presence of one I(2) component can not be rejected. Specifically, 926 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 Table 1 Residual misspecification tests of the model with k=3 Eq. se LB(48) ARCH(3) NORM(3) R2 Drachma–Dollar De 0.0023 42.34 3.35 9.98‡ 0.69 Dm 0.0044 54.67 4.45 5.57 0.77 Dm* 0.0056 44.82 5.56 4.56 0.62 Dy 0.0012 46.98 3.46 8.67‡ 0.73 Dy* 0.0145 61.22 0.99 1.23 0.45 Di 0.0033 53.45 2.32 22.34† 0.46 Di* 0.0125 37.28 1.98 5.97 0.42 Drachma–Mark De 0.0037 47.00 2.68 7.89† 0.72 Dm 0.0032 53.93 2.31 9.23† 0.80 Dm* 0.0127 55.42 2.42 3.38 0.84 Dy 0.0013 34.56 4.28 9.76† 0.85 Dy* 0.0009 51.28 1.34 0.98 0.50 Di 0.0022 26.78 0.21 19.34‡ 0.41 Di* 0.0008 33.24 2.47 5.78 0.49 Notes: LB is the Ljung–Box test statistic for residual autocorrelation, ARCH is the test for heteroscedastic residuals, and NORM the Jarque–Bera test for normality. All test statistics are distributed as c2 with the degrees of freedom given in parentheses, † (‡) denotes significance at the 5% (1%) level. Multivariate Residuals Diagnostics Case L-B(64) LM(1) LM(4) c2 (14) GRD/USD 1933.29 64.09 59.79 35.82 (0.20) (0.07) (0.14) (0.00) GRD/DM 2009.33 66.22 62.78 37.67 (0.15) (0.06) (0.12) (0.00) Notes: L-B is the multivariate version of the Ljung–Box test for autocorrelation based on the estimated auto- and cross-correlations of the first [T/4=64] lags. LM(1) and LM(4) are the tests for first and fourth-order autocorrelation distributed as c2a with 49 degrees of freedom and is a normality test which is a multivariate version of the Shenton–Bowman test. Numbers in parentheses refer to marginal significance levels. Relevance of the conditioning variables in the vector Dt Eq. De Dm Dm* D82 0 0 0.023† (0.0) 0 D83 Dy Dy* Di D∗i 0 0 0.016† (0.00) 0 0.011† 0 0 0 0 0 (0.018†) D85 0.024† 0 0 0 0 0 0 (0.022†) Notes: (†) means significant at the 95% level. Coefficients with a t-value,1.0 take the value zero. Numbers in parentheses apply to the Drachma–Mark exchange rate case. P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 927 Table 2 Testing the rank of the I(2) and the I(1) model. Testing the joint hypothesis p2r r Drachma–Dollar 7 0 6 1 5 2 4 3 3 4 2 5 1 6 s2 Drachma–Mark 7 0 6 1 5 2 4 3 Q(s1>r/H0) 789.1 317.5 638.5 280.2 573.3 240.3 497.8 245.5 432.6 206.8 421.0 171.8 399.4 215.5 328.2 179.0 285.5 145.6 285.8 116.3 323.4 188.4 249.5 154.0 198.5 122.0 165.3 94.7 254.4 166.1 198.1 132.8 120.1 102.7 85.4 76.8 215.2 147.4 151.9 115.4 76.0 86.9 184.9 70.8 64.9 54.5 54.3 36.1 17.2 42.9 9.3 26.0 4.8 12.9 1 7 6 5 4 3 2 1171.8 317.5 948.9 280.2 763.4 240.3 770.7 245.5 583.4 206.8 526.3 171.8 607.8 215.5 412.2 179.0 346.9 145.6 361.3 116.3 460.8 188.4 272.7 154.0 207.6 122.0 210.7 94.7 360.7 166.1 174.0 132.8 111.2 102.7 93.5 76.8 36.5 63.1 287.9 147.4 119.4 115.4 77.6 86.9 50.5 63.1 3 4 191.6 70.8 64.9 54.5 28.9 42.9 2 5 46.636.1 15.7 26.0 1 6 5.8 12.9 7 6 5 3 2 1 s2 Notes: p is the number of variables, r is the rank of the cointegration space s1, is the number of I(1) components and s2 is the number of I(2) components. The numbers in italics are the 95% critical values (Paruolo, 1996, Table 5). The roots of the companion matrix Drachma–Dolllar Drachma–Mark real real 0.9924 1.0027 0.9924 1.0027 0.9859 0.9676 0.9568 0.9676 0.9568 0.9106 0.9500 0.9106 0.6465 0.8147 0.6445 0.4464 Notes: The table shows the real parts of the estimated p×k roots of the companion matrix from the VAR system, p is the number of variables and k is the number of lags of the VAR. We report the first eight roots which are of our interest. For all tests a structure of three lags for both the Drachma– Dollar and the Drachma–Mark exchange rates was chosen according to a likelihood ratio test, corrected for the degrees of freedom (Sims, 1980) and the Ljung–Box Q statistic for detecting serial correlation in the residuals of the equations of the VAR. A model with an unrestricted constant in the VAR equation is estimated for all three cases according to the Johansen (1992b) testing methodology. 928 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 we are unable to reject the hypothesis {r=2, s1=4, s2=1} for both the drachma–dollar and the drachma–mark case.2,3 In addition to the formal test, Juselius (1995) offers further insight into the I(2) and I(1) analysis as well as the correct cointegration rank. She argues that the results of the trace and maximum eigenvalue test statistics of the I(1) analysis, i.e. from the estimation of the model without allowing for I(2) trends, should be interpreted with some caution for two reasons. First, the conditioning on intervention dummies and weakly exogenous variables is likely to change the asymptotic distributions to some (unknown) extent. Second, the asymptotic critical values may not be very close approximations in small samples. Juselius (1995) suggests the use of the additional information contained in the roots of the characteristic polynomial. Table 2 also lists the eight largest roots of the p×k roots of the companion matrix. If there are I(2) components in the vector process, then the number of unit roots in the characteristic polynomial is s1+2s2. Hence, if r=2, implying two cointegrating vectors between the exchange rate and the respective macroeconomic variables, there should be six unit roots in the process, four of which are I(1) components and one of which is the I(2) component. This is consistent with the estimated roots of the companion matrix since there are six roots almost equal to one, and given that we have a system of seven variables, two additional smaller roots are left in the process associated with the two stationary long-run relationships. 4.2. A data transformation from I(2) to I(1) Since the statistical inference of the I(2) model is not yet as developed as that of the I(1) model, a data transformation that allows us to move to the I(1) model will simplify the empirical analysis considerably. A possible hypothesis which could be extracted from the presence of an I(2) component in the system is that the variable {mt2mt∗} is a first-order nonstationary process.4 If accepted, the implication is that the domestic and foreign money aggregates are cointegrating from I(2) to I(1), and use of the transformed data vector zt9=[et, mt2mt*, Dmt, yt*, it, it*], would then allow us to move to the I(1) model. The validity of this transformation is based on the 2 The calculations of all tests as well as the estimation of the eigenvectors have been performed using the program CATS 1.1 in RATS 4.30 developed by Katarina Juselius and Henrik Hansen, Estima Inc., Illinois, 1995. 3 We have also tested for the presence of a linear trend following the work by Dornbusch (1989) who suggests that due to both differing productivity trends in the tradeable and non-tradeable goods sectors and inter-country differences in consumption patterns, a secular decline in domestic prices relative to foreign prices could appear as a linear trend in the purchasing power parity (PPP) relationship underlying the monetary model. To test the monetary model under the twin hypotheses of I(2) components and a linear trend in the cointegrating relations we have applied the recent test by Rahbek et al. (1999) and we find that the linear trend hypothesis is rejected for the sample. 4 The assertion that the nominal money aggregates are I(2) comes from recent empirical work on modeling money demand functions which suggest that nominal money stocks are I(2), (see Johansen, 1992c; Haldrup, 1994; Paruolo, 1996; Rahbek et al., 1999 for UK monetary data and Juselius, 1994 for Danish data). P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 929 assumption that {mt2mt∗}|I(1), {et, yt, yt∗, it, it∗}|I(1), and that {mt2mt∗} is a valid restriction on the long-run structure, but not necessarily on the short-run structure. The hypothesis of long-run proportionality between domestic and foreign money was tested, and the test statistic which is asymptotically distributed as c2(1), is equal to 0.57 and 0.98 for the drachma–dollar and the drachma–mark exchange rate respectively, and hence was not significant. Therefore long-run proportionality between the domestic and foreign money could not be rejected. Furthermore, the I(2) test confirmed that this transformation removes all signs of the I(2) components from the data. The remaining analysis will be performed in the I(1) model, containing long-run but not short-run proportionality between the domestic and foreign money, based on the vector [e, m2m*, Dm, y, y*, i, i*]. Alternatively we could have chosen to analyze the vector [e, m2m*, Dm*, y, y*, i, i*] as it corresponds to the same likelihood function. Since we are interested in how the exchange rate reacts to disequilibrium positions in the domestic money we choose the first alternative. To assess the statistical properties of the chosen variables the test statistics reported in Table 3 are useful. The test of long-run exclusion is a check of the adequacy of the chosen measurements and shows that none of the variables can be excluded from the cointegration space. The test for stationarity indicates that none of the variables can be considered stationary under any reasonable choice of r. Finally, the test of weak exogeneity shows that the foreign variables can be considered weakly exogenous for the long-run parameters b. All three tests are c2 distributed and are constructed following Johansen and Juselius (1990, 1992). Also shown in Table 3 are diagnostics on the residuals from the transformed cointegrated VAR model. These indicate that the results are i.i.d. processes since no evidence of serial correlation was detected.5 This provides further support for the hypothesis of a correctly specified model. 4.3. The empirical analysis of the transformed I(1) model All results discussed in this section are based on the analysis of model (2) with the reduced rank condition on P imposed for k=3 and r=2 applied to the transformed vector z̃t=[e, m2m∗, Dm, y, y∗, i, i∗] and with Dt unaltered. Table 4 reports the unrestricted estimates of the normalized cointegrating vectors which are based upon eigenvectors obtained from an eigenvalue problem resulting from Johansen’s reduced rank regression approach. The estimated parameters, in both cases, carry signs that are in line with those that the transformed monetary model of Section 4.2 predicts. Given the presence of two cointegrating vectors we continue now with the econ- 5 Gonzalo (1994) shows that the performance of the maximum likelihood estimator of the cointegrating vectors is little affected by non-normal errors. Lee and Tse (1996) have shown similar results when conditional heteroskedasticity is present. 930 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 Table 3 Tests for long-run exclusion, stationarity, and weak exogeneity Variable Long-run exclusion Stationarity Weak exogeneity GRD/USD GRD/USD GRD/DM GRD/USD GRD/DM GRD/DM e 9.87† 6.76† 22.21† 12.52† 7.75† 21.86† m–m* 7.98† 7.43† 20.95† 12.64† 33.39† 6.96† Dm 25.90† 105.18† 21.15† 15.06† 32.60† 100.41† y 13.85† 6.10† 20.61† 17.87† 12.07† 9.20† y* 9.50† 7.11† 20.65† 17.96† 1.03 3.22 i 7.14† 8.31† 26.59† 20.83† 9.47† 8.60† i* 12.89† 8.89† 18.68† 15.05† 2.53 2.16 Notes: e, m, y and i are respectively the spot exchange rate, the money supply, the real output and the short-term interest rate, with the foreign magnitudes denoted by an asterisk. The long-run exclusion restriction and the weak exogeneity tests are distributed with two degrees of freedom and the 5% critical level is 5.99, and the stationarity test is a c2 distributed with five degrees of freedom and the 5% critical level is 11.07. Multivariate Residuals Diagnostics Case L-B(64) LM(1) LM(4) x2(14) GRD/USD 3123.12 52.20 61.12 726.29 (0.26) (0.35) (0.09) (0.00) GRD/DM 3068.60 58.71 63.50 288.12 (0.28) (0.09) (0.08) (0.00) Notes: L-B is the multivariate version of the Ljung–Box test for autocorrelation based on the estimated auto- and cross-correlations of the first [T/4=64] lags. LM(1) and LM(4) are the tests for first and fourth-order autocorrelation distributed as a c2 with 49 degrees of freedom and x2 is a normality test which is a multivariate version of the Shenton–Bowman test distributed as a c2 with 14 degrees of freedom. Marginal significance levels in parentheses. omic identification of our system. On the first cointegrating vector we impose four restrictions, namely proportionality between the exchange rate and relative monies and exclusion of the growth of money as well as of the two interest rates. This longrun relationship is necessary to hold in the forward looking solution of the exchange rate when the variables are I(1) processes, the UIP condition is imposed and no bubbles are present in the foreign exchange market (MacDonald and Taylor, 1994b). In fact the imposition of these four restrictions overidentifies this relationship. We could further impose one more restriction on the first cointegrating vector, namely equal and opposite coefficients for the output variables. Identification of the second cointegrating vector requires a set of restrictions that is independent of the one imposed on the first one. This implies that from the accepted cointegrated vectors only one can possibly describe the long-run monetary relationship and this is in variance with the cointegrating results on the monetary model which other researchers report (e.g. MacDonald and Taylor, 1991, 1994a,b), where they conclude that as many as four vectors can be considered as possibly explaining the monetary model (Kouretas, 1997; Diamandis et al., 1998). The second vector can be interpreted 931 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 Table 4 Estimated coefficients and hypothesis testing et=b0+b1(m2m∗)t+b2Dmt+b3yt+b4yt∗+b5it+b6it∗+ut GRD/USD GRD/DM 1.0 1.0 1.0 1.0 10.2 0.11 6.6 0.06 3.45 11.24 1.19 1.77 293.82 286.46 261.23 270.75 27.88 77.87 26.61 225.20 5.34 233.24 6.18 213.64 0.03 0.012 0.025 20.019 20.10 20.11 20.052 0.20 Notes: The eigenvectors have been normalized with respect to the estimated coefficient on the nominal exchange rate, e. (A) Tests for overidentifying restrictions (in vector representation) GRD/DM Q(7)=13.79 [0.07] GRD/US Q(7)=39.42 [0.00] Notes: Q denotes a likelihood ratio test for overidentifying restrictions as suggested by Johansen and Juselius (1994) and is distributed as a c2 with the corresponding degrees of freedom given in parentheses. Numbers in brackets denote marginal significance levels. Numbers in parentheses next to the coefficient estimates report estimated asymptotic standard errors which are the square roots of the computed Wald test statistics developed by Johansen (1991, Theorem 5.2 and Corollary 5.3). (B) Independent tests for the Monetary Model and the UIP Case H1 (b1=1, b2=0, b3=2b4, b5=2b6) H2 (b1=0, b2=1, b3=b4=0, b5=b6) GRD/USD 0.00 0.00 Notes: Numbers correspond to marginal significance levels of the H5 test statistic (Johansen and Juselius, 1992) distributed as a c2 with six (twelve) degrees of freedom for the restrictions implied by the monetary model (UIP). as a particular variant of the UIP condition for a country like Greece. Over the last 15 years Greece has been suffering from chronic budget deficits which led the Greek authorities to offer excess returns in order to increase the capital inflows from abroad, given their inability to finance them through seigniorage (European Union does not allow such a procedure). Such a policy caused an excess demand for drachmas which in turn generated appreciation pressures on the Greek currency against major currencies. At the same time the Bank of Greece had been using the exchange rate as 932 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 a target for the monetary policy in an effort to combat double digit inflation rates. Thus, the Bank of Greece was forced to intervene in the foreign exchange market purchasing foreign exchange and selling drachmas a policy that resulted to an increase in the domestic money supply (the Bank of Greece followed a non-sterilizing intervention policy). This type of monetary and exchange rate policy which has been adopted by the Greek authorities in the last decade has been occassionaly called “Hard drachma policy” (Georgoutsos and Karamouzis, 1994; Papaioannou and Gatzonas, 1997). Imposing the above restrictions on the transformed vector z̃t=[e, m2 m∗, Dm, y, y∗, i, i∗], the matrix of the linear and homogeneous restrictions is the following: F 1 −1 0 b3 b4 0 0 0 0 G b2 0 0 1 −1 (4) where b2 is expected to be negative. The results of the estimated restricted vectors along with the likelihood ratio test for the acceptance of the overidentifying restrictions, for both the drachma/dollar and the drachma/mark exchange rates, are given in Table 4. According to the evidence we reject the joint restrictions for the case of the drachma–dollar bilateral rate but we are unable to reject them for the drachma–mark rate. This outcome is consistent with the adopted monetary and exchange rate policy of the Greek authorities which in the early 1980s changed and linked the drachma with the mark and ECU and placed lesser emphasis on the link between the drachma and dollar. Therefore, while for the drachma–mark exchange rate we are able to identify one long-run relationship as describing the monetary model in its forward-looking solution and in the other a modified UIP condition, we fail to do so for the drachma–dollar case. Given the rejection of the hypothetical joint structure for the drachma–dollar case we further examine whether the monetary model or the UIP condition, as applied in our case, is responsible for this outcome. This task can be accomplished by imposing the same restrictions on both cointegrating vectors (Johansen and Juselius, 1992) and the test statistic is distributed as c2 with (p2s)×r where s is the number of the freely estimated variables. The test results reported in Table 4(B) imply that we reject the coefficient restrictions imposed by the monetary model, or that the UIP condition is encompassed in the cointegrated space we have estimated for the drachma–dollar exchange rate. Figs. 1–4 present the evidence from the Hansen–Johansen (1993) recursive analysis on the sample independence of the Johansen procedure results for the GRD/DM case.6 We choose as a starting point January 1986, the date of the implementation 6 Based on a referee’s suggestion the three tests have been constructed under the “Z” representation of the VAR model, in which case all the parameters of model (2) are re-estimated during the recursions. We present results for the GRD/DM case only since for the GRD/USD case we failed to accept the overidentifying restrictions. However, even in this case the obtained results are favorable to the stability of either the cointegration rank or the eigenvalues and can be obtained upon request from the authors. P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 Fig. 1. Fig. 2. 933 Trace test. Test of known beta eq. to beta. of the gradual liberalization of capital movements in the Greek foreign exchange market (Christodoulakis and Karamouzis, 1993; Papaioannou and Gatzonas, 1997). This liberalization was recently completed when, in May 1994, the Bank of Greece abandoned all capital controls on short term capital. The overall conclusion drawn from the three tests is strongly in favor of the sample independence of the cointe- 934 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 Fig. 3. Test for lambda 1. Fig. 4. Test for lambda 2. gration results. Specifically, Fig. 1 shows that the rank of the cointegration space is independent of the sample size from which it has been estimated, since the null hypothesis of a constant rank (two) could not be rejected. Fig. 2 clearly indicates that we are always unable to reject the null hypothesis for the sample independency of the cointegration space for a given cointegration rank. Finally, the last two figures P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 935 provide substantial evidence in favor of the constancy of the cointegrating vectors since no substantial drift was detected on the time paths of the eigenvalues.7 The last finding seems to indicate that the maximum likelihood estimates do not display considerable instabilities in recursive estimates. These results further reinforce our conclusion that the unrestricted monetary model of exchange rate determination is a valid framework to analyze movements of the Greek drachma from a long-run perspective. 5. Forecasting performance of the monetary model: can we outperform a random walk? In this section we present the forecasting performance of the structural model as compared with the random walk. We use the long-run multivariate relationships derived in the previous section to model the short-run exchange rate dynamics only for the drachma/mark exchange rate. The short-run dynamic equation is then used to construct out-of-sample forecasts, which are compared to a random walk model. Following the general-to-specific modeling strategy, we initially estimated a twelfth-order autoregressive distributed lag of the nominal exchange rate on domestic and foreign money, on the change in domestic money, on domestic and foreign industrial production and short-term interest rates, for the period 1976:01–1995:10. To this equation we added two “error correction” terms (ECMs), which were formed from the two cointegrating vectors which represent the accepted joint specification of Table 4(A). The ECMs terms are given as follows: ECM1t−15et−12(m2m∗)t−115.58yt−123.65yt−1∗26.6, 2 t−1 ECM 5229.16Dmt−11it−12it−1∗10.033. (5) (6) The next stage was to impose statistically insignificant restrictions in order to reduce the dimensions of the parameter space. The final parsimonious specification we arrived at was as follows: −0.085D2mt−3 +0.145D(m−m∗)t−3 +0.066Dyt−2 +0.003Dit−2 −0.087ECM1t−1 +0.002ECM2t−1 −0.003 Det5 (0.029) (0.044) (0.026) (0.002) (0.03) (0.001) (0.001) (7) Eq. (7) passes successfully a set of misspecification tests and most importantly the out-of-sample forecasting test for the 24 months period up to 1997.10.8 Further evi- 7 There appears that the time path of the first eigenvalue of the GRD/DM case has shifted around the time of the reunification of East and West Germany. It should be noticed however that the eigenvalue after 1992 is close to the lower limit, at the 95% significance level, of its value before 1992. At the 99% level of significance the eigenvalue, after 1992, is well inside the bounds of its value before that year. 8 The chosen Eq. (7) passes a number of equation diagnostics easily. The Ljung–Box test for autocorrelation and the ARCH test for conditional heteroscedasticity have a marginal significance level (MSL) of 0.74 and 0.15, respectively. The MSL of a CHOW type test for predictive failure and of a test for misspecification take values of 0.91 and 0.85, respectively. Details on those tests is available upon request from the authors. 936 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 dence of the goodness of fit of our estimated equation is provided in Figs. 5 and 6. Fig. 5 presents the actual and fitted values of the change in the exchange rates over the period 1976:1–1995:10 as well as out-of-sample forecasting periods. As it is shown, the predicted exchange rate change from the model tracks the actual exchange change well and manages to get a considerable number of turning points correct. Moreover, the model is also able to get almost half of the out-of-sample turning points especially for the period 1997.7–1998.12. In Fig. 6 we note that none of the twenty-four observations of the actual exchange rate falls outside the two standard error bars, an outcome which further reinforces the forecasting performance of our model specification. Table 5 reports the results of the out-of-sample forecasts of the structural model and the random walk model and they are very interesting. For short horizon forecasts the random walk model outperforms the structural Eq. (7) but this picture is reversed when the longest horizon forecast is considered. Furthermore, we note that the rate of improvement in the root-mean-square forecast error of the structural model against the random walk increases as the forecasting period is extended. This evidence clearly suggests that while in the short-run the drachma–mark exchange rate is influenced more by factors other than economic fundamentals, it is determined more by economic variables in the long-run. Thus, not only two long-run equilibrium relationships exist that underline the monetary model and the UIP condition, but also the forecasts of the exchange rate based on the combined effects of the two error correction terms are superior to that of the random walk model. Our results are in line Fig. 5. Actual and fitted values. P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 Fig. 6. 937 Forecasts and two standard error bars. Table 5 Out of sample forecasts: random walk vs the monetary model Forecast horizon (months) 1 3 6 9 12 RMSE for random walk RMSE for random walk with drift RMSE for ECM monetary model 0.008 0.024 0.043 0.016 0.070 0.045 0.022 0.136 0.034 0.025 0.202 0.034 0.039 0.259 0.036 with the arguments of Frankel and Froot (1990) and Taylor and Allen (1992) who found in their analyses that while the foreign exchange market dealers rely heavily on charts and current trends for short-run forecasting, they rely more on economic fundamentals as the forecast horizon increases. Statistically, the imposition of cointegrating restrictions produces improved forecasts over long horizon and this evidence is supported by a number of Monte Carlo analyses (e.g. Reinsel and Ahn, 1992; Lin and Tsay, 1996).9 9 A referee has pointed out that recent results by Clements and Hendry (1993, 1998) suggest that one should be careful with using forecasting performance as a model selection criterion. In the presence of structural breaks the out-of-sample performance of econometric models is crucially dependent on whether something unpredictable happens in the forecasting period. However, in our case there do not seem any unpredictable events to have taken place. 938 P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 6. Conclusions In this paper we have re-examined the long-run properties of the monetary exchange rate model using data for the drachma–dollar and drachma–mark exchange rates under the hypothesis that the system contains variables that are I(2). Several recent developments in the econometrics of non-stationarity and cointegration were applied and a number of novel results stem from our analysis. First, this paper makes use of the recently developed testing methodology suggested by Paruolo (1996) to test for the existence of I(2) and I(1) components in a multivariate context. Additionally, we estimated the roots of the companion matrix as suggested by Juselius (1995) in order to make firmer conclusions about the rank of the cointegration space. The joint hypothesis of two cointegration vectors and one I(2) component could not be rejected an outcome that led us to transform the monetary model to contain I(1) variables and in which the rate of growth of domestic money plays a significant role. Second, given that two cointegration vectors were accepted, we formally imposed independent linear restrictions on each vector as suggested by Johansen and Juselius (1994) and Johansen (1995b) in order to identify our system. Based on a likelihood ratio test for overidentifying restrictions (Johansen and Juselius, 1994) we rejected the joint restriction that the system represents the forward looking version of the monetary model for the case of the drachma/dollar bilateral rate but we were unable to reject it for the case of the drachma/mark rate. Furthermore, we tested whether independently the unrestricted version of the monetary model and a version of the UIP condition could be considered and the results show that both hypotheses were again rejected for the drachma/dollar case but not for the drachma/mark rate. These results can be attributed to the monetary and exchange rate policy followed by the Bank of Greece and the Greek government since Greece’s joining of the European Union and in light of the forthcoming European Economic and Monetary Union. Third, we tested for parameter stability and it is shown that the dimension of the cointegration rank is sample independent while the estimated coefficients do not exhibit instabilities in recursive estimations. Finally, using a dynamic error correction model which was shown to pass easily all tests for in-sample and out-of-sample performance, we demonstrate that it outperforms a random walk model (with and without drift) at long forecasting horizons. More importantly, it was also observed that the rate of improvement of the root-mean-square forecasting error over the random walk increased gradually as the forecasting period was risen. This outcome is in variant with Meese and Rogoff’s (1988) argument and it could be attributed to the influence of the error correction term from a statistical point of view while from an economic point of view it could suggest that economic fundamentals are very important in the long run. Acknowledgements An earlier version of this paper was presented at the 54th Econometric Society European Meeting, Santiago de Compostela, 28 August–1 September 1999 and P.F. Diamandis et al. / Journal of International Money and Finance 19 (2000) 917–941 939 thanks are due to Katarina Juselius and other conference participants for many helpful comments and discussions. 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